Two notions of 2-multiplication [Archive]

Thomas Larsson

Sep4-04, 06:30 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nIn recent years there has been some interest in 2-groups and\n2-gauge theories, where the gauge invariant objects are Wilson\nsurfaces rather than Wilson lines, see e.g.\n\nGirelli-Pfeiffer, http://www.arxiv.org/abs/hep-th/0309173\nBaez, http://www.arxiv.org/abs/hep-th/0206130\nSchreiber, http://www-stud.uni-essen.de/~sb0264/p11.pdf\n\nUnderlying these theories is 2-multiplication in the 2-group\nsense. However, a different notion of 2-multiplication is\nimplicit in the Yang-Baxter (YB) equation, which led me to\nformulate 2-gauge theories on the lattice already 15 years ago;\nfor a review and the original reference see\nhttp://www.arxiv.org/abs/math-ph/0205017\n\nSeeing the very complicated and awkward compatibility conditions\nthat are necessary to ensure 2-associativity for 2-groups,\nI\'m becoming increasingly convinced that this is not a fruitful\napproach. To make this point I would like to contrast the\n2-product across a 2x2 square using the two different notions.\n\nIn 2-groups, we have a horizontal product . and a vertical\nproduct o which take two object and construct a new object *of\nthe same type*. Thus, given A and be we can construct products\n(A.B) and (AoB), where A and B should be written on top of each\nother in the second product. The value of a 2x2 square should not\ndepend on whether we do the horizontal or vertical product first,\nso the condition for 2-associativity looks like\n\n(A) (C) (A . C)\n(o).(o) = o\n(B) (D) (B . D)\n\nThis kind of 2-associativity requires some very complicated\nconsistency relations, and it is unclear to me if any\ninteresting solutions to these exist.\n\nIn contrast, one may use the manifestly associative YB product.\nLet V be the basic vector space. An ordinary matrix is an\nelement in End(V), i.e. a two-index quantity. YB matrices A and\nB are instead elements in End(V@V) (@ = tensor product), i.e.\n4-index quantities. We can still define horizontal and vertical\nproducts, but the results are 6-index quantities in End(V@V@V),\nbecause gluing two squares along a common edge results in a\n6-edged rectangle:\n\n__________.__________\n| | |\n| | |\nA . B = | A | B |\n| | |\n__________.__________\n\n__________\n| |\n| |\n| A |\n| |\nA o B = __________\n| |\n| |\n| B |\n| |\n__________\n\nThe 2-associtivity condition for the 2x2 square takes the form\n\n(A).(C) (A . C) A . C\n(o) (o) = o o = o o\n(B).(D) (B . D) B . D\n\nand is automatic. In fact, one of the virtues of associativity\nis that you can write multiple products without parentheses,\nwhich is readily done for the YB product. But this is impossible\nfor the 2-group product for the 2x2 square above, since not even\nthe number of different products are the same on both sides\n(1 . and 2 o versus 2 . and 1 o).\n\nOrdinary multiplication can be seen as the operation of gluing\nlinks at common endpoints. This viewpoint is particularly clear\nin lattice gauge theory where matrices live on links and gluing\namounts to contracting indices. Similarly, the YB objects live\non elementary squares ("plaquettes"), and adjascent plaquettes\ncan be glued together along a shared edge by contracting the\ncorresponding indices. However, in 2-group multiplication one\ndoes something more: two consecutive horizontal (say) edges are\nfused into a single edge. This "fusing operation", which usually\nis drawn\n\n______ ______ __________\n/ \\ / \\ / \\\no o o ==> o o\n\\______/ \\______/ \\__________/\n\nimplies that there must exist a map\n\nV @ V --> V\n\ni.e. the space V associated with edges must itself be equipped\nwith a group structure. Not only does this extra structure lead\nto problems with 2-associativity, but it also leads to a\nnon-minimal situation: a 2-gauge theory built on this kind of\n2-product need a 1-form connection in addition to the expected\n2-form connection. Gerbes, which also require lower-order\nconnections, must be related to the 2-group sense of\nmultiplication.\n\nThe name YB product indicates that notation from the YB equation\nmight be useful, and indeed it is. For the 2x2 square in\nEnd(V@V@V@V), label the four V\'s as V_1@V_2@V_3@V_4 as the\nfigure indicates:\n\nV_3 V_4\n__________.__________\n| | |\n| | |\nV_1 | A_13 | C_14 |\n| | |\n__________.__________\n| | |\n| | |\nV_2 | B_23 | D_24 |\n| | |\n__________.__________\n\n\nSubindices indicate on which V\'s the objects A, B, C, D act\nnon-trivially: A_12 = A@1@1, B_23 = 1@B@1, C_34 = 1@1@C, etc.\n2-associativity for the 2x2 square can now be written as\n\n(A_13 B_23)(C_14 D_24) = (A_13 C_14)(B_23 D_24),\n\nwhich is manifestly true since B_23 and C_14 commute (they act\non different spaces, V_2@V_3 and V_1@V_4, respectively).\n\nNeither definition of a 2-product is more correct than the\nother; a definition is always correct as long as it is\nself-consistent. However, the YB version of 2-multiplication\nseems more fruitful to me, because 2-associativity is automatic,\nno edge objects are needed, and there is a close connection to\nthe physically relevant YB equation.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In recent years there has been some interest in 2-groups and
2-gauge theories, where the gauge invariant objects are Wilson
surfaces rather than Wilson lines, see e.g.

Girelli-Pfeiffer, http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0309173
Baez, http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0206130
Schreiber, http://www-stud.uni-essen.de/~sb0264/p11.pdf

Underlying these theories is 2-multiplication in the 2-group
sense. However, a different notion of 2-multiplication is
implicit in the Yang-Baxter (YB) equation, which led me to
formulate 2-gauge theories on the lattice already 15 years ago;
for a review and the original reference see
http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017

Seeing the very complicated and awkward compatibility conditions
that are necessary to ensure 2-associativity for 2-groups,
I'm becoming increasingly convinced that this is not a fruitful
approach. To make this point I would like to contrast the
2-product across a 2x2 square using the two different notions.

In 2-groups, we have a horizontal product . and a vertical
product o which take two object and construct a new object *of
the same type*. Thus, given A and be we can construct products
(A.B) and (AoB), where A and B should be written on top of each
other in the second product. The value of a 2x2 square should not
depend on whether we do the horizontal or vertical product first,
so the condition for 2-associativity looks like

(A) (C) (A . C)
(o).(o) = o
(B) (D) (B . D)

This kind of 2-associativity requires some very complicated
consistency relations, and it is unclear to me if any
interesting solutions to these exist.

In contrast, one may use the manifestly associative YB product.
Let V be the basic vector space. An ordinary matrix is an
element in End(V), i.e. a two-index quantity. YB matrices A and
B are instead elements in End(V@V) (@ = tensor product), i.e.
4-index quantities. We can still define horizontal and vertical
products, but the results are 6-index quantities in End(V@V@V),
because gluing two squares along a common edge results in a
6-edged rectangle:

__{________}.__{________}
| | || | |A . B = | A | B |
| | |
__{________}.__{________}

__{________}
| || |
| A |
| |A o B = __{________}
| || |
| B |
| |
__{________}

The 2-associtivity condition for the 2x2 square takes the form

(A).(C) (A . C) A . C
(o) (o) = o o = o o
(B).(D) (B . D) B . D

and is automatic. In fact, one of the virtues of associativity
is that you can write multiple products without parentheses,
which is readily done for the YB product. But this is impossible
for the 2-group product for the 2x2 square above, since not even
the number of different products are the same on both sides
(1 . and 2 o versus 2 . and 1 o).

Ordinary multiplication can be seen as the operation of gluing
links at common endpoints. This viewpoint is particularly clear
in lattice gauge theory where matrices live on links and gluing
amounts to contracting indices. Similarly, the YB objects live
on elementary squares ("plaquettes"), and adjascent plaquettes
can be glued together along a shared edge by contracting the
corresponding indices. However, in 2-group multiplication one
does something more: two consecutive horizontal (say) edges are
fused into a single edge. This "fusing operation", which usually
is drawn

__{____} __{____} __{________}
/ \ / \ / \
o o o ==> o o
\__{____}/ \__{____}/ \__{________}/

implies that there must exist a map

V @ V --> V

i.e. the space V associated with edges must itself be equipped
with a group structure. Not only does this extra structure lead
to problems with 2-associativity, but it also leads to a
non-minimal situation: a 2-gauge theory built on this kind of
2-product need a 1-form connection in addition to the expected
2-form connection. Gerbes, which also require lower-order
connections, must be related to the 2-group sense of
multiplication.

The name YB product indicates that notation from the YB equation
might be useful, and indeed it is. For the 2x2 square in
End(V@V@V@V), label the four V's as V_1@V_2@V_3@V_4 as the
figure indicates:

V_3 V_4[/itex]
__{________}.__{________}
| | || | |V_1 | A_{13} | C_{14} || | |
__{________}.__{________}
| | || | |V_2 | B_{23} | D_{24} || | |
__{________}.__{________}

Subindices indicate on which V's the objects A, B, C, D act
non-trivially: A_{12} = A@1@1, B_{23} = 1@B@1, C_{34} = 1@1@C, etc.
2-associativity for the 2x2 square can now be written as

[itex](A_{13} B_{23})(C_{14} D_{24}) = (A_{13} C_{14})(B_{23} D_{24}),

which is manifestly true since B_{23} and C_{14} commute (they act
on different spaces, V_2@V_3 and V_1@V_4, respectively).

Neither definition of a 2-product is more correct than the
other; a definition is always correct as long as it is
self-consistent. However, the YB version of 2-multiplication
seems more fruitful to me, because 2-associativity is automatic,
no edge objects are needed, and there is a close connection to
the physically relevant YB equation.

Urs Schreiber

Sep4-04, 06:35 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409040456.24b87848@pos ting.google.com...\n\n> In recent years there has been some interest in 2-groups and\n> 2-gauge theories, where the gauge invariant objects are Wilson\n> surfaces rather than Wilson lines, see e.g.\n>\n> Girelli-Pfeiffer, http://www.arxiv.org/abs/hep-th/0309173\n> Baez, http://www.arxiv.org/abs/hep-th/0206130\n> Schreiber, http://www-stud.uni-essen.de/~sb0264/p11.pdf\n>\n> Underlying these theories is 2-multiplication in the 2-group\n> sense. However, a different notion of 2-multiplication is\n> implicit in the Yang-Baxter (YB) equation, which led me to\n> formulate 2-gauge theories on the lattice already 15 years ago;\n> for a review and the original reference see\n> http://www.arxiv.org/abs/math-ph/0205017\n>\n> Seeing the very complicated and awkward compatibility conditions\n> that are necessary to ensure 2-associativity for 2-groups,\n> I\'m becoming increasingly convinced that this is not a fruitful\n> approach.\n\nThe problem this approach is addressing is the following:\n\nHow to assign an element of some group to a given surface in a way that is\nwell defined (i.e. depends only on the surface, not on any\nparameterizations).\n\nDoes your approach address this problem?\n\nCan you assign group elements in a unique way, or do you rather assign\nelements of End(V @ V @ .... @V) to a surface?\n\nIs even that assignment unique?\n\nDoesn\'t it depend on the latticization of the surface?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409040456.24b87848@posting.google.c om...

> In recent years there has been some interest in 2-groups and
> 2-gauge theories, where the gauge invariant objects are Wilson
> surfaces rather than Wilson lines, see e.g.
>
> Girelli-Pfeiffer, http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0309173
> Baez, http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0206130
> Schreiber, http://www-stud.uni-essen.de/~sb0264/p11.pdf
>
> Underlying these theories is 2-multiplication in the 2-group
> sense. However, a different notion of 2-multiplication is
> implicit in the Yang-Baxter (YB) equation, which led me to
> formulate 2-gauge theories on the lattice already 15 years ago;
> for a review and the original reference see
> http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017
>
> Seeing the very complicated and awkward compatibility conditions
> that are necessary to ensure 2-associativity for 2-groups,
> I'm becoming increasingly convinced that this is not a fruitful
> approach.

The problem this approach is addressing is the following:

How to assign an element of some group to a given surface in a way that is
well defined (i.e. depends only on the surface, not on any
parameterizations).

Does your approach address this problem?

Can you assign group elements in a unique way, or do you rather assign
elements of End(V @ V @ .... @V) to a surface?

Is even that assignment unique?

Doesn't it depend on the latticization of the surface?

Thomas Larsson

Sep7-04, 11:43 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413a4323\\$1@news.sentex.net>...\n\n> The problem this approach is addressing is the following:\n>\n> How to assign an element of some group to a given surface in a way that is\n> well defined (i.e. depends only on the surface, not on any\n> parameterizations).\n>\n> Does your approach address this problem?\n>\n\nSorry for posting two parallel responses, which might\nconfuse the line of thought. However, since my last post\n(which has not yet appeared on Google) I have been\nthinking more about the continuum limits of 2-gauge\ntheory, using the 2-group and Yang-Baxter versions of\nthe 2-product.\n\nIn http://www-stud.uni-essen.de/~sb0264/p11.pdf,\npage 4, you list yourself the known solutions to the\nconsistency conditions for the 2-group version ("r-flatness").\n(For the benefit of other readers, A is a one-form\ngauge connection and F the corresponding two-form\ncurvature, whereas B is a two-form connection with\nthree-form curvature G. On the lattice, one-forms\nlive on edges, two-forms on elementary squares\n(plaquettes), and three-forms on elementary volumes.)\n\nThere are three known solutions to the consistency\nconditions:\n\n1. F = 0 and B belongs to an abelian ideal. This\nmeans that A is gauge-equivalent to zero and the\ncomponents of B commute. This is essentially our old\nfriend two-form electrodynamics and nothing else.\n\n2. F = 0 and B is covariantly constant wrt A. Again\nA is gauge-equivalent to zero, so B is really constant\nand not only covariantly so. This seems like a trivial\ncase indeed.\n\n3. F + B = 0. This means that B = -F(A) is an\nauxiliary field and dynamics can be formulated in\nterms of the A-field alone.\n\nThe only I used is what you write yourself and my freedom\nto choose a convenient gauge. F = 0 implies A = 0 only\nlocally, of course. There might be topologically nontrivial\nconfigurations, but we want 2-gauge theory to be\ninteresting already locally, right? Thus, if your list of\nsolutions to the consistency relations is exhaustive, we\nmore or less have a no-go theorem stating that no\ninteresting 2-gauge theory can come out of the 2-group\napproach.\n\nIn contrast, the YB product does yield an interesting\n2-gauge theory, at least on the lattice. The weak\npoint is the continuum limit, but I do have a natural\nsuggestion for this, although not a rigorous derivation\n( http://www.arxiv.org/abs/math-ph/0205017 ).\nSince the gauge-covariant objects are Wilson\nsurfaces, we expect to have some string space\nformulation. More precisely, we are interested in\nstring fields, which are functionals f[x(t)] over\nstring space, where x(t) is the string in spacetime.\n\nA key observation is that the lattice version is\nlocal in spacetime. Therefore, it might suffice to\nconsider only fields that are functions of x and\ns = dx/dt. Such truncated string fields can not answer\nall questions about string configurations, but they\ncan answer local questions, e.g. how many strings\nthere are at x with direction s.\n\nNow recall that we want to generalize the expression\nfor the curvature,\n\nF = dA + [A, A],\n\nto a p-form connection A. As it stands, this\nexpression works only for p=1, which is easy to see\nby counting form degrees: F and dA are (p+1)-forms but\n[A,A] is a 2p-form. However, from a trunctated string\n2-form A = A(x,s), we can construct a 1-form s.A by\ncontracting with the vector s. So my suggestion for\nthe 3-form curvature is\n\nF = dA + [s.A, A].\n\nThis expression transforms in a well-defined manner\nunder diffeomorphisms, because all three terms\ntransform as 3-forms. This expression also transforms\nhomogeneously under a kind of s-dependent gauge\ntransformations. Note that the alternative s.[A,A] is\nnot good because it is identically zero in 3D.\n\nThe same expression works of course if A is a p-form\nand s an (p-1)-dimensional surface element, or Jacobian\nmatrix. In particular if p=1 we just get a change in\nthe coupling constant.\n\nThe curvature is the critical ingredient. Once we\nhave that, dynamics follows from the invariant\nYang-Mills-like action\n\nS = \\int dx \\int ds tr (F \\wedge *F).\n\n(* = Hodge star, the metric depends on x but not on s).\nIn 2p+2 dimensions, we can alternatively consider\n\nS = \\int dx \\int ds tr (F \\wedge F),\n\nwhich may be a topological invariant. Probably we can\ndefine something Chern-Simons-like in 2p+1 dimension\nas well.\n\nI admit that this is a bit formal, e.g. I have not\nspecified in which spaces the A\'s are valued.\nNevertheless, I think that the expression for F is\nquite neat, and I am somewhat proud of it.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413a4323$1@news.sentex.net>...

> The problem this approach is addressing is the following:
>
> How to assign an element of some group to a given surface in a way that is
> well defined (i.e. depends only on the surface, not on any
> parameterizations).
>
> Does your approach address this problem?
>

Sorry for posting two parallel responses, which might
confuse the line of thought. However, since my last post
(which has not yet appeared on Google) I have been
thinking more about the continuum limits of 2-gauge
theory, using the 2-group and Yang-Baxter versions of
the 2-product.

In http://www-stud.uni-essen.de/~sb0264/p11.pdf,
page 4, you list yourself the known solutions to the
consistency conditions for the 2-group version ("r-flatness").
(For the benefit of other readers, A is a one-form
gauge connection and F the corresponding two-form
curvature, whereas B is a two-form connection with
three-form curvature G. On the lattice, one-forms
live on edges, two-forms on elementary squares
(plaquettes), and three-forms on elementary volumes.)

There are three known solutions to the consistency
conditions:

1. F = and B belongs to an abelian ideal. This
means that A is gauge-equivalent to zero and the
components of B commute. This is essentially our old
friend two-form electrodynamics and nothing else.

2. F = and B is covariantly constant wrt A. Again
A is gauge-equivalent to zero, so B is really constant
and not only covariantly so. This seems like a trivial
case indeed.

3. F + B = . This means that B = -F(A) is an
auxiliary field and dynamics can be formulated in
terms of the A-field alone.

The only I used is what you write yourself and my freedom
to choose a convenient gauge. F = implies A = only
locally, of course. There might be topologically nontrivial
configurations, but we want 2-gauge theory to be
interesting already locally, right? Thus, if your list of
solutions to the consistency relations is exhaustive, we
more or less have a no-go theorem stating that no
interesting 2-gauge theory can come out of the 2-group
approach.

In contrast, the YB product does yield an interesting
2-gauge theory, at least on the lattice. The weak
point is the continuum limit, but I do have a natural
suggestion for this, although not a rigorous derivation
( http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017 ).
Since the gauge-covariant objects are Wilson
surfaces, we expect to have some string space
formulation. More precisely, we are interested in
string fields, which are functionals f[x(t)] over
string space, where x(t) is the string in spacetime.

A key observation is that the lattice version is
local in spacetime. Therefore, it might suffice to
consider only fields that are functions of x and
s = dx/dt. Such truncated string fields can not answer
all questions about string configurations, but they
can answer local questions, e.g. how many strings
there are at x with direction s.

Now recall that we want to generalize the expression
for the curvature,

F = dA +[/itex] [A, A],

to a p-form connection A. As it stands, this
expression works only for p=1, which is easy to see
by counting form degrees: F and dA are (p+1)-forms but
[A,A] is a 2p-form. However, from a trunctated string
2-form A = A(x,s), we can construct a 1-form s.A by
contracting with the vector s. So my suggestion for
the 3-form curvature is

F = dA + [s.A, A].

This expression transforms in a well-defined manner
under diffeomorphisms, because all three terms
transform as 3-forms. This expression also transforms
homogeneously under a kind of s-dependent gauge
transformations. Note that the alternative s.[A,A] is
not good because it is identically zero in 3D.

The same expression works of course if A is a p-form
and s an (p-1)-dimensional surface element, or Jacobian
matrix. In particular if p=1 we just get a change in
the coupling constant.

The curvature is the critical ingredient. Once we
have that, dynamics follows from the invariant
Yang-Mills-like action

S = \int dx \int ds tr (F \wedge *F).

(* = Hodge star, the metric depends on x but not on s).
In 2p+2 dimensions, we can alternatively consider

[itex]S = \int dx \int ds tr (F \wedge F),

which may be a topological invariant. Probably we can
define something Chern-Simons-like in 2p+1 dimension
as well.

I admit that this is a bit formal, e.g. I have not
specified in which spaces the A's are valued.
Nevertheless, I think that the expression for F is
quite neat, and I am somewhat proud of it.

Urs Schreiber

Sep7-04, 02:58 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409070717.7dfcc450@pos ting.google.com...\n\n> There are three known solutions to the consistency\n> conditions:\n>\n> 1. F = 0 and B belongs to an abelian ideal. This\n> means that A is gauge-equivalent to zero and the\n> components of B commute. This is essentially our old\n> friend two-form electrodynamics and nothing else.\n>\n> 2. F = 0 and B is covariantly constant wrt A. Again\n> A is gauge-equivalent to zero, so B is really constant\n> and not only covariantly so. This seems like a trivial\n> case indeed.\n>\n> 3. F + B = 0. This means that B = -F(A) is an\n> auxiliary field and dynamics can be formulated in\n> terms of the A-field alone.\n\n\nBTW, it turns out that all 2-connections with nonvanishing B+F are\nequivalent to direct products of those with vanishing B+F.\n\n\n> The only I used is what you write yourself and my freedom\n> to choose a convenient gauge. F = 0 implies A = 0 only\n> locally, of course.\n\n\nYes, and that\'s precisely the argument I given in hep-th/0407122, arguing\nthat those solutions with nonvanishing B+F are not that interesting for\nlocal purposes.\n\n\n> There might be topologically nontrivial configurations,\n\n\nThat\'s indeed what Alavarez et al. have considered. There the surface\nholonomy over S^2s plays the role of a conserved charge in an integrable\n2+1d field theory. So this is not unimportant.\n\n\n> but we want 2-gauge theory to be interesting already locally, right?\n\n\nThere are different possibly interesting questions. One is:\n\n- What is the most general consistent notion of surface holonomy that one\ncan come up with?\n\nThat\'s what I am currently concerned with when talking about these\nr-flatness and other consistency conditions. These conditions are there, you\ndon\'t have the freedom to choose an "approach" to consistent surface\nholonomy which does not have them.\n\nAnother interesting question is:\n\n- Is there an interesting generalization of 1-form YM to something like\n2-form YM?\n\nHere, indeed, as we have discussed in a recent thread here on spr, the\nr-flatness condition completely spoils the naively expected 2-form YM\ngeneralization. Hendryk Pfeiffer says he wants to circumvent this by\nconsidering non-differentiable gauge fields. I think this is not really what\npeople naively expected when setting out to do 2-form gauge theory, either.\nBut maybe something interesting can be found there.\n\n\nThen an interesting question is:\n\n- Even if the dynamics of 2-form YM is only that of 1-form YM plus a\nconstraint relating B to A. This still gives us a nontrivial non-abelian\nB-field background. What if we couple anything to it?\n\nIn hep-th/0407122 I consider an approach to couple superstrings to this\nbackground. It is very different to a pure 1-form gauge field background. So\nthis is an interesting physicsal configuration, even though the background\'s\nequation of motion are equivalent to those of 1-form YM.\n\nThis gives rise to the question:\n\n- Can we, instead of trying to postulate a 2-form generlization of YM, come\nup with something more general which in an appropriate limit automatically\ngives rise to non-abelian 2-form dynamics.\n\nThe answer to this seems to be effective field theories on NS5-branes, where\n"little strings" arise as membrane boundaries ending on these branes. I have\nheard from a couple of people about various hints that the resulting 6d\neffective field theories feature a non-abelian 2-form in one way or another.\nBut nobody seems to know yet what happens exactly.\n\n\n> Thus, if your list of\n> solutions to the consistency relations is exhaustive, we\n> more or less have a no-go theorem stating that no\n> interesting 2-gauge theory can come out of the 2-group\n> approach.\n\n\nYes, indeed. And this was discussed before, here in spr and in print for\ninstance in hep-th/0309173 by Girelly&Pfeiffer.\n\nBut please note that the "2-group approach" is not just some randomly chosen\napproach. This is the unique method to get consistent surface holonomies.\nThere is another method, loop space holonomy, but it can be shown to be\nequivalent to 2-group holonomy.\n\nSo what you should conclude is that:\n\nWe have a no-go theorem stating that no "interesting" 2-group gauge theory\ncan come out of a 2-gauge theory which uses a notion of 2-connection that\nassigns unique surface holonomy (unless, maybe, we allow our gauge fields to\nbe non-differentiable, as Henryk Pfeiffer is suggesting).\n\nSo if you wan to to circumvent this you have to deal with a rather serious\nproblem, namely you have to deal with a gauge theory where the very concept\nof holonomy is not well defined. As far as I understand, this is indeed what\nyou are proposing to do.\n\n\n> In contrast, the YB product does yield an interesting\n> 2-gauge theory, at least on the lattice. The weak\n> point is the continuum limit, but I do have a natural\n> suggestion for this, although not a rigorous derivation\n> ( http://www.arxiv.org/abs/math-ph/0205017 ).\n> Since the gauge-covariant objects are Wilson\n> surfaces,\n\n\nIn the light of the above comments this is a strange statement. When using A\nand B such that they don\'t satisfy the r-flateness condition (in your\napproach F=0 and B is arbitrary) you don\'t have a well-defined notion of\nsurface holonomy, i.e. no well-defined notion of Wilson-surface. This is\njust a fact. It has nothing to do with the choice of language (2-groups,\nloop space, etc.) that you choose in order to talk about this stuff.\n\nSo without Wilson surfaces you can\'t say that gauge-covariant objects are\nWilson surfaces.\n\nThis is precisely the reason why the notion of consistent surface holonomy\nis so important and why there is thinking going on concerning the r-flatness\ncondition and its implications.\n\n\n> So my suggestion for the 3-form curvature is\n>\n> F = dA + [s.A, A].\n\n> I admit that this is a bit formal, e.g. I have not\n> specified in which spaces the A\'s are valued.\n> Nevertheless, I think that the expression for F is\n> quite neat, and I am somewhat proud of it.\n\n\nI don\'t quite see yet why this way of defining F is the one we all should\nuse.\n\nI mean, if you are prepared to abandon consistent surface holonomy and to\nwrite your field theory as a field theory on loop space then there is\nnothing more natural than considering the true YM action on loop space.\n\nPick a connection cal{A} on loop space coming from a 1-form A and a 2-form B\nas in equation (1.1) in http://www-stud.uni-essen.de/~sb0264/p11.ps and use\nits field strength (equation (2.6) ) cal(F) to write down the loop space YM\naction ~ F^2 .\n\nNote that your idea of contracting a target space 2-form with the tangent to\nthe loop (what you call s) indeed appears in the connection equation (1.1)\nof the above text. The full expression for s is called X\', there, where X :\n(0,2pi) -> is the loop and the prime indicates its sigma-derivative. This\nworks on full loop space, no truncation as you indicated is needed.\n\nThere are many ways to derive the form of the loop space connection (1.1).\nOne way is to consider a gauge transformation on loop space with the\ngenerating function being the holonomy of A around the loops.\n\nOf course this cal{F} is a 2-form on loop space, as it should be. p-forms on\nloop space correspond to 3-forms on target space.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409070717.7dfcc450@posting.google.c om...

> There are three known solutions to the consistency
> conditions:
>
> 1. F = and B belongs to an abelian ideal. This
> means that A is gauge-equivalent to zero and the
> components of B commute. This is essentially our old
> friend two-form electrodynamics and nothing else.
>
> 2. F = and B is covariantly constant wrt A. Again
> A is gauge-equivalent to zero, so B is really constant
> and not only covariantly so. This seems like a trivial
> case indeed.
>
> 3. F + B = . This means that B = -F(A) is an
> auxiliary field and dynamics can be formulated in
> terms of the A-field alone.

BTW, it turns out that all 2-connections with nonvanishing B+F are
equivalent to direct products of those with vanishing B+F.

> The only I used is what you write yourself and my freedom
> to choose a convenient gauge. F = implies A = only
> locally, of course.

Yes, and that's precisely the argument I given in http://www.arxiv.org/abs/hep-th/0407122, arguing
that those solutions with nonvanishing B+F are not that interesting for
local purposes.

> There might be topologically nontrivial configurations,

That's indeed what Alavarez et al. have considered. There the surface
holonomy over S^{2s} plays the role of a conserved charge in an integrable
2+1d field theory. So this is not unimportant.

> but we want 2-gauge theory to be interesting already locally, right?

There are different possibly interesting questions. One is:

- What is the most general consistent notion of surface holonomy that one
can come up with?

That's what I am currently concerned with when talking about these
r-flatness and other consistency conditions. These conditions are there, you
don't have the freedom to choose an "approach" to consistent surface
holonomy which does not have them.

Another interesting question is:

- Is there an interesting generalization of 1-form YM to something like
2-form YM?

Here, indeed, as we have discussed in a recent thread here on spr, the
r-flatness condition completely spoils the naively expected 2-form YM
generalization. Hendryk Pfeiffer says he wants to circumvent this by
considering non-differentiable gauge fields. I think this is not really what
people naively expected when setting out to do 2-form gauge theory, either.
But maybe something interesting can be found there.

Then an interesting question is:

- Even if the dynamics of 2-form YM is only that of 1-form YM plus a
constraint relating B to A. This still gives us a nontrivial non-abelian
B-field background. What if we couple anything to it?

In http://www.arxiv.org/abs/hep-th/0407122 I consider an approach to couple superstrings to this
background. It is very different to a pure 1-form gauge field background. So
this is an interesting physicsal configuration, even though the background's
equation of motion are equivalent to those of 1-form YM.

This gives rise to the question:

- Can we, instead of trying to postulate a 2-form generlization of YM, come
up with something more general which in an appropriate limit automatically
gives rise to non-abelian 2-form dynamics.

The answer to this seems to be effective field theories on NS5-branes, where
"little strings" arise as membrane boundaries ending on these branes. I have
heard from a couple of people about various hints that the resulting 6d
effective field theories feature a non-abelian 2-form in one way or another.
But nobody seems to know yet what happens exactly.

> Thus, if your list of
> solutions to the consistency relations is exhaustive, we
> more or less have a no-go theorem stating that no
> interesting 2-gauge theory can come out of the 2-group
> approach.

Yes, indeed. And this was discussed before, here in spr and in print for
instance in http://www.arxiv.org/abs/hep-th/0309173 by Girelly&Pfeiffer.

But please note that the "2-group approach" is not just some randomly chosen
approach. This is the unique method to get consistent surface holonomies.
There is another method, loop space holonomy, but it can be shown to be
equivalent to 2-group holonomy.

So what you should conclude is that:

We have a no-go theorem stating that no "interesting" 2-group gauge theory
can come out of a 2-gauge theory which uses a notion of 2-connection that
assigns unique surface holonomy (unless, maybe, we allow our gauge fields to
be non-differentiable, as Henryk Pfeiffer is suggesting).

So if you wan to to circumvent this you have to deal with a rather serious
problem, namely you have to deal with a gauge theory where the very concept
of holonomy is not well defined. As far as I understand, this is indeed what
you are proposing to do.

> In contrast, the YB product does yield an interesting
> 2-gauge theory, at least on the lattice. The weak
> point is the continuum limit, but I do have a natural
> suggestion for this, although not a rigorous derivation
> ( http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017 ).
> Since the gauge-covariant objects are Wilson
> surfaces,

In the light of the above comments this is a strange statement. When using A
and B such that they don't satisfy the r-flateness condition (in your
approach F=0 and B is arbitrary) you don't have a well-defined notion of
surface holonomy, i.e. no well-defined notion of Wilson-surface. This is
just a fact. It has nothing to do with the choice of language (2-groups,
loop space, etc.) that you choose in order to talk about this stuff.

So without Wilson surfaces you can't say that gauge-covariant objects are
Wilson surfaces.

This is precisely the reason why the notion of consistent surface holonomy
is so important and why there is thinking going on concerning the r-flatness
condition and its implications.

> So my suggestion for the 3-form curvature is
>
> F = dA + [s.A, A].

> I admit that this is a bit formal, e.g. I have not
> specified in which spaces the A's are valued.
> Nevertheless, I think that the expression for F is
> quite neat, and I am somewhat proud of it.

I don't quite see yet why this way of defining F is the one we all should
use.

I mean, if you are prepared to abandon consistent surface holonomy and to
write your field theory as a field theory on loop space then there is
nothing more natural than considering the true YM action on loop space.

Pick a connection cal{A} on loop space coming from a 1-form A and a 2-form B
as in equation (1.1) in http://www-stud.uni-essen.de/~sb0264/p11.ps and use
its field strength (equation (2.6) ) cal(F) to write down the loop space YM
action ~ F^2 .

Note that your idea of contracting a target space 2-form with the tangent to
the loop (what you call s) indeed appears in the connection equation (1.1)
of the above text. The full expression for s is called X', there, where X :
(0,2pi) -> is the loop and the prime indicates its \sigma-derivative. This
works on full loop space, no truncation as you indicated is needed.

There are many ways to derive the form of the loop space connection (1.1).
One way is to consider a gauge transformation on loop space with the
generating function being the holonomy of A around the loops.

Of course this cal{F} is a 2-form on loop space, as it should be. p-forms on
loop space correspond to 3-forms on target space.

Thomas Larsson

Sep8-04, 11:18 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413e04de\\$1@news.sentex.net>...\n\n> There are different possibly interesting questions. One is:\n>\n> - What is the most general consistent notion of surface holonomy that one\n> can come up with?\n>\n> That\'s what I am currently concerned with when talking about these\n> r-flatness and other consistency conditions. These conditions are there, you\n> don\'t have the freedom to choose an "approach" to consistent surface\n> holonomy which does not have them.\n\nWell, what can be done depends on your assumptions. It is certainly\npossible to define a consistent surface holonomy on the lattice my way.\n\n>\n> Another interesting question is:\n>\n> - Is there an interesting generalization of 1-form YM to something like\n> 2-form YM?\n>\n> Here, indeed, as we have discussed in a recent thread here on spr, the\n> r-flatness condition completely spoils the naively expected 2-form YM\n> generalization.\n\nI noted the existence of this thread but I didn\'t follow it too closely.\nAnyway, isn\'t this a strong indication that the 2-group approach simply\nfails. On this problem, anyway.\n\nOne thing must be perfectly clear, namely that I have succeeded in\nconstructing the naively expected 2-form generalization of Yang-Mills\ntheory on the lattice. This I did 15 years ago, and it was published in\n\nT. A. Larsson, p-cell gauge theories, manifold space and\nmulti-dimensional integrability, Mod Phys Lett A 5 (1990) 255--264.\n\n(This work built on previous work by J-M Maillet and Frank Nijhoff, who\nworked with similar concepts. They didn\'t explicitly formulate their\nideas as a 2-form lattice gauge theory, presumably because this was of\nno interest to them.)\n\nTo me, this is the crucial difference: I succeed, on the lattice, whereas\nthe 2-group approach simply fails. And I succeeded 15 years ago, before\nthe word "gerbe" was invented and when a 2-group was a discrete group\nwith two elements.\n\n>\n> In the light of the above comments this is a strange statement. When using A\n> and B such that they don\'t satisfy the r-flateness condition (in your\n> approach F=0 and B is arbitrary) you don\'t have a well-defined notion of\n> surface holonomy, i.e. no well-defined notion of Wilson-surface. This is\n> just a fact. It has nothing to do with the choice of language (2-groups,\n> loop space, etc.) that you choose in order to talk about this stuff.\n>\n> So without Wilson surfaces you can\'t say that gauge-covariant objects are\n> Wilson surfaces.\n>\n> This is precisely the reason why the notion of consistent surface holonomy\n> is so important and why there is thinking going on concerning the r-flatness\n> condition and its implications.\n>\n\nSigh. A Wilson line is, by definition, a product of links in lattice\ngauge theory. Ken Wilson came up with this concept when he introduced\nlattice gauge theory in 1974. By a Wilson surface, I mean a 2D surface\n*on the lattice*. There is no doubt that this is a well-defined notion.\n\nThere are admittedly problems is with the continuum limit. One way to\ndeal with that is to let any second-order phase transition define the\nfull quantum theory, in analogy with the non-perturbative definition of\nordinary Yang-Mills theory by lattice methods. This is what we ultimately\nwant, anyway.\n\nBut even a naive, classical continuum limit, may exist. As I pointed out in my\npost which has yet not appeared, there isn\'t really any problem with the\ncontinuum limit as long as the surface is closed. No problem that isn\'t\npresent for ordinary lattice gauge theory, anyway. For a surface with\nboundary, the boundary looks like a barbed wire, with one V on each link.\nOne thus has to make sense of a barbed wire in the limit that the barb\nspacing goes to zero, which requires some new math. In my eprint I\nattempt at a definition of such a surface-ordered integral.\n\nBut then again, an open Wilson line or surface is not gauge invariant.\nTo construct a gauge invariant object from an open Wilson line, you need\nto terminate the ends by matter fields. Similarly you can add "string\nmatter" living on links, to terminate open Wilson surfaces. This gives\nyou a scalar number from the lattice theory.\n\n>\n> > So my suggestion for the 3-form curvature is\n> >\n> > F = dA + [s.A, A].\n>\n> > I admit that this is a bit formal, e.g. I have not\n> > specified in which spaces the A\'s are valued.\n> > Nevertheless, I think that the expression for F is\n> > quite neat, and I am somewhat proud of it.\n>\n>\n> I don\'t quite see yet why this way of defining F is the one we all should\n> use.\n\nYou may use whatever you wish, but using 2-groups doesn\'t seem too\nfruitful.\n\n>\n> I mean, if you are prepared to abandon consistent surface holonomy and to\n> write your field theory as a field theory on loop space then there is\n> nothing more natural than considering the true YM action on loop space.\n\nI don\'t abandon consistent surface holonomy, but I evade one of your\nassumptions. Namely, when you say that you need both 1-form and 2-form\nfields, you assume that they depend on the point x in spacetime only, say\nA = A(x). I propose to use fields which depend on an additional variable\ns which transforms as a vector, A = A(x,s). The way to circumvent a no-go\ntheorem is to relax some of the explicit or implicit assumptions.\n\nThis is different from YM on loop space, since I need a new vector space\nindex for each point on the loop.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413e04de$1@news.sentex.net>...

> There are different possibly interesting questions. One is:
>
> - What is the most general consistent notion of surface holonomy that one
> can come up with?
>
> That's what I am currently concerned with when talking about these
> r-flatness and other consistency conditions. These conditions are there, you
> don't have the freedom to choose an "approach" to consistent surface
> holonomy which does not have them.

Well, what can be done depends on your assumptions. It is certainly
possible to define a consistent surface holonomy on the lattice my way.

>
> Another interesting question is:
>
> - Is there an interesting generalization of 1-form YM to something like
> 2-form YM?
>
> Here, indeed, as we have discussed in a recent thread here on spr, the
> r-flatness condition completely spoils the naively expected 2-form YM
> generalization.

I noted the existence of this thread but I didn't follow it too closely.
Anyway, isn't this a strong indication that the 2-group approach simply
fails. On this problem, anyway.

One thing must be perfectly clear, namely that I have succeeded in
constructing the naively expected 2-form generalization of Yang-Mills
theory on the lattice. This I did 15 years ago, and it was published in

T. A. Larsson, p-cell gauge theories, manifold space and
multi-dimensional integrability, Mod Phys Lett A 5 (1990) 255--264.

(This work built on previous work by J-M Maillet and Frank Nijhoff, who
worked with similar concepts. They didn't explicitly formulate their
ideas as a 2-form lattice gauge theory, presumably because this was of
no interest to them.)

To me, this is the crucial difference: I succeed, on the lattice, whereas
the 2-group approach simply fails. And I succeeded 15 years ago, before
the word "gerbe" was invented and when a 2-group was a discrete group
with two elements.

>
> In the light of the above comments this is a strange statement. When using A
> and B such that they don't satisfy the r-flateness condition (in your
> approach F=0 and B is arbitrary) you don't have a well-defined notion of
> surface holonomy, i.e. no well-defined notion of Wilson-surface. This is
> just a fact. It has nothing to do with the choice of language (2-groups,
> loop space, etc.) that you choose in order to talk about this stuff.
>
> So without Wilson surfaces you can't say that gauge-covariant objects are
> Wilson surfaces.
>
> This is precisely the reason why the notion of consistent surface holonomy
> is so important and why there is thinking going on concerning the r-flatness
> condition and its implications.
>

Sigh. A Wilson line is, by definition, a product of links in lattice
gauge theory. Ken Wilson came up with this concept when he introduced
lattice gauge theory in 1974. By a Wilson surface, I mean a 2D surface
*on the lattice*. There is no doubt that this is a well-defined notion.

There are admittedly problems is with the continuum limit. One way to
deal with that is to let any second-order phase transition define the
full quantum theory, in analogy with the non-perturbative definition of
ordinary Yang-Mills theory by lattice methods. This is what we ultimately
want, anyway.

But even a naive, classical continuum limit, may exist. As I pointed out in my
post which has yet not appeared, there isn't really any problem with the
continuum limit as long as the surface is closed. No problem that isn't
present for ordinary lattice gauge theory, anyway. For a surface with
boundary, the boundary looks like a barbed wire, with one V on each link.
One thus has to make sense of a barbed wire in the limit that the barb
spacing goes to zero, which requires some new math. In my eprint I
attempt at a definition of such a surface-ordered integral.

But then again, an open Wilson line or surface is not gauge invariant.
To construct a gauge invariant object from an open Wilson line, you need
to terminate the ends by matter fields. Similarly you can add "string
matter" living on links, to terminate open Wilson surfaces. This gives
you a scalar number from the lattice theory.

>
> > So my suggestion for the 3-form curvature is
> >
> > F = dA + [s.A, A].
>
> > I admit that this is a bit formal, e.g. I have not
> > specified in which spaces the A's are valued.
> > Nevertheless, I think that the expression for F is
> > quite neat, and I am somewhat proud of it.
>
>
> I don't quite see yet why this way of defining F is the one we all should
> use.

You may use whatever you wish, but using 2-groups doesn't seem too
fruitful.

>
> I mean, if you are prepared to abandon consistent surface holonomy and to
> write your field theory as a field theory on loop space then there is
> nothing more natural than considering the true YM action on loop space.

I don't abandon consistent surface holonomy, but I evade one of your
assumptions. Namely, when you say that you need both 1-form and 2-form
fields, you assume that they depend on the point x in spacetime only, say
A = A(x). I propose to use fields which depend on an additional variable
s which transforms as a vector, A = A(x,s). The way to circumvent a no-go
theorem is to relax some of the explicit or implicit assumptions.

This is different from YM on loop space, since I need a new vector space
index for each point on the loop.

Urs Schreiber

Sep8-04, 11:24 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409080644.2cb428d8@pos ting.google.com...\n\n> By a Wilson surface, I mean a 2D surface *on the lattice*.\n\nYes, I know.\n\n> There is no doubt that this is a well-defined notion.\n\nConsider a surface made up from four plaquettes each labeled by a group\nelement from some group H as in\n\nh1 h1\'\n\nh2 h2\' .\n\nWhat is its surface holonomy?\n\nConsider a closed surface made up from six plaquettes forming a cube, the\nplaquettes being labeled by the group elements h1, h2, ... h6.\n\nWhat is its surface holonomy?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409080644.2cb428d8@posting.google.c om...

> By a Wilson surface, I mean a 2D surface *on the lattice*.

Yes, I know.

> There is no doubt that this is a well-defined notion.

Consider a surface made up from four plaquettes each labeled by a group
element from some group H as in

h1 h1'

h2 h2' .

What is its surface holonomy?

Consider a closed surface made up from six plaquettes forming a cube, the
plaquettes being labeled by the group elements h1, h2, ... h6.

What is its surface holonomy?

Thomas Larsson

Sep9-04, 05:18 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413f2423\\$1@news.sentex.net>...\n\n> Consider a surface made up from four plaquettes each labeled by a group\n> element from some group H as in\n>\n> h1 h1\'\n>\n> h2 h2\' .\n>\n> What is its surface holonomy?\n>\n> Consider a closed surface made up from six plaquettes forming a cube, the\n> plaquettes being labeled by the group elements h1, h2, ... h6.\n>\n> What is its surface holonomy?\n\nLet us start with the simplest surface, namely a single plaquette.\nIt is an element in End(V@V) = V* @ V* @ V @ V (V* = dual space).\nIf we put indices as follows\n\nV_2\n___j____\n| |\nV_1 i | A | k\n| |\n|________|\nl\n\nthen the surface holonomy is the four-index quantity A_12 = (A^kl_ij).\nHere i and j are V* indices and k and l are V indices.\nThis describes parallel transport to the south-east, going from the\nV* links to the V link.\nI also use Yang-Baxter notation which is convenient below.\n\nFor the 2x2 square, also directed to the south-east,\n\nV_3 V_4\n___k____ ___l___\n| | |\nV_1 i | A | B | m\n| | |\n|________|_______|\n| | |\nV_2 j | C | D | n\n| | |\n|________|_______|\np q\n\nthe holonomy becomes\n\nA_13 B_14 C_23 D_24 = ( A^rs_ik B^mt_rl C^up_js D^nq_tu ).\n\nIndices r, s, t, u associated to the internal links are summed over.\n\nIt is unwieldy to write down the holonomy for the cube in explicit\nindex notation, so I only use Yang-Baxter notation. Label the six\nplaquette holonomies by A_12, B_13, C_23, D_21, E_31, F_32. Here\nA_12 is associated with the plaquette in the 12-plane pointing\nsouth-east, and D_21 = (D_12)^-1 with the other such plaquette, which\npoints north-west. The holonomy for the cube, with three links left\nuncontracted, is\n\nA_12 B_13 C_23 D_21 E_31 F_32.\n\nContracting the remaining links by tracing over V_1, V_2 and V_3 gives\nus (a piece of) the action:\n\nS = tr_1 tr_2 tr_3 A_12 B_13 C_23 D_21 E_31 F_32.\n\nSince each plaquette in this cube is uniquely specified by its\nYang-Baxter labels, we may simply write R for A,B,C,D,E,F. The\nzero three-curvature condition then reads\n\nR_12 R_13 R_23 R_21 R_31 R_32 = 1\n\ni.e.\n\nR_12 R_13 R_23 = R_23 R_13 R_12,\n\nwhich is the celebrated Yang-Baxter equation.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413f2423$1@news.sentex.net>...

> Consider a surface made up from four plaquettes each labeled by a group
> element from some group H as in
>
> h1 h1'
>
> h2 h2' .
>
> What is its surface holonomy?
>
> Consider a closed surface made up from six plaquettes forming a cube, the
> plaquettes being labeled by the group elements h1, h2, ... h6.
>
> What is its surface holonomy?

Let us start with the simplest surface, namely a single plaquette.
It is an element in End(V@V) = V* @ V* @ V @ V (V* = dual space).
If we put indices as follows

V_2[/itex]
__{_j____}
| |V_1 i | A | k
| |
|__{______}|
l

then the surface holonomy is the four-index quantity A_{12} = (A^{kl_ij}).
Here i and j are V* indices and k and l are V indices.
This describes parallel transport to the south-east, going from the
V* links to the V link.
I also use Yang-Baxter notation which is convenient below.

For the 2x2 square, also directed to the south-east,

V_3 V_4
__{_k____} __{_l___}
| | |V_1 i | A | B | m
| | |
|__{______}|__{_____}|
| | |V_2 j | C | D | n
| | |
|__{______}|__{_____}|
p q

the holonomy becomes

A_{13} B_{14} C_{23} D_{24} = ( A^{rs_ik} B^{mt_rl} C^{up_js} D^{nq_tu} ).

Indices r, s, t, u associated to the internal links are summed over.

It is unwieldy to write down the holonomy for the cube in explicit
index notation, so I only use Yang-Baxter notation. Label the six
plaquette holonomies by A_{12}, B_{13}, C_{23}, D_{21}, E_{31}, F_{32}. Here
A_{12} is associated with the plaquette in the 12-plane pointing
south-east, and D_{21} = (D_{12})^-1 with the other such plaquette, which
points north-west. The holonomy for the cube, with three links left
uncontracted, is

A_{12} B_{13} C_{23} D_{21} E_{31} F_{32}.

Contracting the remaining links by tracing over V_1, V_2 and V_3 gives
us (a piece of) the action:

S = tr_1 tr_2 tr_3 A_{12} B_{13} C_{23} D_{21} E_{31} F_{32}.

Since each plaquette in this cube is uniquely specified by its
Yang-Baxter labels, we may simply write R for A,B,C,D,E,F. The
zero three-curvature condition then reads

[itex]R_{12} R_{13} R_{23} R_{21} R_{31} R_{32} = 1

i.e.

R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12},

which is the celebrated Yang-Baxter equation.

Urs Schreiber

Sep9-04, 06:43 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409090112.629c7cd4@pos ting.google.com...\n>\n>\n> "Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<413f2423\\$1@news.sentex.net>...\n> \n> > Consider a surface made up from four plaquettes each labeled by a group\n> > element from some group H as in\n> >\n> > h1 h1\'\n> >\n> > h2 h2\' .\n> >\n> > What is its surface holonomy?\n\n> Let us start with the simplest surface, namely a single plaquette.\n> It is an element in End(V@V) = V* @ V* @ V @ V (V* = dual space).\n> If we put indices as follows\n>\n> V_2\n> ___j____\n> | |\n> V_1 i | A | k\n> | |\n> |________|\n> l\n>\n> then the surface holonomy is the four-index quantity A_12 = (A^kl_ij).\n\n\nThe ordinary notion of surface holonomy is not a many-index quantity, but an\nelement of a group. When the surface becomes small and its surface holonomy\n(group element) asymptotically becomes exp(i int B), where B is a 2-form\ntaking values in the algebra of the group and the integral is over the small\nsurface, then we can regard B as a 2-form analogue of the 1-form connection\nA. When you are talking about 2-form gauge theory in which space do these\n2-forms take values and how are they related to the above four-index\nquantities?\n\n\n> For the 2x2 square, also directed to the south-east,\n>\n> V_3 V_4\n> ___k____ ___l___\n> | | |\n> V_1 i | A | B | m\n> | | |\n> |________|_______|\n> | | |\n> V_2 j | C | D | n\n> | | |\n> |________|_______|\n> p q\n>\n> the holonomy becomes\n>\n> A_13 B_14 C_23 D_24 = ( A^rs_ik B^mt_rl C^up_js D^nq_tu ).\n\nSo now the holonomy in your sense is an eight-index quantity. The space in\nwhich a surface holonomy in your sense lives depends on the size of the\nsurface, right? This is another difference to the ordinary notion of\nholonomy.\n\nThe comparison with the 2-group concept of surface holonomy may illustrate\nwhere the conceptual difference lies:\n\nThe starting point is actually identical, up to notation: A 2-group element,\njust like the elementary 4-index quantities that you consider, has two\n"ingoing" and two "outgoing" interfaces. What you call j is the source edge\nof a 2-group element (rather of an element of the groupoid of bigons, but\nI\'ll just sketch the rough idea), l is the target edge, i is the source\n"vertex" (which can be thought of as an edge when due care is taken) and k\nthe target "vertex".\n\nThe difference begins when composition is considered. In your approach the\nhorizontal or vertical composition of two the 4-index quantities associated\nwith an elementary plaquette yields a 6-index object with 3 in- and 3\nout-interfaces.\n\nIt is not only the size of a surface which determines the space (End(V....))\nin which it surface holonomy lives in your approach, but also its shape,\neven its orientation. The surface holonomy in your sense of two vertically\ncomposed plaquettes is a different kind of object than the surface holonomy\nof two horizontally composed plaquettes. The index structure keeps track of\nhow the larger surface is composed from smaller ones.\n\nThe concept of 2-group holonomy however follows the standard notion of\nholonomy in that surfaces of different size, shape and orientation are all\nassigned elements from the same space. This is precisely what makes the\nconsistency condition appear, which you don\'t see in your framework.\n\nNamely if we demand that the composition of two plaquettes labeled by\n4-index quantities is again a surface labeled by a 4-index quantity (as\napposed to a 6-index quantity) and if you then demand that the order of\nhorizontal and vertical compositions must be irrelevant (since surface\nholonomy must only depend on the surface, not on some way it is obtained by\ngluing smaller surfaces) then this imposes the "exchange law" on the\nhorizontal and vertical product operation. And this is where all these\nconstraints come from.\n\n\n> It is unwieldy to write down the holonomy for the cube in explicit\n[...]\n> Contracting the remaining links by tracing over V_1, V_2 and V_3 gives\n> us (a piece of) the action:\n>\n> S = tr_1 tr_2 tr_3 A_12 B_13 C_23 D_21 E_31 F_32.\n\n\nI assume you mean that this is the surface holonomy of the cube in your\nsense?\n\nI see that for closed surfaces your notion of surface holonomy is a 0-index\nobject and hence the same type of object no matter what the size shape and\norientation of the closed surface is.\n\nWhat I don\'t see yet is:\n\nIn which sense is this surface holonomy (non-abelian) group valued?\n\nSeems to me that since it is a contraction of elements of V with elements of\nV* it takes values in some field, like the real or complex numbers.\n\nIn which sense is the result independent of the various choices of\norientation on the lattice? All your computations assume preferred\ndirections (like the south-east convention that you mentioned).\n\nIn 2-group theory there is automatically a self-consistent notion (called\n"whiskering" by Girelli&Pfeiffer) of moving the "in- and out-interfaces"\n(mentioned at the beginning) around, so that not only the order of vertical\nand horizontal composition has no effect on the result but even the\nconvention which direction is horizontal and which is vertical (or anything\nin between) does not influence the result of a total surface holonomy\ncomputed using 2-group technology.\n\nIs anything similar true in the approach that you are following? Is there a\nconsistent way to turn a 4-index quantity describing south-east translation\nto one describing north-east translation? Something like this seems\nnecessary, because if I assign 4-index quantities with a preferred diagonal\ndirection to every plaquette of the lattice, elementary cubes will in\ngeneral not be labeled by 4-index quantities such that all in-interfaces\nmatch out-interfaces and vice versa.\n\nSo my question is: What is a "configuration" in your surface gauge theory?\nI.e. what objects on the lattice do you need to specify so that I can pick\nany composed surface in the lattice and compute its surface holonomy in your\nsense above? If there is no way to "rotate" in- and out interfaces, it seems\nthat a "configuration" of this lattice 2-gauge theory really requires\nspecifying four four-index quantities for each plaquettes.\n\n(To make this point clearer: Suppose I label the plaquettes in a 3\ndimensional lattice with your 4-index quantities in such a way that with\nrespect to some cartesian coordinate system all "interfaces" point in the\ndirection of increasing coordinate values, e.g. all go either "right" or\n"upwards" or "forward" and the preferred "south-east"-direction is the main\ndiagonal going right, up and forward. Then there is not a single elementary\ncube in the lattice which is labeled with 4-index quantities such that they\ncan be composed correctly.)\n\nDue to these issues I currently don\'t understand in which sense you claim to\ndeal with a 2-gauge theory or with non-abelian surface holonomy.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409090112.629c7cd4@posting.google.c om...
>
>
> "Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message
news:<413f2423$1@news.sentex.net>...
>
> > Consider a surface made up from four plaquettes each labeled by a group
> > element from some group H as in
> >
> > h1 h1'
> >
> > h2 h2' .
> >
> > What is its surface holonomy?

> Let us start with the simplest surface, namely a single plaquette.
> It is an element in End(V@V) = V* @ V* @ V @ V (V* = dual space).
> If we put indices as follows
>
> V_2
> __{_j____}
> | |
> V_1 i | A | k
> | |
> |__{______}|
> l
>
> then the surface holonomy is the four-index quantity A_{12} = (A^{kl_ij}).

The ordinary notion of surface holonomy is not a many-index quantity, but an
element of a group. When the surface becomes small and its surface holonomy
(group element) asymptotically becomes \exp(i \int B), where B is a 2-form
taking values in the algebra of the group and the integral is over the small
surface, then we can regard B as a 2-form analogue of the 1-form connection
A. When you are talking about 2-form gauge theory in which space do these
2-forms take values and how are they related to the above four-index
quantities?

> For the 2x2 square, also directed to the south-east,
>
> V_3 V_4
> __{_k____} __{_l___}
> | | |
> V_1 i | A | B | m
> | | |
> |__{______}|__{_____}|
> | | |
> V_2 j | C | D | n
> | | |
> |__{______}|__{_____}|
> p q
>
> the holonomy becomes
>
> A_{13} B_{14} C_{23} D_{24} = ( A^{rs_ik} B^{mt_rl} C^{up_js} D^{nq_tu} ).

So now the holonomy in your sense is an eight-index quantity. The space in
which a surface holonomy in your sense lives depends on the size of the
surface, right? This is another difference to the ordinary notion of
holonomy.

The comparison with the 2-group concept of surface holonomy may illustrate
where the conceptual difference lies:

The starting point is actually identical, up to notation: A 2-group element,
just like the elementary 4-index quantities that you consider, has two
"ingoing" and two "outgoing" interfaces. What you call j is the source edge
of a 2-group element (rather of an element of the groupoid of bigons, but
I'll just sketch the rough idea), l is the target edge, i is the source
"vertex" (which can be thought of as an edge when due care is taken) and k
the target "vertex".

The difference begins when composition is considered. In your approach the
horizontal or vertical composition of two the 4-index quantities associated
with an elementary plaquette yields a 6-index object with 3 in- and 3
out-interfaces.

It is not only the size of a surface which determines the space (End(V....))
in which it surface holonomy lives in your approach, but also its shape,
even its orientation. The surface holonomy in your sense of two vertically
composed plaquettes is a different kind of object than the surface holonomy
of two horizontally composed plaquettes. The index structure keeps track of
how the larger surface is composed from smaller ones.

The concept of 2-group holonomy however follows the standard notion of
holonomy in that surfaces of different size, shape and orientation are all
assigned elements from the same space. This is precisely what makes the
consistency condition appear, which you don't see in your framework.

Namely if we demand that the composition of two plaquettes labeled by
4-index quantities is again a surface labeled by a 4-index quantity (as
apposed to a 6-index quantity) and if you then demand that the order of
horizontal and vertical compositions must be irrelevant (since surface
holonomy must only depend on the surface, not on some way it is obtained by
gluing smaller surfaces) then this imposes the "exchange law" on the
horizontal and vertical product operation. And this is where all these
constraints come from.

> It is unwieldy to write down the holonomy for the cube in explicit
[...]
> Contracting the remaining links by tracing over V_1, V_2 and V_3 gives
> us (a piece of) the action:
>
> S = tr_1 tr_2 tr_3 A_{12} B_{13} C_{23} D_{21} E_{31} F_{32}.

I assume you mean that this is the surface holonomy of the cube in your
sense?

I see that for closed surfaces your notion of surface holonomy is a 0-index
object and hence the same type of object no matter what the size shape and
orientation of the closed surface is.

What I don't see yet is:

In which sense is this surface holonomy (non-abelian) group valued?

Seems to me that since it is a contraction of elements of V with elements of
V* it takes values in some field, like the real or complex numbers.

In which sense is the result independent of the various choices of
orientation on the lattice? All your computations assume preferred
directions (like the south-east convention that you mentioned).

In 2-group theory there is automatically a self-consistent notion (called
"whiskering" by Girelli&Pfeiffer) of moving the "in- and out-interfaces"
(mentioned at the beginning) around, so that not only the order of vertical
and horizontal composition has no effect on the result but even the
convention which direction is horizontal and which is vertical (or anything
in between) does not influence the result of a total surface holonomy
computed using 2-group technology.

Is anything similar true in the approach that you are following? Is there a
consistent way to turn a 4-index quantity describing south-east translation
to one describing north-east translation? Something like this seems
necessary, because if I assign 4-index quantities with a preferred diagonal
direction to every plaquette of the lattice, elementary cubes will in
general not be labeled by 4-index quantities such that all in-interfaces
match out-interfaces and vice versa.

So my question is: What is a "configuration" in your surface gauge theory?
I.e. what objects on the lattice do you need to specify so that I can pick
any composed surface in the lattice and compute its surface holonomy in your
sense above? If there is no way to "rotate" in- and out interfaces, it seems
that a "configuration" of this lattice 2-gauge theory really requires
specifying four four-index quantities for each plaquettes.

(To make this point clearer: Suppose I label the plaquettes in a 3
dimensional lattice with your 4-index quantities in such a way that with
respect to some cartesian coordinate system all "interfaces" point in the
direction of increasing coordinate values, e.g. all go either "right" or
"upwards" or "forward" and the preferred "south-east"-direction is the main
diagonal going right, up and forward. Then there is not a single elementary
cube in the lattice which is labeled with 4-index quantities such that they
can be composed correctly.)

Due to these issues I currently don't understand in which sense you claim to
deal with a 2-gauge theory or with non-abelian surface holonomy.

Thomas Larsson

Sep9-04, 01:59 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<414033cc\\$1@news.sentex.net>...\n\n> The ordinary notion of surface holonomy is not a many-index quantity, but an\n> element of a group. When the surface becomes small and its surface holonomy\n> (group element) asymptotically becomes exp(i int B), where B is a 2-form\n> taking values in the algebra of the group and the integral is over the small\n> surface, then we can regard B as a 2-form analogue of the 1-form connection\n> A. When you are talking about 2-form gauge theory in which space do these\n> 2-forms take values and how are they related to the above four-index\n> quantities?\n\nLet me quote what I said in an earlier post:\n\n> > F = dA + [s.A, A].\n> > I admit that this is a bit formal, e.g. I have not\n> > specified in which spaces the A\'s are valued.\n\nI claim to have constructed, more or less rigorously, a non-abelian\np-form generalization of gauge theory on the lattice. The classical\ncontinuum limit is shaky, and I have not stated otherwise. Nevertheless,\nhere is what I think goes on.\n\nRecall that the zero-curvature condition on the lattice is the\nQYBE (quantum Yang-Baxter equation). So what does F=0 mean? For\nsimplicity, let A be space-independent, A = A(s). Then dA = 0.\nMoreover, let s = (1,1,1). The 1,2,3 component of F becomes\n\nF_123 ~ [A_12, A_13] + [A_12, A_23] + [A_13, A_23] = 0.\n\nThis looks exactly like the CYBE (classical Yang-Baxter equation), which\nis precisely what we want. The CYBE is an equation in V@V@V, but A_ij\nis non-trivial in V_i and V_j. F_123 is non-trivial in all three spaces.\n\nThis strongly suggests that A and F are valued in V^@n, the n:th tensor\npower of V, where n is the dimension of the lattice. Moreover, A_ij\nacts as the unit matrix on each V except V_i and V_j, and F_ijk acts\nnon-trivially on V_i, V_j and V_k. This definition evidently has cubic\nsymmetry, but I am worried about rotation symmetry.\n\nMaybe it\'s there, maybe not. However, we are ultimately interested in the\nquantum theory anyway, which can be defined by a second-order phase\ntransition of the lattice model, if such a transition exists. This does\nnot really depend upon the existence of a classical continuum limit.\n\n>\n>\n> > For the 2x2 square, also directed to the south-east,\n> >\n> > V_3 V_4\n> > ___k____ ___l___\n> > | | |\n> > V_1 i | A | B | m\n> > | | |\n> > |________|_______|\n> > | | |\n> > V_2 j | C | D | n\n> > | | |\n> > |________|_______|\n> > p q\n> >\n> > the holonomy becomes\n> >\n> > A_13 B_14 C_23 D_24 = ( A^rs_ik B^mt_rl C^up_js D^nq_tu ).\n>\n> So now the holonomy in your sense is an eight-index quantity. The space in\n> which a surface holonomy in your sense lives depends on the size of the\n> surface, right? This is another difference to the ordinary notion of\n> holonomy.\n\nThis is what I have said all along. From a previous post: "This is\ndifferent from YM on loop space, since I need a new vector space index\nfor each point on the loop."\n\n>\n> The comparison with the 2-group concept of surface holonomy may illustrate\n> where the conceptual difference lies:\n>\n> The starting point is actually identical, up to notation: A 2-group element,\n> just like the elementary 4-index quantities that you consider, has two\n> "ingoing" and two "outgoing" interfaces. What you call j is the source edge\n> of a 2-group element (rather of an element of the groupoid of bigons, but\n> I\'ll just sketch the rough idea), l is the target edge, i is the source\n> "vertex" (which can be thought of as an edge when due care is taken) and k\n> the target "vertex".\n>\n> The difference begins when composition is considered. In your approach the\n> horizontal or vertical composition of two the 4-index quantities associated\n> with an elementary plaquette yields a 6-index object with 3 in- and 3\n> out-interfaces.\n\nYes. It\'s a feature, not a bug.\n\n>\n> It is not only the size of a surface which determines the space (End(V....))\n> in which it surface holonomy lives in your approach, but also its shape,\n> even its orientation. The surface holonomy in your sense of two vertically\n> composed plaquettes is a different kind of object than the surface holonomy\n> of two horizontally composed plaquettes. The index structure keeps track of\n> how the larger surface is composed from smaller ones.\n>\n> The concept of 2-group holonomy however follows the standard notion of\n> holonomy in that surfaces of different size, shape and orientation are all\n> assigned elements from the same space. This is precisely what makes the\n> consistency condition appear, which you don\'t see in your framework.\n>\n> Namely if we demand that the composition of two plaquettes labeled by\n> 4-index quantities is again a surface labeled by a 4-index quantity (as\n> apposed to a 6-index quantity) and if you then demand that the order of\n> horizontal and vertical compositions must be irrelevant (since surface\n> holonomy must only depend on the surface, not on some way it is obtained by\n> gluing smaller surfaces) then this imposes the "exchange law" on the\n> horizontal and vertical product operation. And this is where all these\n> constraints come from.\n\nI completely agree.\n\n>\n> What I don\'t see yet is:\n>\n> In which sense is this surface holonomy (non-abelian) group valued?\n\nAll holonomies take values in the collection of spaces\n\nG = 1, End(V@V), End(V^@4), End(V^@6), ...\n\nOn a finite lattice, this is a finite collection since there are only\nfinitely many links. Composition of two surfaces stays within G, and each\nelement has an inverse, which describes parallel transport of a string\nacross the surface in the opposite direction. Thus G is a group.\n\n>\n> Seems to me that since it is a contraction of elements of V with elements of\n> V* it takes values in some field, like the real or complex numbers.\n\nIf the surface has a boundary, there are unmatched V\'s and V*\'s.\n\n>\n> In which sense is the result independent of the various choices of\n> orientation on the lattice? All your computations assume preferred\n> directions (like the south-east convention that you mentioned).\n\nYes, and this was my mistake in 1990 which I corrected in my eprint 2002.\nIn fact, I realized the mistake very soon after I published the first\npaper, but never got around writing it up before the discussion on gerbes\nhere on spr.\n\nTwo independent variables must be assigned to each plaquette. Actually,\none assign four variables, valued in V@V@V*@V*, V@V*@V*@V, V*@V*@V@V and\nV*@V@V@V*. The first and third are each other\'s inverses, as are the\nsecond and the fourth, but there is no obvious relation between the two\npairs. IOW, north-west and south-east are inverses, north-east and\nsouth-west are inverses, but north-west and north-east are independent.\nWhen calculating a Wilson surface, one must pick the variable on each\nplaquette so that each internal edge has one V and one V*.\n\nThere are no variables in V@V*@V@V* or V*@V@V*@V (assuming edges are\nlabelled around the plaquette). The boundary of any Wilson surface must be\na string of n V\'s followed by n V*\'s (n even), because otherwise it\nwon\'t be possible to have exactly one V and one V* on each internal edge.\nMore general models are conceivable, which allow for different numbers\nof V and V*, but there seems to be little point in writing them up\nexplicitly. One can introduce additional plaquette variables living in\nV@V@V@V*, V*@V*@V*@V, etc.\n\n>\n> In 2-group theory there is automatically a self-consistent notion (called\n> "whiskering" by Girelli&Pfeiffer) of moving the "in- and out-interfaces"\n> (mentioned at the beginning) around, so that not only the order of vertical\n> and horizontal composition has no effect on the result but even the\n> convention which direction is horizontal and which is vertical (or anything\n> in between) does not influence the result of a total surface holonomy\n> computed using 2-group technology.\n>\n> Is anything similar true in the approach that you are following? Is there a\n> consistent way to turn a 4-index quantity describing south-east translation\n> to one describing north-east translation?\n\nNo. My approach is essentially different from 2-groups.\n\n> Something like this seems\n> necessary, because if I assign 4-index quantities with a preferred diagonal\n> direction to every plaquette of the lattice, elementary cubes will in\n> general not be labeled by 4-index quantities such that all in-interfaces\n> match out-interfaces and vice versa.\n\nCubic symmetry is restored by having two independent variables for\neach plaquette, and eight terms (four pairwise inverses) in the action.\n\nRecall that in 1-LGT, the action is usually written S = tr UUUU + hc,\nwhere hc is the hermitean conjugate (or more generally inverse) of\ntr UUUU. The two terms correspond to parallel transport of a particle\naround the plaquette in the clockwise and counterclockwise direction.\nSimilarly, each of the eight terms in the 2-LGT action transports a\nstring piece around a diagonal in the cube. Figure 3 in my eprint\nillustrates that. Since the cube has eight directed diagonals, the\naction consists of eight terms. Cubic symmetry is restored.\n\n>\n> So my question is: What is a "configuration" in your surface gauge theory?\n> I.e. what objects on the lattice do you need to specify so that I can pick\n> any composed surface in the lattice and compute its surface holonomy in your\n> sense above? If there is no way to "rotate" in- and out interfaces, it seems\n> that a "configuration" of this lattice 2-gauge theory really requires\n> specifying four four-index quantities for each plaquettes.\n>\n> (To make this point clearer: Suppose I label the plaquettes in a 3\n> dimensional lattice with your 4-index quantities in such a way that with\n> respect to some cartesian coordinate system all "interfaces" point in the\n> direction of increasing coordinate values, e.g. all go either "right" or\n> "upwards" or "forward" and the preferred "south-east"-direction is the main\n> diagonal going right, up and forward. Then there is not a single elementary\n> cube in the lattice which is labeled with 4-index quantities such that they\n> can be composed correctly.)\n>\n> Due to these issues I currently don\'t understand in which sense you claim to\n> deal with a 2-gauge theory or with non-abelian surface holonomy.\n\nAs you have figured out by now, the generalization to arbitrary p-form\ngauge theory on the lattice is straightforward. In this family of\ntheories indexed by p, p = 1 is ordinary non-abelian lattice gauge\ntheory, with the ordinary holonomy. Moreover, in the abelian case, where\nV = V* = R or C, we recover p-form electrodynamics on the lattice\n(actually several copies, since there still are several variables on each\nplaquette), again with the ordinary p-holonomy. Any family of theories\nwhich contains the right p=1 and abelian members is a viable version of\np-form lattice gauge theory, isn\'t it?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<414033cc$1@news.sentex.net>...

> The ordinary notion of surface holonomy is not a many-index quantity, but an
> element of a group. When the surface becomes small and its surface holonomy
> (group element) asymptotically becomes \exp(i \int B), where B is a 2-form
> taking values in the algebra of the group and the integral is over the small
> surface, then we can regard B as a 2-form analogue of the 1-form connection
> A. When you are talking about 2-form gauge theory in which space do these
> 2-forms take values and how are they related to the above four-index
> quantities?

Let me quote what I said in an earlier post:

> > F = dA + [s.A, A].
> > I admit that this is a bit formal, e.g. I have not
> > specified in which spaces the A's are valued.

I claim to have constructed, more or less rigorously, a non-abelian
p-form generalization of gauge theory on the lattice. The classical
continuum limit is shaky, and I have not stated otherwise. Nevertheless,
here is what I think goes on.

Recall that the zero-curvature condition on the lattice is the
QYBE (quantum Yang-Baxter equation). So what does F=0 mean? For
simplicity, let A be space-independent, A = A(s). Then dA = .
Moreover, let s = (1,1,1). The 1,2,3 component of F becomes

F_{123} ~ [A_{12}, A_{13}] + [A_{12}, A_{23}] + [A_{13}, A_{23}] = .

This looks exactly like the CYBE (classical Yang-Baxter equation), which
is precisely what we want. The CYBE is an equation in V@V@V, but A_{ij}
is non-trivial in V_i and V_j. F_{123} is non-trivial in all three spaces.

This strongly suggests that A and F are valued in V^@n, the n:th tensor
power of V, where n is the dimension of the lattice. Moreover, A_{ij}
acts as the unit matrix on each V except V_i and V_j, and F_{ijk} acts
non-trivially on V_i, V_j and V_k. This definition evidently has cubic
symmetry, but I am worried about rotation symmetry.

Maybe it's there, maybe not. However, we are ultimately interested in the
quantum theory anyway, which can be defined by a second-order phase
transition of the lattice model, if such a transition exists. This does
not really depend upon the existence of a classical continuum limit.

>
>
> > For the 2x2 square, also directed to the south-east,
> >
> > V_3 V_4
> > __{_k____} __{_l___}
> > | | |
> > V_1 i | A | B | m
> > | | |
> > |__{______}|__{_____}|
> > | | |
> > V_2 j | C | D | n
> > | | |
> > |__{______}|__{_____}|
> > p q
> >
> > the holonomy becomes
> >
> > A_{13} B_{14} C_{23} D_{24} = ( A^{rs_ik} B^{mt_rl} C^{up_js} D^{nq_tu} ).
>
> So now the holonomy in your sense is an eight-index quantity. The space in
> which a surface holonomy in your sense lives depends on the size of the
> surface, right? This is another difference to the ordinary notion of
> holonomy.

This is what I have said all along. From a previous post: "This is
different from YM on loop space, since I need a new vector space index
for each point on the loop."

>
> The comparison with the 2-group concept of surface holonomy may illustrate
> where the conceptual difference lies:
>
> The starting point is actually identical, up to notation: A 2-group element,
> just like the elementary 4-index quantities that you consider, has two
> "ingoing" and two "outgoing" interfaces. What you call j is the source edge
> of a 2-group element (rather of an element of the groupoid of bigons, but
> I'll just sketch the rough idea), l is the target edge, i is the source
> "vertex" (which can be thought of as an edge when due care is taken) and k
> the target "vertex".
>
> The difference begins when composition is considered. In your approach the
> horizontal or vertical composition of two the 4-index quantities associated
> with an elementary plaquette yields a 6-index object with 3 in- and 3
> out-interfaces.

Yes. It's a feature, not a bug.

>
> It is not only the size of a surface which determines the space (End(V....))
> in which it surface holonomy lives in your approach, but also its shape,
> even its orientation. The surface holonomy in your sense of two vertically
> composed plaquettes is a different kind of object than the surface holonomy
> of two horizontally composed plaquettes. The index structure keeps track of
> how the larger surface is composed from smaller ones.
>
> The concept of 2-group holonomy however follows the standard notion of
> holonomy in that surfaces of different size, shape and orientation are all
> assigned elements from the same space. This is precisely what makes the
> consistency condition appear, which you don't see in your framework.
>
> Namely if we demand that the composition of two plaquettes labeled by
> 4-index quantities is again a surface labeled by a 4-index quantity (as
> apposed to a 6-index quantity) and if you then demand that the order of
> horizontal and vertical compositions must be irrelevant (since surface
> holonomy must only depend on the surface, not on some way it is obtained by
> gluing smaller surfaces) then this imposes the "exchange law" on the
> horizontal and vertical product operation. And this is where all these
> constraints come from.

I completely agree.

>
> What I don't see yet is:
>
> In which sense is this surface holonomy (non-abelian) group valued?

All holonomies take values in the collection of spaces

G = 1, End(V@V), End(V^@4), End(V^@6), ...

On a finite lattice, this is a finite collection since there are only
finitely many links. Composition of two surfaces stays within G, and each
element has an inverse, which describes parallel transport of a string
across the surface in the opposite direction. Thus G is a group.

>
> Seems to me that since it is a contraction of elements of V with elements of
> V* it takes values in some field, like the real or complex numbers.

If the surface has a boundary, there are unmatched V's and V*'s.

>
> In which sense is the result independent of the various choices of
> orientation on the lattice? All your computations assume preferred
> directions (like the south-east convention that you mentioned).

Yes, and this was my mistake in 1990 which I corrected in my eprint 2002.
In fact, I realized the mistake very soon after I published the first
paper, but never got around writing it up before the discussion on gerbes
here on spr.

Two independent variables must be assigned to each plaquette. Actually,
one assign four variables, valued in V@V@V*@V*, V@V*@V*@V, V*@V*@V@V and
V*@V@V@V*. The first and third are each other's inverses, as are the
second and the fourth, but there is no obvious relation between the two
pairs. IOW, north-west and south-east are inverses, north-east and
south-west are inverses, but north-west and north-east are independent.
When calculating a Wilson surface, one must pick the variable on each
plaquette so that each internal edge has one V and one V*.

There are no variables in V@V*@V@V* or V*@V@V*@V (assuming edges are
labelled around the plaquette). The boundary of any Wilson surface must be
a string of n V's followed by n V*'s (n even), because otherwise it
won't be possible to have exactly one V and one V* on each internal edge.
More general models are conceivable, which allow for different numbers
of V and V*, but there seems to be little point in writing them up
explicitly. One can introduce additional plaquette variables living in
V@V@V@V*, V*@V*@V*@V, etc.

>
> In 2-group theory there is automatically a self-consistent notion (called
> "whiskering" by Girelli&Pfeiffer) of moving the "in- and out-interfaces"
> (mentioned at the beginning) around, so that not only the order of vertical
> and horizontal composition has no effect on the result but even the
> convention which direction is horizontal and which is vertical (or anything
> in between) does not influence the result of a total surface holonomy
> computed using 2-group technology.
>
> Is anything similar true in the approach that you are following? Is there a
> consistent way to turn a 4-index quantity describing south-east translation
> to one describing north-east translation?

No. My approach is essentially different from 2-groups.

> Something like this seems
> necessary, because if I assign 4-index quantities with a preferred diagonal
> direction to every plaquette of the lattice, elementary cubes will in
> general not be labeled by 4-index quantities such that all in-interfaces
> match out-interfaces and vice versa.

Cubic symmetry is restored by having two independent variables for
each plaquette, and eight terms (four pairwise inverses) in the action.

Recall that in 1-LGT, the action is usually written S = tr UUUU + hc,
where hc is the hermitean conjugate (or more generally inverse) of
tr UUUU. The two terms correspond to parallel transport of a particle
around the plaquette in the clockwise and counterclockwise direction.
Similarly, each of the eight terms in the 2-LGT action transports a
string piece around a diagonal in the cube. Figure 3 in my eprint
illustrates that. Since the cube has eight directed diagonals, the
action consists of eight terms. Cubic symmetry is restored.

>
> So my question is: What is a "configuration" in your surface gauge theory?
> I.e. what objects on the lattice do you need to specify so that I can pick
> any composed surface in the lattice and compute its surface holonomy in your
> sense above? If there is no way to "rotate" in- and out interfaces, it seems
> that a "configuration" of this lattice 2-gauge theory really requires
> specifying four four-index quantities for each plaquettes.
>
> (To make this point clearer: Suppose I label the plaquettes in a 3
> dimensional lattice with your 4-index quantities in such a way that with
> respect to some cartesian coordinate system all "interfaces" point in the
> direction of increasing coordinate values, e.g. all go either "right" or
> "upwards" or "forward" and the preferred "south-east"-direction is the main
> diagonal going right, up and forward. Then there is not a single elementary
> cube in the lattice which is labeled with 4-index quantities such that they
> can be composed correctly.)
>
> Due to these issues I currently don't understand in which sense you claim to
> deal with a 2-gauge theory or with non-abelian surface holonomy.

As you have figured out by now, the generalization to arbitrary p-form
gauge theory on the lattice is straightforward. In this family of
theories indexed by p, p = 1 is ordinary non-abelian lattice gauge
theory, with the ordinary holonomy. Moreover, in the abelian case, where
V = V* = R or C, we recover p-form electrodynamics on the lattice
(actually several copies, since there still are several variables on each
plaquette), again with the ordinary p-holonomy. Any family of theories
which contains the right p=1 and abelian members is a viable version of
p-form lattice gauge theory, isn't it?

Urs Schreiber

Sep9-04, 02:57 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409090946.7a0d6f3@post ing.google.com...\n\n> > In which sense is this surface holonomy (non-abelian) group valued?\n>\n> All holonomies take values in the collection of spaces\n>\n> G = 1, End(V@V), End(V^@4), End(V^@6), ...\n>\n> On a finite lattice, this is a finite collection since there are only\n> finitely many links. Composition of two surfaces stays within G, and each\n> element has an inverse, which describes parallel transport of a string\n> across the surface in the opposite direction. Thus G is a group.\n\n\nLooks like a groupoid to me. You can only compose elements that sit in the\nsame\nEnd(V^@n) space.\n\nYou could argue to regard all End(V^@n) as elements of End(V^@infty).\n\n2-gauge theory with End(V^@infty) as the gauge group and a restriction on\nwhich group element can be assigned to which surface?\n\n\n> > Seems to me that since it is a contraction of elements of V with\nelements of\n> > V* it takes values in some field, like the real or complex numbers.\n>\n> If the surface has a boundary, there are unmatched V\'s and V*\'s.\n\n\nWilson surfaces (closed surfaces) in this approach have holonomy in C\\{0}.\nMore generally, to any kind of surface in this approach there is a\nconstraint saying which elements from the gauge group End(V^@infty) it can\nhave as holonomy.\n\n\n> > In which sense is the result independent of the various choices of\n> > orientation on the lattice? All your computations assume preferred\n> > directions (like the south-east convention that you mentioned).\n>\n> Yes, and this was my mistake in 1990 which I corrected in my eprint 2002.\n> In fact, I realized the mistake very soon after I published the first\n> paper, but never got around writing it up before the discussion on gerbes\n> here on spr.\n>\n> Two independent variables must be assigned to each plaquette. Actually,\n> one assign four variables, valued in V@V@V*@V*, V@V*@V*@V, V*@V*@V@V and\n> V*@V@V@V*. The first and third are each other\'s inverses, as are the\n> second and the fourth, but there is no obvious relation between the two\n> pairs. IOW, north-west and south-east are inverses, north-east and\n> south-west are inverses, but north-west and north-east are independent.\n> When calculating a Wilson surface, one must pick the variable on each\n> plaquette so that each internal edge has one V and one V*.\n\n\nBut there are in general several choices. How do you get a unique surface\nholonomy?\n\nWhat is the surface holonomy of the single plaquette labeled with\n\nA^kl_ij and A^li_kj\n\n?\n\nWhat is the result of composing this at the edge j with a single plaquette\nlabeled by\n\nB^jr_st and B^js_rt\n\n?\n\n\n> > Is anything similar true in the approach that you are following? Is\nthere a\n> > consistent way to turn a 4-index quantity describing south-east\ntranslation\n> > to one describing north-east translation?\n>\n> No. My approach is essentially different from 2-groups.\n\n\nBut there should be a consistent notion of surface holonomy, no?\n\n\n> Cubic symmetry is restored by having two independent variables for\n> each plaquette, and eight terms (four pairwise inverses) in the action.\n\n\nFor closed surfaces you can sum over all ways of using one of the two\nelements on the elementary plaquettes. Each way of doing so gives you a\nholonomy in\n\nC\\{0} < End(V^@infty) .\n\nYou can use the additive structure of R to add these up and get an element\nof C (the complex numbers). This number is well defined for a given labeling\nof a given closed surface. But it is not a holonomy (no longer an invertible\nelement of End(V^@infty) because 0 may occur.)\n\nThis brings me back to a question I asked before: Do you have a notion of\nunique surface holonomy in your approach?\n\n\n> In this family of\n> theories indexed by p, p = 1 is ordinary non-abelian lattice gauge\n> theory, with the ordinary holonomy. Moreover, in the abelian case, where\n> V = V* = R or C, we recover p-form electrodynamics on the lattice\n> (actually several copies, since there still are several variables on each\n> plaquette), again with the ordinary p-holonomy. Any family of theories\n> which contains the right p=1 and abelian members is a viable version of\n> p-form lattice gauge theory, isn\'t it?\n\n\nYou\'d usually want the generalizations to keep some properties of the theory\nbeing generalized. Which properties to keep is of course a matter of taste\nand/or of intended application.\n\nAs far as I understand your generalization does not keep the following\nproperties of 1-form gauge theory:\n\n- free choice of gauge group\n\n- configurations in which all elements from the gauge group may be\nassigned to all objects (paths, surfaces, ...)\n\n- unique holonomy .\n\nDo you agree with that assessment? If not, please bear with me and tell me\nin which sense these properties are preserved in your approach.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409090946.7a0d6f3@posting.google.co m...

> > In which sense is this surface holonomy (non-abelian) group valued?
>
> All holonomies take values in the collection of spaces
>
> G = 1, End(V@V), End(V^@4), End(V^@6), ...
>
> On a finite lattice, this is a finite collection since there are only
> finitely many links. Composition of two surfaces stays within G, and each
> element has an inverse, which describes parallel transport of a string
> across the surface in the opposite direction. Thus G is a group.

Looks like a groupoid to me. You can only compose elements that sit in the
same
End(V^@n) space.

You could argue to regard all End(V^@n) as elements of End(V^@\infty).

2-gauge theory with End(V^@\infty) as the gauge group and a restriction on
which group element can be assigned to which surface?

> > Seems to me that since it is a contraction of elements of V with
elements of
> > V* it takes values in some field, like the real or complex numbers.
>
> If the surface has a boundary, there are unmatched V's and V*'s.

Wilson surfaces (closed surfaces) in this approach have holonomy in C\{0}.
More generally, to any kind of surface in this approach there is a
constraint saying which elements from the gauge group End(V^@\infty) it can
have as holonomy.

> > In which sense is the result independent of the various choices of
> > orientation on the lattice? All your computations assume preferred
> > directions (like the south-east convention that you mentioned).
>
> Yes, and this was my mistake in 1990 which I corrected in my eprint 2002.
> In fact, I realized the mistake very soon after I published the first
> paper, but never got around writing it up before the discussion on gerbes
> here on spr.
>
> Two independent variables must be assigned to each plaquette. Actually,
> one assign four variables, valued in V@V@V*@V*, V@V*@V*@V, V*@V*@V@V and
> V*@V@V@V*. The first and third are each other's inverses, as are the
> second and the fourth, but there is no obvious relation between the two
> pairs. IOW, north-west and south-east are inverses, north-east and
> south-west are inverses, but north-west and north-east are independent.
> When calculating a Wilson surface, one must pick the variable on each
> plaquette so that each internal edge has one V and one V*.

But there are in general several choices. How do you get a unique surface
holonomy?

What is the surface holonomy of the single plaquette labeled with

A^{kl_ij}[/itex] and A^{li_kj}

?

What is the result of composing this at the edge j with a single plaquette
labeled by

B^{jr_st} and B^{js_rt}

?

> > Is anything similar true in the approach that you are following? Is
there a
> > consistent way to turn a 4-index quantity describing south-east
translation
> > to one describing north-east translation?
>
> No. My approach is essentially different from 2-groups.

But there should be a consistent notion of surface holonomy, no?

> Cubic symmetry is restored by having two independent variables for
> each plaquette, and eight terms (four pairwise inverses) in the action.

For closed surfaces you can sum over all ways of using one of the two
elements on the elementary plaquettes. Each way of doing so gives you a
holonomy in

C\{0} < End(V^@\infty) .

You can use the additive structure of R to add these up and get an element
of C (the complex numbers). This number is well defined for a given labeling
of a given closed surface. But it is not a holonomy (no longer an invertible
element of End(V^@\infty) because may occur.)

This brings me back to a question I asked before: Do you have a notion of
unique surface holonomy in your approach?

> In this family of
> theories indexed by p, [itex]p = 1 is ordinary non-abelian lattice gauge
> theory, with the ordinary holonomy. Moreover, in the abelian case, where
> V = V* = R or C, we recover p-form electrodynamics on the lattice
> (actually several copies, since there still are several variables on each
> plaquette), again with the ordinary p-holonomy. Any family of theories
> which contains the right p=1 and abelian members is a viable version of
> p-form lattice gauge theory, isn't it?

You'd usually want the generalizations to keep some properties of the theory
being generalized. Which properties to keep is of course a matter of taste
and/or of intended application.

As far as I understand your generalization does not keep the following
properties of 1-form gauge theory:

- free choice of gauge group

- configurations in which all elements from the gauge group may be
assigned to all objects (paths, surfaces, ...)

- unique holonomy .

Do you agree with that assessment? If not, please bear with me and tell me
in which sense these properties are preserved in your approach.

Thomas Larsson

Sep9-04, 03:56 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413a4323\\$1@news.sentex.net>...\n> How to assign an element of some group to a given surface in a way that is\n> well defined (i.e. depends only on the surface, not on any\n> parameterizations).\n>\n> Does your approach address this problem?\n>\n> Can you assign group elements in a unique way, or do you rather assign\n> elements of End(V @ V @ .... @V) to a surface?\n>\n> Is even that assignment unique?\n>\n> Doesn\'t it depend on the latticization of the surface?\n\n2-multiplication does not narrowly refer to the construction of a\ncontinuum 2-gauge theory. In fact, matrix multiplication is more\nnaturally defined on a lattice than in the continuum. In the\nproduct ABC, the matrices A, B and C can be thought of as links\non a lattice, but the formalism has many other applications as\nwell. The continuum limit converts an infinite string of matrices\ninto a path-ordered integral,which really isn\'t such a natural\nconcept.\n\nNevertheless, for a closed Wilson surface, the result does not\ndepend on latticization any more (or less) than for a closed\nWilson line. If the surface has a boundary, the situation is less\nclear. One has to define a continuum analogue of an n-fold tensor\nproduct in the limit that n -> infinity. To what extent that can\nbe done rigorously (at least as rigorously as the definition of a\npath-ordered integral) is unclear to me.\n\nBut this may be a price we have to pay. The alternative idea,\nto force each boundary to take values in the same space, comes\nwith its own problems. In particular, 2-associativity seems very\nvery shaky in the 2-group approach. At best it can be saved using\nsome geometrically obscure consistency conditions, but it seems\nquite likely that these conditions don\'t have any interesting\nsolutions at all. And even if that is not the case, p-associativity\nfor p > 2 is likely to be even more restrictive; problems are\nlikely to be monotonous in p and for p = 1 there are none.\n\nIn contrast, if you do things my way, p-associativity for any p\nis automatic, which to me seems to be a key property. The Yang-\nBaxter product also has other applications, notably the Yang-Baxter\nequation, which is much more closely related to real physics than\nsome would-be 2-gauge theory.\n\nAnd even for applications to continuum gauge theory in loop\nspace, you may need to distribute vector spaces evenly along\nloops. The reason is continuity under deformations of loops.\nAssume that every simple loop takes values in the same group G.\nThen in what group is a double loop, i.e. a loop which consists\nof two disjoint pieces, valued? G @ G? But what happens if we\nstart from a simple loop, and deform it by pinching it on the\nmiddle. When the waist is tighted, there is a limit point where\nthe loop changes from being simple to being double. Does the\nrelevant group change discontinuously at that point?\n\nIf instead the loop is valued in V^@L (L = length of loop),\neverything changes smoothly and is well-defined on the lattice.\n\nYou may excuse an old man, who successfully constructed a\nwell-defined p-form generalization of non-abelian lattice gauge\ntheory in 1989 (modulo a minor flaw which was later corrected),\nif he is less than impressed by people who try to do the same\nthing fifteen years later. Especially when they run into serious\ntrouble with 2-associativity.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<413a4323$1@news.sentex.net>...
> How to assign an element of some group to a given surface in a way that is
> well defined (i.e. depends only on the surface, not on any
> parameterizations).
>
> Does your approach address this problem?
>
> Can you assign group elements in a unique way, or do you rather assign
> elements of End(V @ V @ .... @V) to a surface?
>
> Is even that assignment unique?
>
> Doesn't it depend on the latticization of the surface?

2-multiplication does not narrowly refer to the construction of a
continuum 2-gauge theory. In fact, matrix multiplication is more
naturally defined on a lattice than in the continuum. In the
product ABC, the matrices A, B and C can be thought of as links
on a lattice, but the formalism has many other applications as
well. The continuum limit converts an infinite string of matrices
into a path-ordered integral,which really isn't such a natural
concept.

Nevertheless, for a closed Wilson surface, the result does not
depend on latticization any more (or less) than for a closed
Wilson line. If the surface has a boundary, the situation is less
clear. One has to define a continuum analogue of an n-fold tensor
product in the limit that n -> infinity. To what extent that can
be done rigorously (at least as rigorously as the definition of a
path-ordered integral) is unclear to me.

But this may be a price we have to pay. The alternative idea,
to force each boundary to take values in the same space, comes
with its own problems. In particular, 2-associativity seems very
very shaky in the 2-group approach. At best it can be saved using
some geometrically obscure consistency conditions, but it seems
quite likely that these conditions don't have any interesting
solutions at all. And even if that is not the case, p-associativity
for p > 2 is likely to be even more restrictive; problems are
likely to be monotonous in p and for p = 1 there are none.

In contrast, if you do things my way, p-associativity for any p
is automatic, which to me seems to be a key property. The Yang-
Baxter product also has other applications, notably the Yang-Baxter
equation, which is much more closely related to real physics than
some would-be 2-gauge theory.

And even for applications to continuum gauge theory in loop
space, you may need to distribute vector spaces evenly along
loops. The reason is continuity under deformations of loops.
Assume that every simple loop takes values in the same group G.
Then in what group is a double loop, i.e. a loop which consists
of two disjoint pieces, valued? G @ G? But what happens if we
start from a simple loop, and deform it by pinching it on the
middle. When the waist is tighted, there is a limit point where
the loop changes from being simple to being double. Does the
relevant group change discontinuously at that point?

If instead the loop is valued in V^@L (L = length of loop),
everything changes smoothly and is well-defined on the lattice.

You may excuse an old man, who successfully constructed a
well-defined p-form generalization of non-abelian lattice gauge
theory in 1989 (modulo a minor flaw which was later corrected),
if he is less than impressed by people who try to do the same
thing fifteen years later. Especially when they run into serious
trouble with 2-associativity.

Thomas Larsson

Sep10-04, 04:40 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nthomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0409050917.5fd66be3@posting.google. com>...\n\na lot of stuff which have been better dealt with in later posts,\nwhich however appeared earlier on Google.\n\n\n[Moderator\'s note: With multiple moderators the order of appearance of posts may deviate\nfrom the order of their submission. If a submitted message which has not yet appeared\nis wished to be discarded a note should be sent to physics-research-request@ncar.ucar.edu,\nto which all administrative requests concerning all moderators are to be addressed.\nSee also http://math.ucr.edu/home/baez/spr.html -usc]\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0409050917.5fd66be3@posting.google.com>...

a lot of stuff which have been better dealt with in later posts,
which however appeared earlier on Google.

[Moderator's note: With multiple moderators the order of appearance of posts may deviate
from the order of their submission. If a submitted message which has not yet appeared
is wished to be discarded a note should be sent to physics-research-request@ncar.ucar.edu,
to which all administrative requests concerning all moderators are to be addressed.
See also http://math.ucr.edu/home/baez/spr.html -usc]

Hendryk Pfeiffer

Sep12-04, 03:22 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi,\n\nThomas Larsson wrote:\n\n> One thing must be perfectly clear, namely that I have succeeded in\n> constructing the naively expected 2-form generalization of Yang-Mills\n> theory on the lattice. This I did 15 years ago, and it was published in\n>\n> T. A. Larsson, p-cell gauge theories, manifold space and\n> multi-dimensional integrability, Mod Phys Lett A 5 (1990) 255--264.\n>\n> (This work built on previous work by J-M Maillet and Frank Nijhoff, who\n> worked with similar concepts. They didn\'t explicitly formulate their\n> ideas as a 2-form lattice gauge theory, presumably because this was of\n> no interest to them.)\n>\n> To me, this is the crucial difference: I succeed, on the lattice, whereas\n> the 2-group approach simply fails. And I succeeded 15 years ago, before\n> the word "gerbe" was invented and when a 2-group was a discrete group\n> with two elements.\n\nIf you study ordinary gauge theory, there is a particularly simple\nspecial case in which the connection is required to be flat. In the\ndiscrete formulation of the theory on a triangulation [lattice], you\nwould just impose for each triangle [plaquette] a condition which\nrequires the holonomy around the triangle [plaquette] to be trivial.\n\nIn simple cases (for example when you work with the triangulation of a\ncompact manifold and when the gauge group G is a finite group), the\npartition function is well defined,\n\n(1) Z = #Hom (pi_1(M), G)/G\n\nIt is the number of homomorphisms from the fundamental group of the\nmanifold you have triangulated into the gauge group G (up to conjugation\nat the base point of the loops). In summary: the theory for flat\nconnections sees only the topology of your manifold.\n\nIf you study higher gauge theory with 2-groups on the lattice, there is\nstill a notion of flatness for a generalized connection and a result\ngeneralizing (1). In three dimensions, this special case is much older\nthan higher gauge theory,\n\nDN Yetter, TQFTs from homotopy 2-types,\nJ Knot Theory Ramif 2 No 1 (1993) 113-123\n\nI find this result rather reassuring. It shows that even though there\nmay be some subtle questions regarding the precise definition of the\nconfigurations of higher gauge theory, at least the flat special case is\nprecisely what we expect.\n\nI wonder whether in your approach, you also have a generalization of the\nnotion of flatness and what your analogy of (1) is.\n\n> There are admittedly problems is with the continuum limit. One way to\n> deal with that is to let any second-order phase transition define the\n> full quantum theory, in analogy with the non-perturbative definition of\n> ordinary Yang-Mills theory by lattice methods. This is what we ultimately\n> want, anyway.\n\nBy the way, if you study higher gauge theory with 2-groups in the\ndiscrete formulation, there is indeed a generalization of Wilson\'s\naction, i.e. in some sense a higher level analogue of the Yang-Mills\naction (hep-th/0304074). If Urs and others say that a straightforward\ngeneraliztion of Yang-Mills theory is missing, this refers to the\ncontinuum formulation.\n\nI guess that the question is really the naive continuum limit, i.e. the\nquestion of whether there is a refinement limit in which the discrete\naction goes to some expression which you would call the continuum\naction. I think the problem is actually the action of the generalized\nsymmetry transformations on the continuum fields.\n\nBest wishes\n\nHendryk\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi,

Thomas Larsson wrote:

> One thing must be perfectly clear, namely that I have succeeded in
> constructing the naively expected 2-form generalization of Yang-Mills
> theory on the lattice. This I did 15 years ago, and it was published in
>
> T. A. Larsson, p-cell gauge theories, manifold space and
> multi-dimensional integrability, Mod Phys Lett A 5 (1990) 255--264.
>
> (This work built on previous work by J-M Maillet and Frank Nijhoff, who
> worked with similar concepts. They didn't explicitly formulate their
> ideas as a 2-form lattice gauge theory, presumably because this was of
> no interest to them.)
>
> To me, this is the crucial difference: I succeed, on the lattice, whereas
> the 2-group approach simply fails. And I succeeded 15 years ago, before
> the word "gerbe" was invented and when a 2-group was a discrete group
> with two elements.

If you study ordinary gauge theory, there is a particularly simple
special case in which the connection is required to be flat. In the
discrete formulation of the theory on a triangulation [lattice], you
would just impose for each triangle [plaquette] a condition which
requires the holonomy around the triangle [plaquette] to be trivial.

In simple cases (for example when you work with the triangulation of a
compact manifold and when the gauge group G is a finite group), the
partition function is well defined,

(1) Z = #Hom (\pi_1(M), G)/G

It is the number of homomorphisms from the fundamental group of the
manifold you have triangulated into the gauge group G (up to conjugation
at the base point of the loops). In summary: the theory for flat
connections sees only the topology of your manifold.

If you study higher gauge theory with 2-groups on the lattice, there is
still a notion of flatness for a generalized connection and a result
generalizing (1). In three dimensions, this special case is much older
than higher gauge theory,

DN Yetter, TQFTs from homotopy 2-types,
J Knot Theory Ramif 2 No 1 (1993) 113-123

I find this result rather reassuring. It shows that even though there
may be some subtle questions regarding the precise definition of the
configurations of higher gauge theory, at least the flat special case is
precisely what we expect.

I wonder whether in your approach, you also have a generalization of the
notion of flatness and what your analogy of (1) is.

> There are admittedly problems is with the continuum limit. One way to
> deal with that is to let any second-order phase transition define the
> full quantum theory, in analogy with the non-perturbative definition of
> ordinary Yang-Mills theory by lattice methods. This is what we ultimately
> want, anyway.

By the way, if you study higher gauge theory with 2-groups in the
discrete formulation, there is indeed a generalization of Wilson's
action, i.e. in some sense a higher level analogue of the Yang-Mills
action (http://www.arxiv.org/abs/hep-th/0304074). If Urs and others say that a straightforward
generaliztion of Yang-Mills theory is missing, this refers to the
continuum formulation.

I guess that the question is really the naive continuum limit, i.e. the
question of whether there is a refinement limit in which the discrete
action goes to some expression which you would call the continuum
action. I think the problem is actually the action of the generalized
symmetry transformations on the continuum fields.

Best wishes

Hendryk

Thomas Larsson

Sep12-04, 03:23 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<4140a79f\\$1@news.sentex.net>...\n\n> But there are in general several choices. How do you get a unique surface\n> holonomy?\n\nSurfaces with the topology of an open disk seem to be the critical case.\nSuch a surface must be equipped with extra data to make the holonomy\nunique. If the boundary has length 2L, we must specify L "in" (V*) edges\nand L "out" (V) edges. Moreover, the in edges must lie consecutively. In\nparticular, a single plaquette corresponds to four plaquettes thus\ndecorated:\n\n++++++ ++++++ ------ ------\n+ | | + + | | +\n+ | | + + | | +\n+ | | + + | | +\n------ ------ ++++++ ++++++\n\nThis kind of decoration actually takes place already in 1-form gauge\ntheory. To each link we assign two matrices, U and U^-1, depending on\norientation. Each directed link has an "in" end and an "out" end.\n\nWith this kind of decoration, the surface holonomy is unique, with the\nopen disk topology. If the topology is a sphere, we must specify two\npunctures, where an exterior string is attached to the sphere. Then we\nhave two well-defined holonomies, which transport the string around the\nsphere, without moving the endpoints. To get an endpoint-independent\nnumber, we finally need to sum over all endpoints. Again I suggest that\nyou look at figure 3 in my eprint, which illustrates the surface holonomy\nfor a cube with two punctures.\n\nThe open disk also has two punctures, where the in and out links meet.\n\nOther topologies are trickier, and I haven\'t really thought about\nthem.\n\n>\n> What is the surface holonomy of the single plaquette labeled with\n>\n> A^kl_ij and A^li_kj\n>\n> ?\n>\n> What is the result of composing this at the edge j with a single plaquette\n> labeled by\n>\n> B^jr_st and B^js_rt\n>\n> ?\n\nIt depends on how the labels correspond to links. One-dimensional\nnotation is inconvenient and ambigious, and these relation should\nreally be drawn in two or three dimensions. However, lower indices\nalways correspond to in links and upper to out links, and plaquettes\nwith different assignments of in and out links should be thought of\nas different.\n\n>\n>\n> > > Is anything similar true in the approach that you are following? Is\n> there a\n> > > consistent way to turn a 4-index quantity describing south-east\n> translation\n> > > to one describing north-east translation?\n> >\n> > No. My approach is essentially different from 2-groups.\n>\n>\n> But there should be a consistent notion of surface holonomy, no?\n\nOnce you specify the in and out edges (or the punctures), the surface\nholonomy is unique and well-defined.\n\n>\n>\n> > Cubic symmetry is restored by having two independent variables for\n> > each plaquette, and eight terms (four pairwise inverses) in the action.\n>\n>\n> For closed surfaces you can sum over all ways of using one of the two\n> elements on the elementary plaquettes. Each way of doing so gives you a\n> holonomy in\n>\n> C\\{0} < End(V^@infty) .\n>\n> You can use the additive structure of R to add these up and get an element\n> of C (the complex numbers). This number is well defined for a given labeling\n> of a given closed surface. But it is not a holonomy (no longer an invertible\n> element of End(V^@infty) because 0 may occur.)\n\nSorry, I was not clear here. For a closed surface you need to specify the\ntwo punctures to get a holonomy. The same surface with the punctures\ninterchanged gives you the inverse. Summing over punctures gives you an\ninvariant, which does not need to be invertible.\n\nThe action may vanish already in abelian 1-gauge theory. Assign numbers\n(1D matrices) to the links around a plaquette as follows:\n\n1\n________\n| |\n| |\n1 | | 1\n| |\n|________|\ni\n\nThe holonomy around this plaquette = +i counterclockwise and = -i\nclockwise. If we sum these we get zero.\n\n>\n> This brings me back to a question I asked before: Do you have a notion of\n> unique surface holonomy in your approach?\n>\n>\n> > In this family of\n> > theories indexed by p, p = 1 is ordinary non-abelian lattice gauge\n> > theory, with the ordinary holonomy. Moreover, in the abelian case, where\n> > V = V* = R or C, we recover p-form electrodynamics on the lattice\n> > (actually several copies, since there still are several variables on each\n> > plaquette), again with the ordinary p-holonomy. Any family of theories\n> > which contains the right p=1 and abelian members is a viable version of\n> > p-form lattice gauge theory, isn\'t it?\n>\n>\n> You\'d usually want the generalizations to keep some properties of the theory\n> being generalized. Which properties to keep is of course a matter of taste\n> and/or of intended application.\n>\n> As far as I understand your generalization does not keep the following\n> properties of 1-form gauge theory:\n>\n> - free choice of gauge group\n\nFinite-dimensional non-abelian groups don\'t work. However, there is still\na large freedom, since the space V can have any dimension. The same\nphenomenon appears in 2D statphys, when you compute the transfer matrix.\nIf each row has L spins, the transfer matrix acts on V^@L, so it does not\nbelong to some finite-dimensional group when L -> 0.\n\n>\n> - configurations in which all elements from the gauge group may be\n> assigned to all objects (paths, surfaces, ...)\n\nIn 1-gauge theory, we have the assignments:\n\n0D node: vector space\nOpen 1D path with in and out ends specified: unique holonomy\nClosed 1D path: action\n2D surface: nothing which isn\'t fully specified by the 1D boundary.\n\nIn 2-gauge theory, we have:\n\n1D link: vector space\n1D path: string of vector spaces, which I call "barbed wire"\nOpen 2D disk with in and out edges specified: unique holonomy\nClosed 2D sphere: action\n3D surface: nothing which isn\'t fully specified by the 2D boundary.\n\nIn particular, in both 1- and 2-gauge theory, the holonomy for an open\nmanifold whose boundary has size 2L takes values in V^@L @ V*^@L. The\ndifference is that a 1D string always has 2L = 0 (if it\'s closed) or\n2L = 2 (if it\'s open), whereas the boundary of a 2D surface can have any\nlength. One may lament that this geometrical fact makes life harder, but\nthere is not much one can do to change it.\n\n>\n> - unique holonomy .\n\nThe holonomy is unique once in and out edges have been specified on the\nboundary.\n\n>\n> Do you agree with that assessment? If not, please bear with me and tell me\n> in which sense these properties are preserved in your approach.\n\nOne should add that a number of important properties of 1-gauge theory\nare generalized as expected:\n\n- 2-cell connection, 3-cell curvature.\n\n- 1-cell gauge invariance.\n\n- Cubic symmetry.\n\n- Manifest locality.\n\n- Member of a family of p-cell theories, whose p=1 member is nonabelian\nlattice gauge theory and whose abelian members are essentially p-form\nelectrodynamics on the lattice.\n\n- Unique surface holonomy once in and out edges have been specified on\nthe boundary.\n\n- No extra conditions which rule out all interesting examples.\n\nManifest locality is something that Yang-Mills theory in loop space\nlacks. That theory may (or may not) be local anyway, but without a local\nformulation we cannot really know. A local formulation is also crucial\nfor practical purposes, since loop space is really too unwieldy for\ncalculations. Btw, I am pretty sure that my model is not equivalent to YM\nin ordinary loop space, since my loops are made of barbed wire rather\nthan smooth rope. A smooth string can only have something attached to\nits endpoints, whereas a barbed wire can hook into a surface along its\nentire length. I think this is crucial for locality.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<4140a79f$1@news.sentex.net>...

> But there are in general several choices. How do you get a unique surface
> holonomy?

Surfaces with the topology of an open disk seem to be the critical case.
Such a surface must be equipped with extra data to make the holonomy
unique. If the boundary has length 2L, we must specify L "in" (V*) edges
and L "out" (V) edges. Moreover, the in edges must lie consecutively. In
particular, a single plaquette corresponds to four plaquettes thus
decorated:

++++++ ++++++[/itex] ------ ------
+ | | + + | | ++ | | + + | | ++ | | + + | | +
------ ------ ++++++ ++++++

This kind of decoration actually takes place already in 1-form gauge
theory. To each link we assign two matrices, U and U^-1, depending on
orientation. Each directed link has an "in" end and an "out" end.

With this kind of decoration, the surface holonomy is unique, with the
open disk topology. If the topology is a sphere, we must specify two
punctures, where an exterior string is attached to the sphere. Then we
have two well-defined holonomies, which transport the string around the
sphere, without moving the endpoints. To get an endpoint-independent
number, we finally need to sum over all endpoints. Again I suggest that
you look at figure 3 in my eprint, which illustrates the surface holonomy
for a cube with two punctures.

The open disk also has two punctures, where the in and out links meet.

Other topologies are trickier, and I haven't really thought about
them.

>
> What is the surface holonomy of the single plaquette labeled with
>
> A^{kl_ij} and A^{li_kj}
>
> ?
>
> What is the result of composing this at the edge j with a single plaquette
> labeled by
>
> B^{jr_st} and B^{js_rt}
>
> ?

It depends on how the labels correspond to links. One-dimensional
notation is inconvenient and ambigious, and these relation should
really be drawn in two or three dimensions. However, lower indices
always correspond to in links and upper to out links, and plaquettes
with different assignments of in and out links should be thought of
as different.

>
>
> > > Is anything similar true in the approach that you are following? Is
> there a
> > > consistent way to turn a 4-index quantity describing south-east
> translation
> > > to one describing north-east translation?
> >
> > No. My approach is essentially different from 2-groups.
>
>
> But there should be a consistent notion of surface holonomy, no?

Once you specify the in and out edges (or the punctures), the surface
holonomy is unique and well-defined.

>
>
> > Cubic symmetry is restored by having two independent variables for
> > each plaquette, and eight terms (four pairwise inverses) in the action.
>
>
> For closed surfaces you can sum over all ways of using one of the two
> elements on the elementary plaquettes. Each way of doing so gives you a
> holonomy in
>
> C\{0} < End(V^@\infty) .
>
> You can use the additive structure of R to add these up and get an element
> of C (the complex numbers). This number is well defined for a given labeling
> of a given closed surface. But it is not a holonomy (no longer an invertible
> element of End(V^@\infty) because may occur.)

Sorry, I was not clear here. For a closed surface you need to specify the
two punctures to get a holonomy. The same surface with the punctures
interchanged gives you the inverse. Summing over punctures gives you an
invariant, which does not need to be invertible.

The action may vanish already in abelian 1-gauge theory. Assign numbers
(1D matrices) to the links around a plaquette as follows:

1
__{______}
| || |1 | | 1| |
|__{______}|
i

The holonomy around this plaquette = +i counterclockwise and = -i
clockwise. If we sum these we get zero.

>
> This brings me back to a question I asked before: Do you have a notion of
> unique surface holonomy in your approach?
>
>
> > In this family of
> > theories indexed by p, [itex]p = 1 is ordinary non-abelian lattice gauge
> > theory, with the ordinary holonomy. Moreover, in the abelian case, where
> > V = V* = R or C, we recover p-form electrodynamics on the lattice
> > (actually several copies, since there still are several variables on each
> > plaquette), again with the ordinary p-holonomy. Any family of theories
> > which contains the right p=1 and abelian members is a viable version of
> > p-form lattice gauge theory, isn't it?
>
>
> You'd usually want the generalizations to keep some properties of the theory
> being generalized. Which properties to keep is of course a matter of taste
> and/or of intended application.
>
> As far as I understand your generalization does not keep the following
> properties of 1-form gauge theory:
>
> - free choice of gauge group

Finite-dimensional non-abelian groups don't work. However, there is still
a large freedom, since the space V can have any dimension. The same
phenomenon appears in 2D statphys, when you compute the transfer matrix.
If each row has L spins, the transfer matrix acts on V^@L, so it does not
belong to some finite-dimensional group when L -> .

>
> - configurations in which all elements from the gauge group may be
> assigned to all objects (paths, surfaces, ...)

In 1-gauge theory, we have the assignments:

0D node: vector space
Open 1D path with in and out ends specified: unique holonomy
Closed 1D path: action
2D surface: nothing which isn't fully specified by the 1D boundary.

In 2-gauge theory, we have:

1D link: vector space
1D path: string of vector spaces, which I call "barbed wire"
Open 2D disk with in and out edges specified: unique holonomy
Closed 2D sphere: action
3D surface: nothing which isn't fully specified by the 2D boundary.

In particular, in both 1- and 2-gauge theory, the holonomy for an open
manifold whose boundary has size 2L takes values in V^@L @ V*^@L. The
difference is that a 1D string always has 2L = (if it's closed) or
2L = 2 (if it's open), whereas the boundary of a 2D surface can have any
length. One may lament that this geometrical fact makes life harder, but
there is not much one can do to change it.

>
> - unique holonomy .

The holonomy is unique once in and out edges have been specified on the
boundary.

>
> Do you agree with that assessment? If not, please bear with me and tell me
> in which sense these properties are preserved in your approach.

One should add that a number of important properties of 1-gauge theory
are generalized as expected:

- 2-cell connection, 3-cell curvature.

- 1-cell gauge invariance.

- Cubic symmetry.

- Manifest locality.

- Member of a family of p-cell theories, whose p=1 member is nonabelian
lattice gauge theory and whose abelian members are essentially p-form
electrodynamics on the lattice.

- Unique surface holonomy once in and out edges have been specified on
the boundary.

- No extra conditions which rule out all interesting examples.

Manifest locality is something that Yang-Mills theory in loop space
lacks. That theory may (or may not) be local anyway, but without a local
formulation we cannot really know. A local formulation is also crucial
for practical purposes, since loop space is really too unwieldy for
calculations. Btw, I am pretty sure that my model is not equivalent to YM
in ordinary loop space, since my loops are made of barbed wire rather
than smooth rope. A smooth string can only have something attached to
its endpoints, whereas a barbed wire can hook into a surface along its
entire length. I think this is crucial for locality.

Urs Schreiber

Sep14-04, 07:28 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Hendryk Pfeiffer" <H.Pfeiffer@nospam.damtp.cam.ac.uk> schrieb im\nNewsbeitrag news:chp2d9\\$4aq\\$1@pegasus.csx.cam.ac.uk...\n\n > By the way, if you study higher gauge theory with 2-groups in the\n> discrete formulation, there is indeed a generalization of Wilson\'s\n> action, i.e. in some sense a higher level analogue of the Yang-Mills\n> action (hep-th/0304074). If Urs and others say that a straightforward\n> generaliztion of Yang-Mills theory is missing, this refers to the\n> continuum formulation.\n\nTrue. On the other hand, while your integral formulaiton using large gauge\ntransformations is no doubt interesting, it still has "less degrees of\nfreedom" than expected from a starightforward generalization of YM in its\nrespective integral formulation. Wouldn\'t you agree?\n\nI think that irrespective of subtle work-arounds, the constraint dt(B)+F=0\ntells us that there is some discrepancy between the "naive" expectation of\nwhat higher order YM should be and reality.\n\nThis might tell us two things:\n\n1) On the one hand we should better try to understand why field theories\nwith non-abelian 2-form fields seem to have no more degrees of freedom than\nthose with a non-abelian 1-form.\n\n2) On the other hand we should think about which assumptions went into the\nline of thought that derives dt(B)+F = 0. Maybe not all these assumptions\nare desirable.\n\n\nConcerning 1) one should maybe open the "Big Dictionary of Effective Field\nTheories" (TM) and see if it knows about any field theories where a\nnon-abelian B-field comes up all by itself, so that we can read off instead\nof guessing what is going on.\n\nIt turns out that this dictionary (aka string theory) tells us that\nnon-abelian B-fields are expected to arise in the worldvolume theory of\nsolitonic NS 5-branes. The reason is simple: There are open membranes ending\non these (and only on these) and their string-like boundary generalizes the\npoint-like boundary of open strings (which gives rise to ordinary gauge\ntheory) up one step in the dimensional ladder and in particular couples to a\n2-form field for precisely the same reasons as the open string boundary\ncouples to a 1-form field.\n\nFor some literature on how this works see the very early\n\nA. Strominger,\nOpen p-branes,\nhep-th/9512059\n\nWitten\'s comment in section 1.1 of\n\nE. Witten\nSome Comments on String Dynamics,\nhep-th/9507121\n\nor, for some more explicit formulas, for instance\n\nBergshoeff et al.\nThe M5-brane and non-commutative loop space\nCQG 18 (2001) 3265-3273 .\n\nLots of other references are given in\n\nE. Witten,\nBranes and the Dynamics of QCD,\nhep-th/9706109 .\n\nMembranes can only end on (M or NS) 5-branes, so this fixes the number of\ndimensions of effective field theories that should have a non-abelian\nB-field to 5+1.\n\nThis is important, because in 6 dimensions a 3-form field strength can be\nself-dual, and indeed in these theories it is.\n\nBy considering a compactification of the 1+5 dimensional on a circle we get\na gauge field A_i = B_i6 from the 2-form field, and one can easily see that\nthe degrees of freedom contained in B_ij are precisely captured by the\n1-form A alone (section 1.1. of hep-th/9507121). It follows that these\n2-form field theories should have a dynamics very similar to ordinary 1-form\ngauge theories. (Neither world-volume theory of (NS/M) 5-branes is known\nexplicitly (be it "little string theory" or "N=(2,0) field theory"), they\nare so far defined only implicitly by the given limits of the dynamics of\nthese branes).\n\nOf course this proves nothing for the general formalism of non-abelian\n2-form gauge theory, but it harmonizes well with the result that on general\ngrounds 2-form gauge theories are highly constrained and seem to allow only\ndegrees of freedom that are already encoded in a 1-form.\n\n\nConcerning the second point, 2) (generalizing assumptions), I would like to\nmake the following comment:\n\nIt seems vital to me to preserve the assumption that elements from the gauge\ngroup (like SU(N) or something) is associated with a surface elemt in 2-form\ngauge theory (as opposed to assigning objects of different nature) but one\nassumption of 2-group gauge theory is at least on first sight definitely\nstronger than what we really need:\n\nIn order to be able to uniquely compute the surface holonomy of some surface\nall we need is that the exchange law between horizontal and vertical\ncomposition holds for bigons that are horizontally adjacent. (Vertical\nadjacency is built in the rules of 2-group multiplication, but horizontal\nadjacency of bigons is not). But the exchange law of a 2-group really\nrequires that given any 4 bigons in the manifold their associated 2-group\nelements must satisfy the exchange law. But in actually computing surface\nholonomy we only need this law for sets of 4 bigons which are all adjacent.\n\nIn technical terms: The exchange law of the 2-groupoid of bigons is already\nsufficient to ensure unique surface holoomy. If we take a 2-connection to be\na 2-functor from that strict 2-groupoid of bigons into a strict 2-group, the\nexchange law of the 2-group governs the admissable 2-functors. If, on the\nother hand, we use a weaker notion of 2-group where the exchange law is not\nsupposed to hold on the nose as an identity (but possibly up to\nisomorphisms) then there are more 2-functors from the 2-groupoid of bigons\nwith that weak target category, and they still guarantee unique surface\nholonomy.\n\nI think I can show that the consistency condition for these weak\n2-connections is equivalent to the r-flatness conditon known from the loop\nspace approach, and how the solutions found by Alvarez et al to r-flatness\nwhich are not of the form dt(B)+F = 0 are either isomorphic to strict\n2-connections or are weak 2-connections, even thoug rather "uninteresting"\nones. But I also think that I am able to give further "interesting"\nsolutions to r-flatness, i.e. interesting weak 2-connections, which don\'t\nhave dt(B)+F=0.\n\n\n> I guess that the question is really the naive continuum limit, i.e. the\n> question of whether there is a refinement limit in which the discrete\n> action goes to some expression which you would call the continuum\n> action. I think the problem is actually the action of the generalized\n> symmetry transformations on the continuum fields.\n\nIn which sense do you think the continuum limit is problematic? If you take\nyour lattice 2-connections, being strict 2-functors from the lattice\ngroupoid to the 2-group and replace the source category with the 2-groupoid\nof bigons, would you encounter any problems? Aren\'t you already doing\nessentially that in section 3 of hep-th/0309173.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Hendryk Pfeiffer" <H.Pfeiffer@nospam.damtp.cam.ac.uk> schrieb im
Newsbeitrag news:chp2d9$4aq$1@pegasus.csx.cam.ac.uk...

> By the way, if you study higher gauge theory with 2-groups in the
> discrete formulation, there is indeed a generalization of Wilson's
> action, i.e. in some sense a higher level analogue of the Yang-Mills
> action (http://www.arxiv.org/abs/hep-th/0304074). If Urs and others say that a straightforward
> generaliztion of Yang-Mills theory is missing, this refers to the
> continuum formulation.

True. On the other hand, while your integral formulaiton using large gauge
transformations is no doubt interesting, it still has "less degrees of
freedom" than expected from a starightforward generalization of YM in its
respective integral formulation. Wouldn't you agree?

I think that irrespective of subtle work-arounds, the constraint dt(B)+F=0
tells us that there is some discrepancy between the "naive" expectation of
what higher order YM should be and reality.

This might tell us two things:

1) On the one hand we should better try to understand why field theories
with non-abelian 2-form fields seem to have no more degrees of freedom than
those with a non-abelian 1-form.

2) On the other hand we should think about which assumptions went into the
line of thought that derives dt(B)+F = . Maybe not all these assumptions
are desirable.

Concerning 1) one should maybe open the "Big Dictionary of Effective Field
Theories" (TM) and see if it knows about any field theories where a
non-abelian B-field comes up all by itself, so that we can read off instead
of guessing what is going on.

It turns out that this dictionary (aka string theory) tells us that
non-abelian B-fields are expected to arise in the worldvolume theory of
solitonic NS 5-branes. The reason is simple: There are open membranes ending
on these (and only on these) and their string-like boundary generalizes the
point-like boundary of open strings (which gives rise to ordinary gauge
theory) up one step in the dimensional ladder and in particular couples to a
2-form field for precisely the same reasons as the open string boundary
couples to a 1-form field.

For some literature on how this works see the very early

A. Strominger,
Open p-branes,
http://www.arxiv.org/abs/hep-th/9512059

Witten's comment in section 1.1 of

E. Witten
Some Comments on String Dynamics,
http://www.arxiv.org/abs/hep-th/9507121

or, for some more explicit formulas, for instance

Bergshoeff et al.
The M5-brane and non-commutative loop space
CQG 18 (2001) 3265-3273 .

Lots of other references are given in

E. Witten,
Branes and the Dynamics of QCD,
http://www.arxiv.org/abs/hep-th/9706109 .

Membranes can only end on (M or NS) 5-branes, so this fixes the number of
dimensions of effective field theories that should have a non-abelian
B-field to 5+1.

This is important, because in 6 dimensions a 3-form field strength can be
self-dual, and indeed in these theories it is.

By considering a compactification of the 1+5 dimensional on a circle we get
a gauge field A_i = B_{i6} from the 2-form field, and one can easily see that
the degrees of freedom contained in B_{ij} are precisely captured by the
1-form A alone (section 1.1. of http://www.arxiv.org/abs/hep-th/9507121). It follows that these
2-form field theories should have a dynamics very similar to ordinary 1-form
gauge theories. (Neither world-volume theory of (NS/M) 5-branes is known
explicitly (be it "little string theory" or "N=(2,0) field theory"), they
are so far defined only implicitly by the given limits of the dynamics of
these branes).

Of course this proves nothing for the general formalism of non-abelian
2-form gauge theory, but it harmonizes well with the result that on general
grounds 2-form gauge theories are highly constrained and seem to allow only
degrees of freedom that are already encoded in a 1-form.

Concerning the second point, 2) (generalizing assumptions), I would like to
make the following comment:

It seems vital to me to preserve the assumption that elements from the gauge
group (like SU(N) or something) is associated with a surface elemt in 2-form
gauge theory (as opposed to assigning objects of different nature) but one
assumption of 2-group gauge theory is at least on first sight definitely
stronger than what we really need:

In order to be able to uniquely compute the surface holonomy of some surface
all we need is that the exchange law between horizontal and vertical
composition holds for bigons that are horizontally adjacent. (Vertical
adjacency is built in the rules of 2-group multiplication, but horizontal
adjacency of bigons is not). But the exchange law of a 2-group really
requires that given any 4 bigons in the manifold their associated 2-group
elements must satisfy the exchange law. But in actually computing surface
holonomy we only need this law for sets of 4 bigons which are all adjacent.

In technical terms: The exchange law of the 2-groupoid of bigons is already
sufficient to ensure unique surface holoomy. If we take a 2-connection to be
a 2-functor from that strict 2-groupoid of bigons into a strict 2-group, the
exchange law of the 2-group governs the admissable 2-functors. If, on the
other hand, we use a weaker notion of 2-group where the exchange law is not
supposed to hold on the nose as an identity (but possibly up to
isomorphisms) then there are more 2-functors from the 2-groupoid of bigons
with that weak target category, and they still guarantee unique surface
holonomy.

I think I can show that the consistency condition for these weak
2-connections is equivalent to the r-flatness conditon known from the loop
space approach, and how the solutions found by Alvarez et al to r-flatness
which are not of the form dt(B)+F = are either isomorphic to strict
2-connections or are weak 2-connections, even thoug rather "uninteresting"
ones. But I also think that I am able to give further "interesting"
solutions to r-flatness, i.e. interesting weak 2-connections, which don't
have dt(B)+F=0.

> I guess that the question is really the naive continuum limit, i.e. the
> question of whether there is a refinement limit in which the discrete
> action goes to some expression which you would call the continuum
> action. I think the problem is actually the action of the generalized
> symmetry transformations on the continuum fields.

In which sense do you think the continuum limit is problematic? If you take
your lattice 2-connections, being strict 2-functors from the lattice
groupoid to the 2-group and replace the source category with the 2-groupoid
of bigons, would you encounter any problems? Aren't you already doing
essentially that in section 3 of http://www.arxiv.org/abs/hep-th/0309173.

Thomas Larsson

Sep14-04, 01:35 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hendryk Pfeiffer <H.Pfeiffer@nospam.damtp.cam.ac.uk> wrote in message news:<chp2d9\\$4aq\\$1@pegasus.csx.cam.ac.uk>...\n \n> If you study ordinary gauge theory, there is a particularly simple\n> special case in which the connection is required to be flat. In the\n> discrete formulation of the theory on a triangulation [lattice], you\n> would just impose for each triangle [plaquette] a condition which\n> requires the holonomy around the triangle [plaquette] to be trivial.\n>\n> In simple cases (for example when you work with the triangulation of a\n> compact manifold and when the gauge group G is a finite group), the\n> partition function is well defined,\n>\n> (1) Z = #Hom (pi_1(M), G)/G\n>\n> It is the number of homomorphisms from the fundamental group of the\n> manifold you have triangulated into the gauge group G (up to conjugation\n> at the base point of the loops). In summary: the theory for flat\n> connections sees only the topology of your manifold.\n>\n> If you study higher gauge theory with 2-groups on the lattice, there is\n> still a notion of flatness for a generalized connection and a result\n> generalizing (1). In three dimensions, this special case is much older\n> than higher gauge theory,\n>\n> DN Yetter, TQFTs from homotopy 2-types,\n> J Knot Theory Ramif 2 No 1 (1993) 113-123\n>\n> I find this result rather reassuring. It shows that even though there\n> may be some subtle questions regarding the precise definition of the\n> configurations of higher gauge theory, at least the flat special case is\n> precisely what we expect.\n>\n> I wonder whether in your approach, you also have a generalization of the\n> notion of flatness and what your analogy of (1) is.\n\nFlatness is a localized version of the Yang-Baxter equation.\n\nConsider a spatially homogeneous configuration of 1-gauge theory. To\neach horizontal link (pointing left) we assign a matrix U, and to each\nvertical link (pointing up) a matrix V, and U and V are the same\nthroughout the lattice. Zero curvature then becomes UV = VU, i.e. U\nand V commute. Analogously, assign a matrix R_ij to a plaquette in the\nij-plane. Zero 3-curvature around an elementary 123-cube then becomes\n\nR_12 R_13 R_23 = R_23 R_13 R_12\n\nwhich is the Yang-Baxter (YB) equation. One has to be a bit careful to\nmake sure that one picks the right variable at each plaquette, but it\ndoes work out right. More generally, one obtain new zero-curvature\nsolutions by edge gauge transformations.\n\nAs you undoubtedly know, the YB (or triangle) equation is a sufficient\n(and in practice necessary) condition for integrability of 2D statistical\nmodels. The analogous condition in p dimensions is known as the p-simplex\nequation; the next equation in the hierarchy is the tetrahedron equation,\nwhich was formulated by Zamolodchikov in 1981. So ordinary commuting\nmatrices is the zero-curvature condition "square = 1" in 1-gauge theory,\nthe YB equation is the zero-curvature conditon "cube = 1" in 2-gauge\ntheory, etc. In fact, one of the two standard illustrations of the YB\nequation is the equality of two half cubes. Thus,\n\np=1 Commuting matrices "square = 1"\np=2 Yang-Baxter equation "cube = 1"\np=3 Tetrahedron equation "4-cube = 1"\n...\np p-simplex equation "(p+1)-cube = 1"\n\nThis hierarchy was my main motivation for constructing my model, inspired\nby Maillet and Nijhoff who argued along similar lines. My hope was that\nby viewing the simplex equations as zero-curvature conditions, I might be\nable to find new solutions. It would be particularly interesting to find\na solution in the 3D Ising or Ising gauge universality class, of course.\nIt is also a very natural idea, since integrability conditions, like Lax\npairs and Sacharov-Shabat equations, can be formulated as zero curvature.\n\nAlas, in this respect I failed. I think that the p-cell models are neat,\nand the p=1 case is ordinary lattice gauge theory, but they didn\'t give\nme any new insights into integrability, and eventually I dropped the idea.\n\nI don\'t really understand your question about the partition function. If\nyou only sum over configurations where the holonomy is zero everywhere,\nthe partition function must be unity, mustn\'t it?\n\n>\n> > There are admittedly problems is with the continuum limit. One way to\n> > deal with that is to let any second-order phase transition define the\n> > full quantum theory, in analogy with the non-perturbative definition of\n> > ordinary Yang-Mills theory by lattice methods. This is what we ultimately\n> > want, anyway.\n>\n> By the way, if you study higher gauge theory with 2-groups in the\n> discrete formulation, there is indeed a generalization of Wilson\'s\n> action, i.e. in some sense a higher level analogue of the Yang-Mills\n> action (hep-th/0304074). If Urs and others say that a straightforward\n> generaliztion of Yang-Mills theory is missing, this refers to the\n> continuum formulation.\n\nIf I understand you correctly, you assign to each triangular face (V,E,F)\nthree maps\n\nF_0: V -> {*}\nF_1: E -> G\nF_2: F -> H x| G\n\nSo in your construction it is the edges that act on the faces rather than\nthe other way around. Your group G lives on edges, and acts on the face\ngroup. In contrast, I assign nothing to vertices, a vector space to\nedges, and a four-index Yang-Baxter matrix to square plaquettes. My face\nmatrices act on the edge spaces, just as a edge matrix acts on the vertex\nspace in ordinary lattice gauge theory. So we do very different things,\nbut nevertheless we both seem to succeed in generalizing Yang-Mills on\nthe lattice.\n\nBut it seems to me that requiring the face group to be of the semi-direct\nform H x| G is quite restrictive. You give some solutions in Section 5,\nbut it is unclear to me how general those are. Is it not this that leads\nto the continuum condition F(A)+B = 0, which makes the 2-form field B an\nauxiliary field? In ordinary gauge theory, as well as in my 2-gauge\ntheory, there are really no conditions on the highest-cell variables; any\nYang-Baxter matrix works for me.\n\nAnyway, the crucial difference is that you demand that all boundaries (of\nsurface holonomies) be must valued in the same group G. If you make that\nassumption, I agree that the your construction is presumably the only\npossible. But my boundaries take values in different spaces V^@L\ndepending on their length L. Both versions generalize a 1-holonomy, whose\nboundary always has the same length.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hendryk Pfeiffer <H.Pfeiffer@nospam.damtp.cam.ac.uk> wrote in message news:<chp2d9$4aq$1@pegasus.csx.cam.ac.uk>...

> If you study ordinary gauge theory, there is a particularly simple
> special case in which the connection is required to be flat. In the
> discrete formulation of the theory on a triangulation [lattice], you
> would just impose for each triangle [plaquette] a condition which
> requires the holonomy around the triangle [plaquette] to be trivial.
>
> In simple cases (for example when you work with the triangulation of a
> compact manifold and when the gauge group G is a finite group), the
> partition function is well defined,
>
> (1) Z = #Hom (\pi_1(M), G)/G
>
> It is the number of homomorphisms from the fundamental group of the
> manifold you have triangulated into the gauge group G (up to conjugation
> at the base point of the loops). In summary: the theory for flat
> connections sees only the topology of your manifold.
>
> If you study higher gauge theory with 2-groups on the lattice, there is
> still a notion of flatness for a generalized connection and a result
> generalizing (1). In three dimensions, this special case is much older
> than higher gauge theory,
>
> DN Yetter, TQFTs from homotopy 2-types,
> J Knot Theory Ramif 2 No 1 (1993) 113-123
>
> I find this result rather reassuring. It shows that even though there
> may be some subtle questions regarding the precise definition of the
> configurations of higher gauge theory, at least the flat special case is
> precisely what we expect.
>
> I wonder whether in your approach, you also have a generalization of the
> notion of flatness and what your analogy of (1) is.

Flatness is a localized version of the Yang-Baxter equation.

Consider a spatially homogeneous configuration of 1-gauge theory. To
each horizontal link (pointing left) we assign a matrix U, and to each
vertical link (pointing up) a matrix V, and U and V are the same
throughout the lattice. Zero curvature then becomes UV = VU, i.e. U
and V commute. Analogously, assign a matrix R_{ij} to a plaquette in the
ij-plane. Zero 3-curvature around an elementary 123-cube then becomes

R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}

which is the Yang-Baxter (YB) equation. One has to be a bit careful to
make sure that one picks the right variable at each plaquette, but it
does work out right. More generally, one obtain new zero-curvature
solutions by edge gauge transformations.

As you undoubtedly know, the YB (or triangle) equation is a sufficient
(and in practice necessary) condition for integrability of 2D statistical
models. The analogous condition in p dimensions is known as the p-simplex
equation; the next equation in the hierarchy is the tetrahedron equation,
which was formulated by Zamolodchikov in 1981. So ordinary commuting
matrices is the zero-curvature condition "square = 1" in 1-gauge theory,
the YB equation is the zero-curvature conditon "cube = 1" in 2-gauge
theory, etc. In fact, one of the two standard illustrations of the YB
equation is the equality of two half cubes. Thus,

p=1 Commuting matrices "square = 1"
p=2 Yang-Baxter equation "cube = 1"
p=3 Tetrahedron equation "4-cube = 1"
...
p p-simplex equation "(p+1)-cube = 1"

This hierarchy was my main motivation for constructing my model, inspired
by Maillet and Nijhoff who argued along similar lines. My hope was that
by viewing the simplex equations as zero-curvature conditions, I might be
able to find new solutions. It would be particularly interesting to find
a solution in the 3D Ising or Ising gauge universality class, of course.
It is also a very natural idea, since integrability conditions, like Lax
pairs and Sacharov-Shabat equations, can be formulated as zero curvature.

Alas, in this respect I failed. I think that the p-cell models are neat,
and the p=1 case is ordinary lattice gauge theory, but they didn't give
me any new insights into integrability, and eventually I dropped the idea.

I don't really understand your question about the partition function. If
you only sum over configurations where the holonomy is zero everywhere,
the partition function must be unity, mustn't it?

>
> > There are admittedly problems is with the continuum limit. One way to
> > deal with that is to let any second-order phase transition define the
> > full quantum theory, in analogy with the non-perturbative definition of
> > ordinary Yang-Mills theory by lattice methods. This is what we ultimately
> > want, anyway.
>
> By the way, if you study higher gauge theory with 2-groups in the
> discrete formulation, there is indeed a generalization of Wilson's
> action, i.e. in some sense a higher level analogue of the Yang-Mills
> action (http://www.arxiv.org/abs/hep-th/0304074). If Urs and others say that a straightforward
> generaliztion of Yang-Mills theory is missing, this refers to the
> continuum formulation.

If I understand you correctly, you assign to each triangular face (V,E,F)
three maps

F_0: V -> {*}F_1: E -> GF_2: F -> H x| G

So in your construction it is the edges that act on the faces rather than
the other way around. Your group G lives on edges, and acts on the face
group. In contrast, I assign nothing to vertices, a vector space to
edges, and a four-index Yang-Baxter matrix to square plaquettes. My face
matrices act on the edge spaces, just as a edge matrix acts on the vertex
space in ordinary lattice gauge theory. So we do very different things,
but nevertheless we both seem to succeed in generalizing Yang-Mills on
the lattice.

But it seems to me that requiring the face group to be of the semi-direct
form H x| G is quite restrictive. You give some solutions in Section 5,
but it is unclear to me how general those are. Is it not this that leads
to the continuum condition F(A)+B = 0, which makes the 2-form field B an
auxiliary field? In ordinary gauge theory, as well as in my 2-gauge
theory, there are really no conditions on the highest-cell variables; any
Yang-Baxter matrix works for me.

Anyway, the crucial difference is that you demand that all boundaries (of
surface holonomies) be must valued in the same group G. If you make that
assumption, I agree that the your construction is presumably the only
possible. But my boundaries take values in different spaces V^@L
depending on their length L. Both versions generalize a 1-holonomy, whose
boundary always has the same length.

Urs Schreiber

Sep15-04, 04:45 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409100745.516ff2e3@pos ting.google.com...\n> "Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<4140a79f\\$1@news.sentex.net>...\n> \n> > But there are in general several choices. How do you get a unique\nsurface\n> > holonomy?\n\n[...]\n\n> This kind of decoration actually takes place already in 1-form gauge\n> theory. To each link we assign two matrices, U and U^-1, depending on\n> orientation. Each directed link has an "in" end and an "out" end.\n\nTrue. Moreover a closed curve has a holonomy depending on the choice of base\npoint. But different base points are related by conjugation with a\nwell-defined element.\n\nThis generalizes in 2-group theory, as described very nicely in section 2.6\nof Girelli&Pfeiffer\'s hep-th/0309173. The 2-holonomy depends on on a choice\nof base edge, much as in your formalism, but different choices of base edges\nare related by operations generalizing the above conjugation.\n\nSimilarly the four different labels which you assign to a plaquette\nessentially also appear in 2-group theory, but there they are all related in\na well defined way using the operation of "whiskering". Given one, all\nothers are specified.\n\n"Whiskering" drops out of the 2-group formalism and is not a new ingredient.\nIt follows from the fact tha composing two objects (plaquettes) gives an\nobject of the same kind and that one of the objects may be a degenerate\nsurface.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409100745.516ff2e3@posting.google.c om...
> "Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message
news:<4140a79f$1@news.sentex.net>...
>
> > But there are in general several choices. How do you get a unique
surface
> > holonomy?

[...]

> This kind of decoration actually takes place already in 1-form gauge
> theory. To each link we assign two matrices, U and U^-1, depending on
> orientation. Each directed link has an "in" end and an "out" end.

True. Moreover a closed curve has a holonomy depending on the choice of base
point. But different base points are related by conjugation with a
well-defined element.

This generalizes in 2-group theory, as described very nicely in section 2.6
of Girelli&Pfeiffer's http://www.arxiv.org/abs/hep-th/0309173. The 2-holonomy depends on on a choice
of base edge, much as in your formalism, but different choices of base edges
are related by operations generalizing the above conjugation.

Similarly the four different labels which you assign to a plaquette
essentially also appear in 2-group theory, but there they are all related in
a well defined way using the operation of "whiskering". Given one, all
others are specified.

"Whiskering" drops out of the 2-group formalism and is not a new ingredient.
It follows from the fact tha composing two objects (plaquettes) gives an
object of the same kind and that one of the objects may be a degenerate
surface.

Thomas Larsson

Sep15-04, 08:44 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2qo2teFvrkvqU1-100000@uni-berlin.de>...\n\n> True. On the other hand, while your integral formulaiton using large gauge\n> transformations is no doubt interesting, it still has "less degrees of\n> freedom" than expected from a starightforward generalization of YM in its\n> respective integral formulation. Wouldn\'t you agree?\n>\n> I think that irrespective of subtle work-arounds, the constraint dt(B)+F=0\n> tells us that there is some discrepancy between the "naive" expectation of\n> what higher order YM should be and reality.\n\nThis is exactly my impression too, although you probably expressed it\nclearer than I did.\n\n> This might tell us two things:\n>\n> 1) On the one hand we should better try to understand why field theories\n> with non-abelian 2-form fields seem to have no more degrees of freedom\nthan\n> those with a non-abelian 1-form.\n>\n> 2) On the other hand we should think about which assumptions went into the\n> line of thought that derives dt(B)+F = 0. Maybe not all these assumptions\n> are desirable.\n\nOne assumption is that the connection is a function (or form, rather)\nthat depends on spacetime only, A = A(x). However, A is geometrically\nresponsible for parallel transport of a string. Unlike a particle, a\nstring has both a position x and a direction s. Real strings have more\nstructure, but giving the position and direction is a minimal\ncharacterization. One may therefore expect that the two-form connection\ndepends on both x and s, A = A(x,s).\n\nNote the difference to the form components, which also indicate\ndirection:\n\nThe 1-form A_i(x) transports a particle at x in the i-direction.\n\nThe 2-form A_ij(x,s) transports a string at x, with direction s, in the\nij-plane. s must lie in the ij-plane, but there is no reason to expect\nthat transporting an i-directed string in the j-direction has anything\nto do with transporting a j-directed string in the i-direction.\n\nWith this extra data one can at least formally write down a nice\nnon-abelian 3-curvature (Urs knows this, but maybe not Hendryk):\n\nF(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],\n\nwhere the dot indicates contraction of the 2-form A and the vector s.\nForm degrees match because all three terms are 3-forms.\n\n> Concerning the second point, 2) (generalizing assumptions), I would like\nto\n> make the following comment:\n>\n> It seems vital to me to preserve the assumption that elements from the\ngauge\n> group (like SU(N) or something) is associated with a surface elemt in\n2-form\n> gauge theory (as opposed to assigning objects of different nature)\n\nOnly two assumptions are truly non-negotiable for something that calls\nitself non-abelian p-form gauge theory: it must reduce to 1-form gauge\ntheory and p-form electrodynamics in appropriate limits, and interesting\nexamples beyond these must exist. I think that locality in spacetime is\nalso highly desirable. My model seems to be the only known model which\nsatisfies these three conditions. On the lattice, admittedly.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2qo2teFvrkvqU1-100000@uni-berlin.de>...

> True. On the other hand, while your integral formulaiton using large gauge
> transformations is no doubt interesting, it still has "less degrees of
> freedom" than expected from a starightforward generalization of YM in its
> respective integral formulation. Wouldn't you agree?
>
> I think that irrespective of subtle work-arounds, the constraint dt(B)+F=0
> tells us that there is some discrepancy between the "naive" expectation of
> what higher order YM should be and reality.

This is exactly my impression too, although you probably expressed it
clearer than I did.

> This might tell us two things:
>
> 1) On the one hand we should better try to understand why field theories
> with non-abelian 2-form fields seem to have no more degrees of freedom
than
> those with a non-abelian 1-form.
>
> 2) On the other hand we should think about which assumptions went into the
> line of thought that derives dt(B)+F = . Maybe not all these assumptions
> are desirable.

One assumption is that the connection is a function (or form, rather)
that depends on spacetime only, A = A(x). However, A is geometrically
responsible for parallel transport of a string. Unlike a particle, a
string has both a position x and a direction s. Real strings have more
structure, but giving the position and direction is a minimal
characterization. One may therefore expect that the two-form connection
depends on both x and s, A = A(x,s).

Note the difference to the form components, which also indicate
direction:

The 1-form A_i(x) transports a particle at x in the i-direction.

The 2-form A_{ij}(x,s) transports a string at x, with direction s, in the
ij-plane. s must lie in the ij-plane, but there is no reason to expect
that transporting an i-directed string in the j-direction has anything
to do with transporting a j-directed string in the i-direction.

With this extra data one can at least formally write down a nice
non-abelian 3-curvature (Urs knows this, but maybe not Hendryk):

F(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],

where the dot indicates contraction of the 2-form A and the vector s.
Form degrees match because all three terms are 3-forms.

> Concerning the second point, 2) (generalizing assumptions), I would like
to
> make the following comment:
>
> It seems vital to me to preserve the assumption that elements from the
gauge
> group (like SU(N) or something) is associated with a surface elemt in
2-form
> gauge theory (as opposed to assigning objects of different nature)

Only two assumptions are truly non-negotiable for something that calls
itself non-abelian p-form gauge theory: it must reduce to 1-form gauge
theory and p-form electrodynamics in appropriate limits, and interesting
examples beyond these must exist. I think that locality in spacetime is
also highly desirable. My model seems to be the only known model which
satisfies these three conditions. On the lattice, admittedly.

Urs Schreiber

Sep15-04, 12:21 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409150200.2a351028-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2qo2teFvrkvqU1-100000@uni-berlin.de>...\n\n> One may therefore expect that the two-form connection\n> depends on both x and s, A = A(x,s).\n\nOk, now I get what you mean. That\'s saying that we could have a 2-form on\nloop space which explicitly depends on dX/dsigma. Actually the ordinary\nconnection on loop space already does so, but only in the form\n\nint W^{-1}(sigma) B_mn(X(sigma)) W(sigma) dX^m/dsigma dX^n .\n\nOne could in principle have B=B(X,dX/dsigma) instead. This would be a 2-form\non loop space which does not \'lift\' from one in target space.\n\nIn fact, this addresses a general issue that needs to be thought about:\nThere are a-priori lots and lots of different forms of 1-forms on loop space\ncoming from 2-forms in one way or another. For them to produce consistent\nsurface holonomy they need to be r-flat. The question is if maybe every\nr-flat connection is of the above form or if there are lots and lots of\ndifferent r-flat connections of different forms.\n\n> With this extra data one can at least formally write down a nice\n> non-abelian 3-curvature (Urs knows this, but maybe not Hendryk):\n>\n> F(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],\n\nI know that you said this before, but it seems to me that there is some work\nneeded to give this expression a well-defined meaning. The space with\ncoordinates {(x,s)} *is* (a simplified version of) loop space, and that\'s\nwhere the F must hence live. So we should go to the full thing and replace s\nby dX/dsigma and include appropriate integrals and all. Then I\'d need to\nrethink if this expression is the one to consider.\n\n\n> I think that locality in spacetime is also highly desirable.\n\n\nThere is an interesting paper arguing that a theory of non-abelian 2-forms\n(in particular a theory arising as the worldvolume theory of a stack of\n5-branes) cannot be a local field theory. That\'s\n\nX. Bekaert & M. Henneux & A. Sevrin:\nChiral Forms and their deformations,\nhep-th/0004049 .\n\nActually they are only considering chiral=self-dual 2-forms, because the\nnon-abelian 2-forms living on 5-branes happen to be self-dual\n\nIt\'s not too surprising: A 2-form gauge theory must be about strings and\nhence it is not natural to assume locality to be preserved, even more so\nwhen these strings are not weakly coupled (and the self-dual strings living\non 5-branes never are weakly coupled. Being their own weak/strong\nelectric/magnetic dual they have unit coupling).\n\nI have come across another maybe intersting approach to non-abelian 2-form\ntheories:\n\nIn\n\nO. Ganor,\nSix-dimensional tensionless strings in the large N limit,\nhep-th/9605201\n\nthe author tries to do something about the fact that no Lagrangian\ndescription for 5-brane worldvolumes are known (or are there meanwhile?\nLubos might know, he has written papers on that I think), but what he does\nis interesting irrespective of the string theory context:\n\nNamely Ganor tries to construct higher dimensional analogs of the "loop\nequations" that arise when Yang-Mills is formulated in loop space. He\ndiscusses "surface equations" and tries to guess/derive/motivate their\nnature and properties. I haven\'t yet followed his construction in detail,\nbut that might be an interesting perspective.\n\nBTW, there is a well-known puzzle concerning worldvolume theories of stacks\nof M/NS5-branes: When the number N of branes in the stack is increased\n(which for ordinary D-brane scenarios means increasing the size of the gauge\ngroup U(N)) various quantities in the theory scale as N^3, while the\ndimension of semi-simple Lie algebras never scales as fast as N^3, with N\nthe size of the Cartan sub-algebra.\n\nA very recent paper addressing this issue is\n\nD. Berman & J. Harvey:\nThe self-dual string and anomalies in the M5-brane,\nhep-th/0408198\n\nThis almost seems to call for a construction like yours, where not ordinary\nsemi-simple Lie-group elements colour the surfaces, but something else. On\nthe other hand, the non-abelian 2-forms expected to arise on stacks of\nM/SN-5 branes are related by a host of compactifications, limits,\nreformulations and dualities to non-abelian theories of ordinary gauge\ngroups, so that it seems very unlikely that the non-abelian 2-form takes\nvalues in something that is not an ordinary semi-simple Lie group.\n\nIndeed, in hep-th/9905018 and math.dg/9907034 (which I have not looked at\nyet) it is argued that the situations is clarified by using 1-gerbes.\n\nI need to better understand the relation between 1-gerbes and 2-groups. As\nfar as I can see I am not the only one, but nobody understands this yet\n(corrections are very welcome).\n\nThere is an old approach that seems to have been constructed in parallel to\nBaez\' 2-groups, namely\n\nI. Cheplev,\nNon-abelian Wilson surfaces,\nJHEP 02 (2002) 013 .\n\nBoth Baez and Cheplec are inspired/motivated by the apparently\ngroundbreaking work\n\nL. Breen & W. Messing,:\nDifferential Geometry of Gerbes,\nmath.AG/0106083\n\nbut Cheplev\'s notion of a 2-connection is orthogonal (in a surprisingly\nliteral sense) to that of 2-group theory and instead comes from 1-gerbes.\n\nIn 1-gerbe theory a connection is a set of functors from categorified fibers\nof the gauge bundle to another such categoriefied fiber. This is supposed to\nbe the obvious generalization of the fact that an ordinary connection can be\nregarded as a set of functors between ordinary bundle fibers.\n\nCheplev deduces from that a notion of surface holonomy which looks quite\ndifferent, but can be seen to be pretty much the same as that of 2-group\ntheory, using the same definition (secertly) of horizontal and parallal\ncomposition.\n\nBut in 2-group theory a 2-connection is a functor not "horizontally" between\nfibers, but instead from the 2-groupoid of bigons (or their lattice version\nor something similar) to the respective 2-group. That "vertical" in a way.\n\nBut checking the citations it seems that Chepelev\'s constructin was not very\ninfluential. Indeed that might be related to my suspicion that it is just\n2-group theory formulated in a more convoluted way. But I haven\'t checked\nthis in detail.\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409150200.2a351028-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2qo2teFvrkvqU1-100000@uni-berlin.de>...

> One may therefore expect that the two-form connection
> depends on both x and s, A = A(x,s).

Ok, now I get what you mean. That's saying that we could have a 2-form on
loop space which explicitly depends on dX/dsigma. Actually the ordinary
connection on loop space already does so, but only in the form

\int W^{-1}(\sigma) B_{mn}(X(\sigma)) W(\sigma) dX^m/dsigma dX^n .

One could in principle have B=B(X,dX/dsigma) instead. This would be a 2-form
on loop space which does not 'lift' from one in target space.

In fact, this addresses a general issue that needs to be thought about:
There are a-priori lots and lots of different forms of 1-forms on loop space
coming from 2-forms in one way or another. For them to produce consistent
surface holonomy they need to be r-flat. The question is if maybe every
r-flat connection is of the above form or if there are lots and lots of
different r-flat connections of different forms.

> With this extra data one can at least formally write down a nice
> non-abelian 3-curvature (Urs knows this, but maybe not Hendryk):
>
> F(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],

I know that you said this before, but it seems to me that there is some work
needed to give this expression a well-defined meaning. The space with
coordinates {(x,s)} *is* (a simplified version of) loop space, and that's
where the F must hence live. So we should go to the full thing and replace s
by dX/dsigma and include appropriate integrals and all. Then I'd need to
rethink if this expression is the one to consider.

> I think that locality in spacetime is also highly desirable.

There is an interesting paper arguing that a theory of non-abelian 2-forms
(in particular a theory arising as the worldvolume theory of a stack of
5-branes) cannot be a local field theory. That's

X. Bekaert & M. Henneux & A. Sevrin:
Chiral Forms and their deformations,
http://www.arxiv.org/abs/hep-th/0004049 .

Actually they are only considering chiral=self-dual 2-forms, because the
non-abelian 2-forms living on 5-branes happen to be self-dual

It's not too surprising: A 2-form gauge theory must be about strings and
hence it is not natural to assume locality to be preserved, even more so
when these strings are not weakly coupled (and the self-dual strings living
on 5-branes never are weakly coupled. Being their own weak/strong
electric/magnetic dual they have unit coupling).

I have come across another maybe intersting approach to non-abelian 2-form
theories:

In

O. Ganor,
Six-dimensional tensionless strings in the large N limit,
http://www.arxiv.org/abs/hep-th/9605201

the author tries to do something about the fact that no Lagrangian
description for 5-brane worldvolumes are known (or are there meanwhile?
Lubos might know, he has written papers on that I think), but what he does
is interesting irrespective of the string theory context:

Namely Ganor tries to construct higher dimensional analogs of the "loop
equations" that arise when Yang-Mills is formulated in loop space. He
discusses "surface equations" and tries to guess/derive/motivate their
nature and properties. I haven't yet followed his construction in detail,
but that might be an interesting perspective.

BTW, there is a well-known puzzle concerning worldvolume theories of stacks
of M/NS5-branes: When the number N of branes in the stack is increased
(which for ordinary D-brane scenarios means increasing the size of the gauge
group U(N)) various quantities in the theory scale as N^3, while the
dimension of semi-simple Lie algebras never scales as fast as N^3, with N
the size of the Cartan sub-algebra.

A very recent paper addressing this issue is

D. Berman & J. Harvey:
The self-dual string and anomalies in the M5-brane,
http://www.arxiv.org/abs/hep-th/0408198

This almost seems to call for a construction like yours, where not ordinary
semi-simple Lie-group elements colour the surfaces, but something else. On
the other hand, the non-abelian 2-forms expected to arise on stacks of
M/SN-5 branes are related by a host of compactifications, limits,
reformulations and dualities to non-abelian theories of ordinary gauge
groups, so that it seems very unlikely that the non-abelian 2-form takes
values in something that is not an ordinary semi-simple Lie group.

Indeed, in http://www.arxiv.org/abs/hep-th/9905018 and math.dg/9907034 (which I have not looked at
yet) it is argued that the situations is clarified by using 1-gerbes.

I need to better understand the relation between 1-gerbes and 2-groups. As
far as I can see I am not the only one, but nobody understands this yet
(corrections are very welcome).

There is an old approach that seems to have been constructed in parallel to
Baez' 2-groups, namely

I. Cheplev,
Non-abelian Wilson surfaces,
JHEP 02 (2002) 013 .

Both Baez and Cheplec are inspired/motivated by the apparently
groundbreaking work

L. Breen & W. Messing,:
Differential Geometry of Gerbes,
math.AG/0106083

but Cheplev's notion of a 2-connection is orthogonal (in a surprisingly
literal sense) to that of 2-group theory and instead comes from 1-gerbes.

In 1-gerbe theory a connection is a set of functors from categorified fibers
of the gauge bundle to another such categoriefied fiber. This is supposed to
be the obvious generalization of the fact that an ordinary connection can be
regarded as a set of functors between ordinary bundle fibers.

Cheplev deduces from that a notion of surface holonomy which looks quite
different, but can be seen to be pretty much the same as that of 2-group
theory, using the same definition (secertly) of horizontal and parallal
composition.

But in 2-group theory a 2-connection is a functor not "horizontally" between
fibers, but instead from the 2-groupoid of bigons (or their lattice version
or something similar) to the respective 2-group. That "vertical" in a way.

But checking the citations it seems that Chepelev's constructin was not very
influential. Indeed that might be related to my suspicion that it is just
2-group theory formulated in a more convoluted way. But I haven't checked
this in detail.

Thomas Larsson

Sep16-04, 08:09 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<ci8ves\\$fch\\$1@lfa222122.richmond.edu>...\ n\n> True. Moreover a closed curve has a holonomy depending on the choice of base\n> point. But different base points are related by conjugation with a\n> well-defined element.\n>\n> This generalizes in 2-group theory, as described very nicely in section 2.6\n> of Girelli&Pfeiffer\'s hep-th/0309173. The 2-holonomy depends on on a choice\n> of base edge, much as in your formalism, but different choices of base edges\n> are related by operations generalizing the above conjugation.\n\nOne can consider different operations, some related by conjugation and\nothers not.\n\nThink of the zero meridian (through Greenwich) as an open string with\nendpoints at the poles. Rotating this meridian 24 hours gives you a globe\ncut open at the zero meridian. This 2-holonomy is conjugate to what you\nget if you rotate the dateline around the same axis. Gluing the sphere\ntogether gives you a unique answer, irrespective of the longitude of the\ncut.\n\nHowever, if we introduce an axis through two antipodes on the equator,\nand then rotate half the equator around this new axis, we get a\ncompletely unrelated 2-holonomy, even though the same surface is\ngenerated. This is because the punctures, i.e. the string\'s endpoints,\nmatter from my viewpoint.\n\nSimilarly, an open disk has two reference points where the in and out\nedges meet, i.e. the string\'s endpoint. Rotating the reference points\ngives me a completely new object. Transport of a vertical string in the\nhorizontal direction, and transport of a horizontal string in the\nvertical direction, are two distinct processes even though they happen\nto sweep out the same world surface. Therefore their 2-holonomies should\ndiffer.\n\nHowever, although rotating the reference points generically gives you\nan unrelated object, there is one exception. A 180 degree rotation\ninterchanges the in and out strings, so the 2-holonomy it replaced by\nits inverse.\n\nMust not a 180 degree rotation invert Girelli-Pfeiffer\'s 2-holonomy as\nwell? How is this compatible with conjugation?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<ci8ves$fch$1@lfa222122.richmond.edu>...

> True. Moreover a closed curve has a holonomy depending on the choice of base
> point. But different base points are related by conjugation with a
> well-defined element.
>
> This generalizes in 2-group theory, as described very nicely in section 2.6
> of Girelli&Pfeiffer's http://www.arxiv.org/abs/hep-th/0309173. The 2-holonomy depends on on a choice
> of base edge, much as in your formalism, but different choices of base edges
> are related by operations generalizing the above conjugation.

One can consider different operations, some related by conjugation and
others not.

Think of the zero meridian (through Greenwich) as an open string with
endpoints at the poles. Rotating this meridian 24 hours gives you a globe
cut open at the zero meridian. This 2-holonomy is conjugate to what you
get if you rotate the dateline around the same axis. Gluing the sphere
together gives you a unique answer, irrespective of the longitude of the
cut.

However, if we introduce an axis through two antipodes on the equator,
and then rotate half the equator around this new axis, we get a
completely unrelated 2-holonomy, even though the same surface is
generated. This is because the punctures, i.e. the string's endpoints,
matter from my viewpoint.

Similarly, an open disk has two reference points where the in and out
edges meet, i.e. the string's endpoint. Rotating the reference points
gives me a completely new object. Transport of a vertical string in the
horizontal direction, and transport of a horizontal string in the
vertical direction, are two distinct processes even though they happen
to sweep out the same world surface. Therefore their 2-holonomies should
differ.

However, although rotating the reference points generically gives you
an unrelated object, there is one exception. A 180 degree rotation
interchanges the in and out strings, so the 2-holonomy it replaced by
its inverse.

Must not a 180 degree rotation invert Girelli-Pfeiffer's 2-holonomy as
well? How is this compatible with conjugation?

John Baez

Sep16-04, 08:09 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nHi, Urs - I\'m back in Riverside! The trip from London was long,\nand the flight was delayed by an hour... but it\'s done now, thank god.\n\nIn article <2qr8ejF124mdlU1-100000@uni-berlin.de>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>I need to better understand the relation between 1-gerbes and 2-groups. As\n>far as I can see I am not the only one, but nobody understands this yet\n>(corrections are very welcome).\n\nI understand this pretty well... I developed the theory of\ncategorified bundles with a 2-group as "gauge group" because\ngerbes are a categorification of sheaves rather than bundles,\nand personally I\'m more comfortable with bundles than sheaves.\n\nSo, maybe you can ask a question. Now that I\'m done being a\ntourist in London, we should really be working on our paper,\nnot chatting... but this gerbe stuff is good to understand,\nso it\'d be worth talking about.\n\nThey\'re not very quick to download, but you might have a look at\nmy Winter 2002 and Spring 2002 quantum gravity seminar notes\non categorified gauge theory - Alissa Crans just scanned them\nin:\n\nhttp://math.ucr.edu/home/baez/qg-winter2002/\nhttp://math.ucr.edu/home/baez/qg-spring2002/\n\nThese don\'t talk about gerbes, but they do talk about 2-bundles\nwith a *strict* 2-group as gauge group - that\'s the only sort of\n2-group we\'ve ever discussed. Toby Bartels has generalized a\nlot of this stuff to the case of a *coherent* 2-group - that\'s\na more general sort of 2-group, described here:\n\nhttp://arXiv.org/abs/math.QA/0307200\n\nAnyway, it\'s bedtime now!\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi, Urs - I'm back in Riverside! The trip from London was long,
and the flight was delayed by an hour... but it's done now, thank god.

In article <2qr8ejF124mdlU1-100000@uni-berlin.de>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

>I need to better understand the relation between 1-gerbes and 2-groups. As
>far as I can see I am not the only one, but nobody understands this yet
>(corrections are very welcome).

I understand this pretty well... I developed the theory of
categorified bundles with a 2-group as "gauge group" because
gerbes are a categorification of sheaves rather than bundles,
and personally I'm more comfortable with bundles than sheaves.

So, maybe you can ask a question. Now that I'm done being a
tourist in London, we should really be working on our paper,
not chatting... but this gerbe stuff is good to understand,
so it'd be worth talking about.

They're not very quick to download, but you might have a look at
my Winter 2002 and Spring 2002 quantum gravity seminar notes
on categorified gauge theory - Alissa Crans just scanned them
in:

http://math.ucr.edu/home/baez/qg-winter2002/
http://math.ucr.edu/home/baez/qg-spring2002/

These don't talk about gerbes, but they do talk about 2-bundles
with a *strict* 2-group as gauge group - that's the only sort of
2-group we've ever discussed. Toby Bartels has generalized a
lot of this stuff to the case of a *coherent* 2-group - that's
a more general sort of 2-group, described here:

http://arXiv.org/abs/math.QA/0307200

Anyway, it's bedtime now!

Urs Schreiber

Sep16-04, 08:43 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409160201.486ab612@pos ting.google.com...\n\n> Must not a 180 degree rotation invert Girelli-Pfeiffer\'s 2-holonomy as\n> well?\n\nIt does!\n\n> How is this compatible with conjugation?\n\nDepending on which operation exactly you have in mind (there are a couple of\npossibilities that lead to the same result) you can easily work it out using\nthe formalism in their paper. The exchange law ensures that no matter which\nway you arrive at a "directed plaquette" you get the same result.\n\nFor triangular surfaces you can find this demonstrated in a lot of detail in\nsection 4.2 of Hendryk Pfeiffer\'s solo paper preceding the one with Girelli:\n\nHendryk Pfeiffer:\nHigher gauge theory and a non-Abelian generalization of 2-form\nelectrodynamics,\nhep-th/0304074\n\nas well as on p. 7 of hep-th/0309173 .\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409160201.486ab612@posting.google.c om...

> Must not a 180 degree rotation invert Girelli-Pfeiffer's 2-holonomy as
> well?

It does!

> How is this compatible with conjugation?

Depending on which operation exactly you have in mind (there are a couple of
possibilities that lead to the same result) you can easily work it out using
the formalism in their paper. The exchange law ensures that no matter which
way you arrive at a "directed plaquette" you get the same result.

For triangular surfaces you can find this demonstrated in a lot of detail in
section 4.2 of Hendryk Pfeiffer's solo paper preceding the one with Girelli:

Hendryk Pfeiffer:
Higher gauge theory and a non-Abelian generalization of 2-form
electrodynamics,
http://www.arxiv.org/abs/hep-th/0304074

as well as on p. 7 of http://www.arxiv.org/abs/hep-th/0309173 .

Urs Schreiber

Sep16-04, 09:03 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag\nnews:cibhse\\$5qf\\$1@glue.ucr.edu... >\n\nIn article <2qr8ejF124mdlU1-100000@uni-berlin.de>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n> >I need to better understand the relation between 1-gerbes and 2-groups.\nAs\n> >far as I can see I am not the only one, but nobody understands this yet\n> >(corrections are very welcome).\n>\n> I understand this pretty well...\n\nOh, good. Sorry for the dumb comment then. It was just that I was looking at\n\nIouri Chepelev,\nNon-abelian Wilson surfaces,\nJHEP 02 (2002) 013\n\nseeing that the author is talking about 1-gerbes but essentially using the\nsame mechanisms as in 2-groups that made me wonder if it is known how to\ntranslate from one concept to the other.\n\n> I developed the theory of\n> categorified bundles with a 2-group as "gauge group" because\n> gerbes are a categorification of sheaves rather than bundles,\n> and personally I\'m more comfortable with bundles than sheaves.\n\n> So, maybe you can ask a question. Now that I\'m done being a\n> tourist in London, we should really be working on our paper,\n> not chatting...\n\nSorry, I guess I got carried away.\n\nBut, here is a questions, which I\'ll pose anyway:\n\nIn the above paper (which is essentially the only paper on gerbes that I\nhave ever looked at so far) it is said that a connection on a non-abelian\n1-gerbe "is a functor" (I guess a whole set of functors, really), from one\ncategoified fiber to another.\n\n1) Is that about right?\n2) Has it been studied how from such a connection one gets a 2-connection in\nthe 2-group sense?\n\n(I guess the reason why I said I believe that nobody seems to understand\nthis is that I seem to recall you saying that the concept of 2-connection\nstill needs to be formulated.)\n\n\n> but this gerbe stuff is good to understand, so it\'d be worth talking\nabout.\n\n\nI hope I find the time to read more on this. I\'ll start here:\n\n> http://math.ucr.edu/home/baez/qg-winter2002/\n> http://math.ucr.edu/home/baez/qg-spring2002/\n\n> These don\'t talk about gerbes, but they do talk about 2-bundles\n> with a *strict* 2-group as gauge group - that\'s the only sort of\n> 2-group we\'ve ever discussed. Toby Bartels has generalized a\n> lot of this stuff to the case of a *coherent* 2-group - that\'s\n> a more general sort of 2-group, described here:\n\n\nI once looked at that paper. There it is the notion of group inverse that is\nweakened, i.e the equation\n\n(g)^-1 = g^-1\n\nis weakened to\n\n(g)^-1 x g ~ 1,\n\nwhere "~" denotes an isomorphism. (Right?)\n\nAs you know, I am currently thinking that in order to capture the notion of\nsurface holonomy one instead needs to weaken the exchange law, i.e. replace\nits equality sign with something weaker. I have detailed reasons for believing\nso and have tried to express them, but maybe not in appropriate language.\nIt would be great if you could comment on that. But we need not do that here in\npublic...\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
news:cibhse$5qf$1@glue.ucr.edu...>

In article <2qr8ejF124mdlU1-100000@uni-berlin.de>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
> >I need to better understand the relation between 1-gerbes and 2-groups.
As
> >far as I can see I am not the only one, but nobody understands this yet
> >(corrections are very welcome).
>
> I understand this pretty well...

Oh, good. Sorry for the dumb comment then. It was just that I was looking at

Iouri Chepelev,
Non-abelian Wilson surfaces,
JHEP 02 (2002) 013

seeing that the author is talking about 1-gerbes but essentially using the
same mechanisms as in 2-groups that made me wonder if it is known how to
translate from one concept to the other.

> I developed the theory of
> categorified bundles with a 2-group as "gauge group" because
> gerbes are a categorification of sheaves rather than bundles,
> and personally I'm more comfortable with bundles than sheaves.

> So, maybe you can ask a question. Now that I'm done being a
> tourist in London, we should really be working on our paper,
> not chatting...

Sorry, I guess I got carried away.

But, here is a questions, which I'll pose anyway:

In the above paper (which is essentially the only paper on gerbes that I
have ever looked at so far) it is said that a connection on a non-abelian
1-gerbe "is a functor" (I guess a whole set of functors, really), from one
categoified fiber to another.

1) Is that about right?
2) Has it been studied how from such a connection one gets a 2-connection in
the 2-group sense?

(I guess the reason why I said I believe that nobody seems to understand
this is that I seem to recall you saying that the concept of 2-connection
still needs to be formulated.)

> but this gerbe stuff is good to understand, so it'd be worth talking
about.

I hope I find the time to read more on this. I'll start here:

> http://math.ucr.edu/home/baez/qg-winter2002/
> http://math.ucr.edu/home/baez/qg-spring2002/

> These don't talk about gerbes, but they do talk about 2-bundles
> with a *strict* 2-group as gauge group - that's the only sort of
> 2-group we've ever discussed. Toby Bartels has generalized a
> lot of this stuff to the case of a *coherent* 2-group - that's
> a more general sort of 2-group, described here:

I once looked at that paper. There it is the notion of group inverse that is
weakened, i.e the equation

(g)^-1 = g^-1

is weakened to

(g)^-1 x g ~ 1,

where "~" denotes an isomorphism. (Right?)

As you know, I am currently thinking that in order to capture the notion of
surface holonomy one instead needs to weaken the exchange law, i.e. replace
its equality sign with something weaker. I have detailed reasons for believing
so and have tried to express them, but maybe not in appropriate language.
It would be great if you could comment on that. But we need not do that here in
public...

Urs Schreiber

Sep16-04, 12:09 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409160552.6a765518-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2qr8ejF124mdlU1-100000@uni-berlin.de>...\n>\n> > > With this extra data one can at least formally write down a nice\n> > > non-abelian 3-curvature (Urs knows this, but maybe not Hendryk):\n> > >\n> > > F(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],\n> >\n> > I know that you said this before, but it seems to me that there is some\nwork\n> > needed to give this expression a well-defined meaning.\n>\n> I agree. That\'s why I keep falling back on the well-defined lattice\n> formulation.\n\nHm, ok, let\'s stick to the lattice. Then this is still assuming a space with\ncoordinates x in Z^D and s in (I assume that\'s what you have in mind)\n{+1,,-1}^D or something like that. That\'s a crude version of a discretized\nloop space. In the language that Eric Forgy and have been using\n(http://golem.ph.utexas.edu/string/archives/000394.html#c001472)\nthis is the "2-gon space with target the lattice Z^D". I think that if you\nwant these objects F and Ato depend on s you have to admit that they live on\nsuch a space.\n\nThe next question would be what you want to understand under "d". Is the\ndiscrete exterior derivative on Z^D or on that 2-gon space?\n\nBTW, I know of two papers which consider strings on lattices in maybe\nroughly a way that you are thinking of. One is\n\nR. Easther, B. Greene, M. Jackson, Cosmological String Gas on Orbifolds,\nhep-th/0204099\n\n(in which the Brandenberger-Vafa mechanism is studied for the bosonic string\nusing a numeric lattice simulation to see how classical bosonic strings\nwrapped on an orbifold propagate and unwarp eventually, allowing (that\'s the\nidea of that mechanism) some spatial dimensions to become large)\n\nthe other, which the first is based on is\n\nA. G. Smith and A. Vilenkin, Phys. Rev. D36, 990 (1987).\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409160552.6a765518-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2qr8ejF124mdlU1-100000@uni-berlin.de>...
>
> > > With this extra data one can at least formally write down a nice
> > > non-abelian 3-curvature (Urs knows this, but maybe not Hendryk):
> > >
> > > F(x,s) = dA(x,s) + [s.A(x,s), A(x,s)],
> >
> > I know that you said this before, but it seems to me that there is some
work
> > needed to give this expression a well-defined meaning.
>
> I agree. That's why I keep falling back on the well-defined lattice
> formulation.

Hm, ok, let's stick to the lattice. Then this is still assuming a space with
coordinates x in Z^D and s in (I assume that's what you have in mind)
{+1,,-1}^D or something like that. That's a crude version of a discretized
loop space. In the language that Eric Forgy and have been using
(http://golem.ph.utexas.edu/string/archives/000394.html#c001472)
this is the "2-gon space with target the lattice Z^D". I think that if you
want these objects F and Ato depend on s you have to admit that they live on
such a space.

The next question would be what you want to understand under "d". Is the
discrete exterior derivative on Z^D or on that 2-gon space?

BTW, I know of two papers which consider strings on lattices in maybe
roughly a way that you are thinking of. One is

R. Easther, B. Greene, M. Jackson, Cosmological String Gas on Orbifolds,
http://www.arxiv.org/abs/hep-th/0204099

(in which the Brandenberger-Vafa mechanism is studied for the bosonic string
using a numeric lattice simulation to see how classical bosonic strings
wrapped on an orbifold propagate and unwarp eventually, allowing (that's the
idea of that mechanism) some spatial dimensions to become large)

the other, which the first is based on is

A. G. Smith and A. Vilenkin, Phys. Rev. D36, 990 (1987).

Thomas Larsson

Sep22-04, 10:22 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2r5ce5F162omcU1-100000@uni-berlin.de>...\n> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\n> news:24a23f36.0409182307.677df5d7-100000@posting.google.com...\n>\n> > Recall that (x,s)-space is a poor man\'s version of string space\n> > (with the advantage of being finite-dimensional and thus more\n> > tractable). A 2-connection in ordinary x-space corresponds to a\n> > 1-connection A_i(x,s) in string space. For clarity, I\'ll write out\n> > arguments and indices explicitly. The 1-connection transforms in\n> > the usual ways under gauge transformations in (x,s)-space and\n> > diffeomorphisms in x-space, extended to (x,s)-space by requiring\n> > that s transforms as a vector. This makes the 2-form curvature\n> >\n> > F_{ij}(x,s) = d_i A_j(x,s) - d_j A_i(x,s) + [A_i(x,s), A_j(x,s)]\n> >\n> > (d_i = d/dx^i) well-defined, and we can use it to write down a\n> > nice invariant action which generalizes Yang-Mills.\n>\n> When you go to the continuum limit with this you see that the surface\n> holonomy this induces is independent of the path in loop space or path space\n> (what you call string space) precisely if your A (which is usually called B)\n> is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986) 63).\n\nSorry if I have confused you. I have obviously been pretty confused\nmyself about the continuum limit, by advocating different continuum\nformulations. Let us now leave the lattice model aside - that it is a\np-form generalization of non-abelian lattice gauge theory is completely\nclear, including a well-defined notion of surface holonomy - and turn\nthe continuum theory, as I now think that it should be formulated.\n\nFirst of all: the expression in (x,s)-space is a already a continuum\nformulation. By x-space I mean the N-dimensional space R^N with\ncoordinates x^i, and by (x,s)-space the 2N-dimensional space R^2N with\ncoordinate (x^i, s^i). The gauge transformations generate the usual\nalgebra of gauge transformations in 2N dimensions - nothing new here.\nHowever, we are always interested in objects which also transform\ncovariantly under some group of diffeomorphisms, and the choice of\ndiffeo group matters. There are a few new things, which differ from\nTeitelboim\'s assuptions.\n\nOne is that I only consider diffeomorphisms in N dimensions - arbitrary\ndiffeomorphisms in x, extended to (x,s) by assuming that s transforms as\na vector. If f = f^i(x) d_i is a vector field in x-space, then the\ngenerator of x-diffeos is\n\nL_f = f^i(x) d_i + d_j f^i(x) s^j d/ds^i.\n\nThe second term is new, but it is easy to check that [L_f, L_g] = L_[f,g],\nso we have a realization of the x-diffeo algebra.\n\nX = X_a(x,s) J^a generate gauge transformations in (x,s)-space, where\nJ^a are the generators of a finite-dimensional Lie algebra g; this is\njust the usual gauge generator in 2N dimensions. Something very striking\nis that the exterior derivative and connection A_i(x,s) only have N\ncomponents, although (x,s)-space has 2N dimensions; the choice of diffeo\ngroup really did matter!\n\nAlthough this setup is a well-defined gauge theory, which differs from\n1-gauge theory in 2N dimensions, there is probably still no good surface\nholonomy. The geometrical reason is that a field phi(x,s) should not\ndepend on both x and s, but only on the string it describes. So we\nshould impose the condition s^i d_i phi(x,s) = 0 for all string fields.\nIn particular, we should only consider the subalgebra of gauge\ntransformations which satisfy\n\ns^i d_i X_a(x,s) = 0.\n\nIt is easy to check that this is indeed a subalgebra. The extra\ncondition is not compatible with all diffeos, but only with igl(N)\n(inhomogeneous gl). This is easily fixed: we must covariatize d_i by\nmeans of the Levi-Civita connection, i.e.\n\nd_i -> d_i + Gamma^j_ki s^k d/ds^j.\n\nIgnoring this subtlety, we can now consistenly (i.e. in a way which\ncommutes with the gauge and diffeo symmetries) impose\n\ns^i d_i A_jk(x,s) = s^i d_i F_jkl(x,s) = 0,\n\ns^i A_ij(x,s) = s^i F_ijk(x,s) = 0.\n\nI haven\'t checked Teitelboim\'s paper, but I am pretty sure that he\ndoes not impose these extra conditions. Probably they may make surface\nholonomy unique. Anyway, it is clear that if we don\'t impose these\nconditions, there is an unphysical longitudinal dependence.\n\nOne reason why I think that the condition s^i d_i phi(x,s) = 0 should\narise naturally from the lattice model is that it has the distribution\nsolution\n\nphi(x,s) = delta(s^2) delta(s^3) f(x^1)@1@1\n+ delta(s^1) delta(s^3) 1@g(x^2)@1\n+ delta(s^1) delta(s^2) 1@1@h(x^3),\n\nif g = End(V@V@V). This is precisely the kind of objects that arise\nin my lattice model.\n\n>\n> You can then augment your connection by the adjoint action of a target space\n> 1-form as in hep-th/9710147, hep-th/0207017, hep-th/0407122 (which is\n> implied by gauge transformations on loop space) and find that now the\n> surface holonomy is independent of the path in loop space precisely if the\n> non-abelian 1-form and the 2-form together satisfy a certain condition,\n> which is precisely the condition that these forms define a "weak\n> 2-connection", i.e. a functor from the strict 2-groupoid of bigons to a\n> sesqui-group. A special case of this is a slightly stromger condition which\n> makes this a strict 2-connection (hep-th/0309173), i.e. a functor to a\n> strict 2-group.\n\n2-groups define some kind of gauge symmetry, perhaps similar to BF-theories.\nI thought we agreed that they cannot really define 2-Yang-Mills, because\nthere are too few solutions to the consistency conditions.\n\nHowever, the two notions of 2-gauge theory are closely related. In both\napproaches, one has points x, a 2-form connection B, and a 1-index\nquantity, s or A. The difference is that I consider x and s to be\ncoordinates that describe a string, and the 2-form B is the operator\nthat transports the string at (x,s). In 2-groups, one starts with a\nparticle at x, and transports it with the 1-form field A. One then\nreinterprets the trajectory as a string, which is transported using the\n2-form B. However, one must not forget that this string is really a\nparticle trajectory, and there is already an operator that compares\nadjascent trajectories, namely the curvature F(A). We should therefore\nexpect that B ~ F(A), which is precisely what the consistency conditions\ngive use, except in the somewhat singular abelian case.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2r5ce5F162omcU1-100000@uni-berlin.de>...
> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
> news:24a23f36.0409182307.677df5d7-100000@posting.google.com...
>
> > Recall that (x,s)-space is a poor man's version of string space
> > (with the advantage of being finite-dimensional and thus more
> > tractable). A 2-connection in ordinary x-space corresponds to a
> > 1-connection A_i(x,s) in string space. For clarity, I'll write out
> > arguments and indices explicitly. The 1-connection transforms in
> > the usual ways under gauge transformations in (x,s)-space and
> > diffeomorphisms in x-space, extended to (x,s)-space by requiring
> > that s transforms as a vector. This makes the 2-form curvature
> >
> > F_{ij}(x,s) = d_i A_j(x,s) - d_j A_i(x,s) + [A_i(x,s), A_j(x,s)]
> >
> > (d_i = d/dx^i) well-defined, and we can use it to write down a
> > nice invariant action which generalizes Yang-Mills.
>
> When you go to the continuum limit with this you see that the surface
> holonomy this induces is independent of the path in loop space or path space
> (what you call string space) precisely if your A (which is usually called B)
> is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986) 63).

Sorry if I have confused you. I have obviously been pretty confused
myself about the continuum limit, by advocating different continuum
formulations. Let us now leave the lattice model aside - that it is a
p-form generalization of non-abelian lattice gauge theory is completely
clear, including a well-defined notion of surface holonomy - and turn
the continuum theory, as I now think that it should be formulated.

First of all: the expression in (x,s)-space is a already a continuum
formulation. By x-space I mean the N-dimensional space R^N with
coordinates x^i, and by (x,s)-space the 2N-dimensional space R^{2N} with
coordinate (x^i, s^i). The gauge transformations generate the usual
algebra of gauge transformations in 2N dimensions - nothing new here.
However, we are always interested in objects which also transform
covariantly under some group of diffeomorphisms, and the choice of
diffeo group matters. There are a few new things, which differ from
Teitelboim's assuptions.

One is that I only consider diffeomorphisms in N dimensions - arbitrary
diffeomorphisms in x, extended to (x,s) by assuming that s transforms as
a vector. If f = f^i(x) d_i is a vector field in x-space, then the
generator of x-diffeos is

L_f = f^i(x) d_i + d_j f^i(x) s^j d/ds^i[/itex].

The second term is new, but it is easy to check that [L_f, L_g] = L_[f,g],
so we have a realization of the x-diffeo algebra.

X = X_a(x,s) J^a generate gauge transformations in (x,s)-space, where
J^a are the generators of a finite-dimensional Lie algebra g; this is
just the usual gauge generator in 2N dimensions. Something very striking
is that the exterior derivative and connection A_i(x,s) only have N
components, although (x,s)-space has 2N dimensions; the choice of diffeo
group really did matter!

Although this setup is a well-defined gauge theory, which differs from
1-gauge theory in 2N dimensions, there is probably still no good surface
holonomy. The geometrical reason is that a field \phi(x,s) should not
depend on both x and s, but only on the string it describes. So we
should impose the condition s^i d_i \phi(x,s) = for all string fields.
In particular, we should only consider the subalgebra of gauge
transformations which satisfy

s^i d_i X_a(x,s) = .

It is easy to check that this is indeed a subalgebra. The extra
condition is not compatible with all diffeos, but only with igl(N)
(inhomogeneous gl). This is easily fixed: we must covariatize d_i by
means of the Levi-Civita connection, i.e.

d_i -> d_i + \Gamma^j_{ki} s^k d/ds^j.

Ignoring this subtlety, we can now consistenly (i.e. in a way which
commutes with the gauge and diffeo symmetries) impose

s^i d_i A_{jk}(x,s) = s^i d_i F_{jkl}(x,s) = 0,s^i A_{ij}(x,s) = s^i F_{ijk}(x,s) = .

I haven't checked Teitelboim's paper, but I am pretty sure that he
does not impose these extra conditions. Probably they may make surface
holonomy unique. Anyway, it is clear that if we don't impose these
conditions, there is an unphysical longitudinal dependence.

One reason why I think that the condition s^i d_i \phi(x,s) = should
arise naturally from the lattice model is that it has the distribution
solution

[itex]\phi(x,s) = \delta(s^2) \delta(s^3) f(x^1)@1@1+ \delta(s^1) \delta(s^3) 1@g(x^2)@1+ \delta(s^1) \delta(s^2) 1@1@h(x^3),

if g = End(V@V@V). This is precisely the kind of objects that arise
in my lattice model.

>
> You can then augment your connection by the adjoint action of a target space
> 1-form as in http://www.arxiv.org/abs/hep-th/9710147, http://www.arxiv.org/abs/hep-th/0207017, http://www.arxiv.org/abs/hep-th/0407122 (which is
> implied by gauge transformations on loop space) and find that now the
> surface holonomy is independent of the path in loop space precisely if the
> non-abelian 1-form and the 2-form together satisfy a certain condition,
> which is precisely the condition that these forms define a "weak
> 2-connection", i.e. a functor from the strict 2-groupoid of bigons to a
> sesqui-group. A special case of this is a slightly stromger condition which
> makes this a strict 2-connection (http://www.arxiv.org/abs/hep-th/0309173), i.e. a functor to a
> strict 2-group.

2-groups define some kind of gauge symmetry, perhaps similar to BF-theories.
I thought we agreed that they cannot really define 2-Yang-Mills, because
there are too few solutions to the consistency conditions.

However, the two notions of 2-gauge theory are closely related. In both
approaches, one has points x, a 2-form connection B, and a 1-index
quantity, s or A. The difference is that I consider x and s to be
coordinates that describe a string, and the 2-form B is the operator
that transports the string at (x,s). In 2-groups, one starts with a
particle at x, and transports it with the 1-form field A. One then
reinterprets the trajectory as a string, which is transported using the
2-form B. However, one must not forget that this string is really a
particle trajectory, and there is already an operator that compares
adjascent trajectories, namely the curvature F(A). We should therefore
expect that B ~ F(A), which is precisely what the consistency conditions
give use, except in the somewhat singular abelian case.

Thomas Larsson

Sep23-04, 01:01 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2r5ce5F162omcU1-100000@uni-berlin.de>...\n> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\n> news:24a23f36.0409182307.677df5d7-100000@posting.google.com...\n>\n> > Recall that (x,s)-space is a poor man\'s version of string space\n> > (with the advantage of being finite-dimensional and thus more\n> > tractable). A 2-connection in ordinary x-space corresponds to a\n> > 1-connection A_i(x,s) in string space. For clarity, I\'ll write out\n> > arguments and indices explicitly. The 1-connection transforms in\n> > the usual ways under gauge transformations in (x,s)-space and\n> > diffeomorphisms in x-space, extended to (x,s)-space by requiring\n> > that s transforms as a vector. This makes the 2-form curvature\n> >\n> > F_{ij}(x,s) = d_i A_j(x,s) - d_j A_i(x,s) + [A_i(x,s), A_j(x,s)]\n> >\n> > (d_i = d/dx^i) well-defined, and we can use it to write down a\n> > nice invariant action which generalizes Yang-Mills.\n>\n> When you go to the continuum limit with this you see that the surface\n> holonomy this induces is independent of the path in loop space or path\nspace\n> (what you call string space) precisely if your A (which is usually called\nB)\n> is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986)\n63).\n\nSorry if I have confused you. I have obviously been pretty confused\nmyself about the continuum limit, by advocating different continuum\nformulations. Let us now leave the lattice model aside - that it is a\np-form generalization of non-abelian lattice gauge theory is completely\nclear, including a well-defined notion of surface holonomy - and turn\nthe continuum theory, as I now think that it should be formulated.\n\nFirst of all: the expression in (x,s)-space is a already a continuum\nformulation. By x-space I mean the N-dimensional space R^N with\ncoordinates x^i, and by (x,s)-space the 2N-dimensional space R^2N with\ncoordinate (x^i, s^i). The gauge transformations generate the usual\nalgebra of gauge transformations in 2N dimensions - nothing new here.\nHowever, we are always interested in objects which also transform\ncovariantly under some group of diffeomorphisms, and the choice of\ndiffeo group matters. There are a few new things, which differ from\nTeitelboim\'s assuptions.\n\nOne is that I only consider diffeomorphisms in N dimensions - arbitrary\ndiffeomorphisms in x, extended to (x,s) by assuming that s transforms as\na vector. If f = f^i(x) d_i is a vector field in x-space, then the\ngenerator of x-diffeos is\n\nL_f = f^i(x) d_i + d_j f^i(x) s^j d/ds^i.\n\nThe second term is new, but it is easy to check that [L_f, L_g] = L_[f,g],\nso we have a realization of the x-diffeo algebra.\n\nX = X_a(x,s) J^a generate gauge transformations in (x,s)-space, where\nJ^a are the generators of a finite-dimensional Lie algebra g; this is\njust the usual gauge generator in 2N dimensions. Something very striking\nis that the exterior derivative and connection A_i(x,s) only have N\ncomponents, although (x,s)-space has 2N dimensions; the choice of diffeo\ngroup really did matter!\n\nAlthough this setup is a well-defined gauge theory, which differs from\n1-gauge theory in 2N dimensions, there is probably still no good surface\nholonomy. The geometrical reason is that a field phi(x,s) should not\ndepend on both x and s, but only on the string it describes. So we\nshould impose the condition s^i d_i phi(x,s) = 0 for all string fields.\nIn particular, we should only consider the subalgebra of gauge\ntransformations which satisfy\n\ns^i d_i X_a(x,s) = 0.\n\nIt is easy to check that this is indeed a subalgebra. The extra\ncondition is not compatible with all diffeos, but only with igl(N)\n(inhomogeneous gl). This is easily fixed: we must covariatize d_i by\nmeans of the Levi-Civita connection, i.e.\n\nd_i -> d_i + Gamma^j_ki s^k d/ds^j.\n\nIgnoring this subtlety, we can now consistenly (i.e. in a way which\ncommutes with the gauge and diffeo symmetries) impose\n\ns^i d_i A_jk(x,s) = s^i d_i F_jkl(x,s) = 0,\n\ns^i A_ij(x,s) = s^i F_ijk(x,s) = 0.\n\nI haven\'t checked Teitelboim\'s paper, but I am pretty sure that he\ndoes not impose these extra conditions. Probably they may make surface\nholonomy unique. Anyway, it is clear that if we don\'t impose these\nconditions, there is an unphysical longitudinal dependence.\n\nOne reason why I think that the condition s^i d_i phi(x,s) = 0 should\narise naturally from the lattice model is that it has the distribution\nsolution\n\nphi(x,s) = delta(s^2) delta(s^3) f(x^1)@1@1\n+ delta(s^1) delta(s^3) 1@g(x^2)@1\n+ delta(s^1) delta(s^2) 1@1@h(x^3),\n\nif g = End(V@V@V). This is precisely the kind of objects that arise\nin my lattice model.\n\n>\n> You can then augment your connection by the adjoint action of a target\nspace\n> 1-form as in hep-th/9710147, hep-th/0207017, hep-th/0407122 (which is\n> implied by gauge transformations on loop space) and find that now the\n> surface holonomy is independent of the path in loop space precisely if the\n> non-abelian 1-form and the 2-form together satisfy a certain condition,\n> which is precisely the condition that these forms define a "weak\n> 2-connection", i.e. a functor from the strict 2-groupoid of bigons to a\n> sesqui-group. A special case of this is a slightly stromger condition\nwhich\n> makes this a strict 2-connection (hep-th/0309173), i.e. a functor to a\n> strict 2-group.\n\n2-groups define some kind of gauge symmetry, perhaps similar to BF-theories.\nI thought we agreed that they cannot really define 2-Yang-Mills, because\nthere are too few solutions to the consistency conditions.\n\nHowever, the two notions of 2-gauge theory are closely related. In both\napproaches, one has points x, a 2-form connection B, and a 1-index\nquantity, s or A. The difference is that I consider x and s to be\ncoordinates that describe a string, and the 2-form B is the operator\nthat transports the string at (x,s). In 2-groups, one starts with a\nparticle at x, and transports it with the 1-form field A. One then\nreinterprets the trajectory as a string, which is transported using the\n2-form B. However, one must not forget that this string is really a\nparticle trajectory, and there is already an operator that compares\nadjascent trajectories, namely the curvature F(A). We should therefore\nexpect that B ~ F(A), which is precisely what the consistency conditions\ngive use, except in the somewhat singular abelian case.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2r5ce5F162omcU1-100000@uni-berlin.de>...
> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
> news:24a23f36.0409182307.677df5d7-100000@posting.google.com...
>
> > Recall that (x,s)-space is a poor man's version of string space
> > (with the advantage of being finite-dimensional and thus more
> > tractable). A 2-connection in ordinary x-space corresponds to a
> > 1-connection A_i(x,s) in string space. For clarity, I'll write out
> > arguments and indices explicitly. The 1-connection transforms in
> > the usual ways under gauge transformations in (x,s)-space and
> > diffeomorphisms in x-space, extended to (x,s)-space by requiring
> > that s transforms as a vector. This makes the 2-form curvature
> >
> > F_{ij}(x,s) = d_i A_j(x,s) - d_j A_i(x,s) + [A_i(x,s), A_j(x,s)]
> >
> > (d_i = d/dx^i) well-defined, and we can use it to write down a
> > nice invariant action which generalizes Yang-Mills.
>
> When you go to the continuum limit with this you see that the surface
> holonomy this induces is independent of the path in loop space or path
space
> (what you call string space) precisely if your A (which is usually called
B)
> is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986)
63).

Sorry if I have confused you. I have obviously been pretty confused
myself about the continuum limit, by advocating different continuum
formulations. Let us now leave the lattice model aside - that it is a
p-form generalization of non-abelian lattice gauge theory is completely
clear, including a well-defined notion of surface holonomy - and turn
the continuum theory, as I now think that it should be formulated.

First of all: the expression in (x,s)-space is a already a continuum
formulation. By x-space I mean the N-dimensional space R^N with
coordinates x^i, and by (x,s)-space the 2N-dimensional space R^{2N} with
coordinate (x^i, s^i). The gauge transformations generate the usual
algebra of gauge transformations in 2N dimensions - nothing new here.
However, we are always interested in objects which also transform
covariantly under some group of diffeomorphisms, and the choice of
diffeo group matters. There are a few new things, which differ from
Teitelboim's assuptions.

One is that I only consider diffeomorphisms in N dimensions - arbitrary
diffeomorphisms in x, extended to (x,s) by assuming that s transforms as
a vector. If f = f^i(x) d_i is a vector field in x-space, then the
generator of x-diffeos is

L_f = f^i(x) d_i + d_j f^i(x) s^j d/ds^i[/itex].

The second term is new, but it is easy to check that [L_f, L_g] = L_[f,g],
so we have a realization of the x-diffeo algebra.

X = X_a(x,s) J^a generate gauge transformations in (x,s)-space, where
J^a are the generators of a finite-dimensional Lie algebra g; this is
just the usual gauge generator in 2N dimensions. Something very striking
is that the exterior derivative and connection A_i(x,s) only have N
components, although (x,s)-space has 2N dimensions; the choice of diffeo
group really did matter!

Although this setup is a well-defined gauge theory, which differs from
1-gauge theory in 2N dimensions, there is probably still no good surface
holonomy. The geometrical reason is that a field \phi(x,s) should not
depend on both x and s, but only on the string it describes. So we
should impose the condition s^i d_i \phi(x,s) = for all string fields.
In particular, we should only consider the subalgebra of gauge
transformations which satisfy

s^i d_i X_a(x,s) = .

It is easy to check that this is indeed a subalgebra. The extra
condition is not compatible with all diffeos, but only with igl(N)
(inhomogeneous gl). This is easily fixed: we must covariatize d_i by
means of the Levi-Civita connection, i.e.

d_i -> d_i + \Gamma^j_{ki} s^k d/ds^j.

Ignoring this subtlety, we can now consistenly (i.e. in a way which
commutes with the gauge and diffeo symmetries) impose

s^i d_i A_{jk}(x,s) = s^i d_i F_{jkl}(x,s) = 0,s^i A_{ij}(x,s) = s^i F_{ijk}(x,s) = .

I haven't checked Teitelboim's paper, but I am pretty sure that he
does not impose these extra conditions. Probably they may make surface
holonomy unique. Anyway, it is clear that if we don't impose these
conditions, there is an unphysical longitudinal dependence.

One reason why I think that the condition s^i d_i \phi(x,s) = should
arise naturally from the lattice model is that it has the distribution
solution

[itex]\phi(x,s) = \delta(s^2) \delta(s^3) f(x^1)@1@1+ \delta(s^1) \delta(s^3) 1@g(x^2)@1+ \delta(s^1) \delta(s^2) 1@1@h(x^3),

if g = End(V@V@V). This is precisely the kind of objects that arise
in my lattice model.

>
> You can then augment your connection by the adjoint action of a target
space
> 1-form as in http://www.arxiv.org/abs/hep-th/9710147, http://www.arxiv.org/abs/hep-th/0207017, http://www.arxiv.org/abs/hep-th/0407122 (which is
> implied by gauge transformations on loop space) and find that now the
> surface holonomy is independent of the path in loop space precisely if the
> non-abelian 1-form and the 2-form together satisfy a certain condition,
> which is precisely the condition that these forms define a "weak
> 2-connection", i.e. a functor from the strict 2-groupoid of bigons to a
> sesqui-group. A special case of this is a slightly stromger condition
which
> makes this a strict 2-connection (http://www.arxiv.org/abs/hep-th/0309173), i.e. a functor to a
> strict 2-group.

2-groups define some kind of gauge symmetry, perhaps similar to BF-theories.
I thought we agreed that they cannot really define 2-Yang-Mills, because
there are too few solutions to the consistency conditions.

However, the two notions of 2-gauge theory are closely related. In both
approaches, one has points x, a 2-form connection B, and a 1-index
quantity, s or A. The difference is that I consider x and s to be
coordinates that describe a string, and the 2-form B is the operator
that transports the string at (x,s). In 2-groups, one starts with a
particle at x, and transports it with the 1-form field A. One then
reinterprets the trajectory as a string, which is transported using the
2-form B. However, one must not forget that this string is really a
particle trajectory, and there is already an operator that compares
adjascent trajectories, namely the curvature F(A). We should therefore
expect that B ~ F(A), which is precisely what the consistency conditions
give use, except in the somewhat singular abelian case.

Urs Schreiber

Sep23-04, 03:17 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0409230806.5ffb268d-100000@posting.google.com...\n> Thomas Larsson <thomas_larsson_01@hotmail.com> wrote in message\nnews:<24a23f36.0409220540.2b54d4dc-100000@posting.google.com>...\n>\n> > Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2r5ce5F162omcU1-100000@uni-berlin.de>...\n> > > When you go to the continuum limit with this you see that the surface\n> > > holonomy this induces is independent of the path in loop space or path\nspace\n> > > (what you call string space) precisely if your A (which is usually\ncalled B)\n> > > is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986)\n63).\n> >\n> > Sorry if I have confused you. I have obviously been pretty confused\n> > myself about the continuum limit, by advocating different continuum\n> > formulations. Let us now leave the lattice model aside - that it is a\n> > p-form generalization of non-abelian lattice gauge theory is completely\n> > clear, including a well-defined notion of surface holonomy - and turn\n> > the continuum theory, as I now think that it should be formulated.\n>\n> Aha, now I realize what you are saying. Yes of course, my\n> 2-holonomy does depend on the path in string space, even on the\n> lattice. In fact, even a single plaquette has four 2-holonomies\n> (two inverse pairs), which I call NW, NE, SW and SE. The\n> continuum formulation also depends on s; I can write\n> A_i(x,s) = B_ij(x) s^j, but this is not a gauge-invariant\n> statement.\n>\n> However, as I pointed out before, we have exactly the same kind\n> of dependence in 1-gauge theory. A Wilson line does not only\n> depend on the line, but also on direction. Hence we must specify\n> the in side and the out side of the line. The two directed Wilson\n> lines are *not* related by conjugation, but rather by inversion.\n> Conjugation does not change the determinant (if the endpoints are\n> at the same place) but inversion does, unless det = +-1.\n\n\nThere really is a problem here which should not be disucssed away: A\n1-holonomy does not depend on the *reparameterization* of the loop/path. The\nissues with orientation and choice of base point are quite independent of\nthat.\n\nIn your proposal the 2-holonomies will depend on the very parameterization\nof the surfaces, even when their base edge and orientation is kept fix. This\nis what Teitelboim showed for the simple loop/path space connection with\njust the 2-form in it.\n\nIf you are not thinking in terms of this connection considered by Teitelboim\nbut about something that lives on a space {(x,s) \\in R^n x R^n} or something\nlike that then we are talking past each other, because that has no obvious\nrelation to ordinary surface holonomy.\n\nIn constructing a 1-holonomy it is very easy to ensure reparameterization\ninvariance. For 2-holonomies one runs into surprisingly strong constraints.\n\n\n> Similarly, my 2-holonomy depends not only on the surface itself,\n> but on the division of the boundary into in and out sides. But\n> this is all, on the lattice at least. So for a surface decorated\n> with an in side, the 2-holonomy is unique. I always talk about\n> such decorate surfaces, just as I always talk about decorated, or\n> directed, Wilson lines. In this way we may have talked past each\n> other.\n\n\nIt seems that here you are again talking about your lattice model. I can\'t\nsee in which sense it is related to the (x,s)-space constructions that we\ntalked about in the previous posts and in fact, frankly speaking, I have the\nimpression that it is not very well defined itself.\n\n\n> But I think you are right when 2-groups define a unique\n> 2-holonomy in the stronger sense that it depends only on the\n> surface and not the decoration. But this is surely too strong a\n> condition. You said before that moving the reference points on\n> the boundary only gives a conjugation. This is problematic,\n> because moving the reference points 180 degrees should give\n> inversion, and conjugation does not give you inversion. Again,\n> check the determinants!\n\n\nI have answered that before, please see\n\nhttp://groups.google.de/groups?selm=cic1pd%24h0g%241%40lfa222122.richmond. edu .\n\nThe 180 degrees roation of course gives the inverse not of the original\nbigon but of its vertical mirror image, as it should be, because in and out\nvertices are exchanged. Have a look at Hendryk Pfeiffer\'s papers where lots\nof figures make this very clear.\n\nA very similar "recentering" is considered in gerbe theory, see for instance\np. 29 of the very nice paper\n\nM. Mackaay & R. Picken:\nHolonomy and parallel transport for Abelian gerbes,\nmath.DG0007053.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409230806.5ffb268d-100000@posting.google.com...
> Thomas Larsson <thomas_larsson_01@hotmail.com> wrote in message
news:<24a23f36.0409220540.2b54d4dc-100000@posting.google.com>...
>
> > Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2r5ce5F162omcU1-100000@uni-berlin.de>...
> > > When you go to the continuum limit with this you see that the surface
> > > holonomy this induces is independent of the path in loop space or path
space
> > > (what you call string space) precisely if your A (which is usually
called B)
> > > is abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986)
63).
> >
> > Sorry if I have confused you. I have obviously been pretty confused
> > myself about the continuum limit, by advocating different continuum
> > formulations. Let us now leave the lattice model aside - that it is a
> > p-form generalization of non-abelian lattice gauge theory is completely
> > clear, including a well-defined notion of surface holonomy - and turn
> > the continuum theory, as I now think that it should be formulated.
>
> Aha, now I realize what you are saying. Yes of course, my
> 2-holonomy does depend on the path in string space, even on the
> lattice. In fact, even a single plaquette has four 2-holonomies
> (two inverse pairs), which I call NW, NE, SW and SE. The
> continuum formulation also depends on s; I can write
> A_i(x,s) = B_{ij}(x) s^j, but this is not a gauge-invariant
> statement.
>
> However, as I pointed out before, we have exactly the same kind
> of dependence in 1-gauge theory. A Wilson line does not only
> depend on the line, but also on direction. Hence we must specify
> the in side and the out side of the line. The two directed Wilson
> lines are *not* related by conjugation, but rather by inversion.
> Conjugation does not change the determinant (if the endpoints are
> at the same place) but inversion does, unless det = +-1.

There really is a problem here which should not be disucssed away: A
1-holonomy does not depend on the *reparameterization* of the loop/path. The
issues with orientation and choice of base point are quite independent of
that.

In your proposal the 2-holonomies will depend on the very parameterization
of the surfaces, even when their base edge and orientation is kept fix. This
is what Teitelboim showed for the simple loop/path space connection with
just the 2-form in it.

If you are not thinking in terms of this connection considered by Teitelboim
but about something that lives on a space {(x,s) \in R^n x R^n} or something
like that then we are talking past each other, because that has no obvious
relation to ordinary surface holonomy.

In constructing a 1-holonomy it is very easy to ensure reparameterization
invariance. For 2-holonomies one runs into surprisingly strong constraints.

> Similarly, my 2-holonomy depends not only on the surface itself,
> but on the division of the boundary into in and out sides. But
> this is all, on the lattice at least. So for a surface decorated
> with an in side, the 2-holonomy is unique. I always talk about
> such decorate surfaces, just as I always talk about decorated, or
> directed, Wilson lines. In this way we may have talked past each
> other.

It seems that here you are again talking about your lattice model. I can't
see in which sense it is related to the (x,s)-space constructions that we
talked about in the previous posts and in fact, frankly speaking, I have the
impression that it is not very well defined itself.

> But I think you are right when 2-groups define a unique
> 2-holonomy in the stronger sense that it depends only on the
> surface and not the decoration. But this is surely too strong a
> condition. You said before that moving the reference points on
> the boundary only gives a conjugation. This is problematic,
> because moving the reference points 180 degrees should give
> inversion, and conjugation does not give you inversion. Again,
> check the determinants!

I have answered that before, please see

http://groups.google.de/groups?selm=cic1pd%24h0g%241%40lfa222122.richmond. edu .

The 180 degrees roation of course gives the inverse not of the original
bigon but of its vertical mirror image, as it should be, because in and out
vertices are exchanged. Have a look at Hendryk Pfeiffer's papers where lots
of figures make this very clear.

A very similar "recentering" is considered in gerbe theory, see for instance
p. 29 of the very nice paper

M. Mackaay & R. Picken:
Holonomy and parallel transport for Abelian gerbes,
math.DG0007053.