Re: Two notions of 2-multiplication [Archive]

Thomas Larsson

Sep15-04, 06:10 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:<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 than\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 to\n> make the following comment:\n>\n> It seems vital to me to preserve the assumption that elements from the gauge\n> group (like SU(N) or something) is associated with a surface elemt in 2-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.

Thomas Larsson

Sep16-04, 09:44 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 news:<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 work\n> needed to give this expression a well-defined meaning.\n\nI agree. That\'s why I keep falling back on the well-defined lattice\nformulation.\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:<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.

Thomas Larsson

Sep17-04, 07:10 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:<2qtrtfF1381kpU1-100000@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\n> > > work 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\nLet me retract a little. In a given coordinate system, the above\nequation is completely well-defined - more so than expressions in\nloop space. Loop space is infinite-dimensional and giving a precise\nmeaning to the space of functionals over it is nasty, especially\nif you want to field theory which is delicate even for fields\nover spacetime.\n\nBut here we deal with ordinary functions of 2n variables (x^i, s^i).\nIn components, the equation above reads\n\nF_ijk = d_i A_jk + [s^l A_il, A_jk] + anti-symmetrization,\n\nwhere d_i = d/dx^i. Moreover, we can define a covariant derivative D\n= d + s.A. Thus F = DA and [D,D] = s.F. I haven\'t seen a notion of a\ncovariant derivative in 2-group theory.\n\nIn section 4 of http://www.arxiv.org/abs/math-ph/0205017 I also\ndiscussed the transformation laws under diffeomorphisms and string\ngauge transformations. The algebras are written down in (4.6) and (4.7).\nNote that understanding the transformation properties is the only\nthing that is missing; once we have these rules and the equations in\none coordinate system above, the theory is well defined.\n\nHowever, I was careless when I wrote down these algebras. The problem\nis that A_ij takes values in V_i @ V_j, and F_ijk in V_i @ V_j @ V_k,\nso A_ij acts as the unit matrix in V_k. Hence one must somehow make\nthe the gauge transformations act in these different factors, and I\ndon\'t see how to do that right now.\n\n> Hm, ok, let\'s stick to the lattice. Then this is still assuming a space with\n> coordinates x in Z^D and s in (I assume that\'s what you have in mind)\n> {+1,,-1}^D or something like that.\n\nI thought we had reached agreement on what how lattice model looks.\nTo take the continuum limit, the plaquette variable is assumed to\nbe of the form\n\nU_ij = exp(i a^2 A_ij)\n\nSince A_ij is a matrix in End(V@V), the exponential is well defined\n(namely the exponential of a matrix), and U_ji = (U_ij)^-1. An\nelementary cube involves not only plaquettes at x, but also\nthe neighbors at x^i + a e^i (unit vector)\n\nU_ij(x + ae_k) = exp(i a^2 (A_ij + a e^k d_k A_ij)).\n\nNow one can compute the 3-curvature along the (1,1,1)-diagonal\n\nU_12(x) U_13(x) U_23(x) U_21(x+ae^3) U_31(x+ae^2) U_32(x+ae^1)\n\nusing the BHC formula. If everything is done right one should get\nexp(i a^3 F_123), where\n\nF_123 ~ d_1 A_23 + d_2 A_31 + d_3 A_12\n+ [A_12, A_23] + [A_23, A_31] + [A_31, A_12].\n\nOr something very similar, at least. This is the formula above\nfor s = (1,1,1), the direction of the relevant diagonal.\n\nSo this shows how to get the continuum limit, and also that\nA in End(V@V) and F in End(V@V@V).\n\n> BTW, I know of two papers which consider strings on lattices in maybe\n> roughly a way that you are thinking of. One is\n>\n> R. Easther, B. Greene, M. Jackson, Cosmological String Gas on Orbifolds,\n> hep-th/0204099\n\nThere are many lattice models, most of which are as unrelated as\ndifferent continuum models. I don\'t see any relation here. E.g.,\nwhere is the two-form connection?\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:<2qtrtfF1381kpU1-100000@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.

Let me retract a little. In a given coordinate system, the above
equation is completely well-defined - more so than expressions in
loop space. Loop space is infinite-dimensional and giving a precise
meaning to the space of functionals over it is nasty, especially
if you want to field theory which is delicate even for fields
over spacetime.

But here we deal with ordinary functions of 2n variables (x^i, s^i).
In components, the equation above reads

F_{ijk} = d_i A_{jk} + [s^l A_{il}, A_{jk}] +[/itex] anti-symmetrization,

where d_i = d/dx^i. Moreover, we can define a covariant derivative D
= d + s.A. Thus F = DA and [D,D] = s.F. I haven't seen a notion of a
covariant derivative in 2-group theory.

In section 4 of http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017 I also
discussed the transformation laws under diffeomorphisms and string
gauge transformations. The algebras are written down in (4.6) and (4.7).
Note that understanding the transformation properties is the only
thing that is missing; once we have these rules and the equations in
one coordinate system above, the theory is well defined.

However, I was careless when I wrote down these algebras. The problem
is that A_{ij} takes values in V_i @ V_j, and F_{ijk} in V_i @ V_j @ V_k,
so A_{ij} acts as the unit matrix in V_k. Hence one must somehow make
the the gauge transformations act in these different factors, and I
don't see how to do that right now.

> 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.

I thought we had reached agreement on what how lattice model looks.
To take the continuum limit, the plaquette variable is assumed to
be of the form

[itex]U_{ij} = \exp(i a^2 A_{ij})

Since A_{ij} is a matrix in End(V@V), the exponential is well defined
(namely the exponential of a matrix), and U_{ji} = (U_{ij})^-1. An
elementary cube involves not only plaquettes at x, but also
the neighbors at x^i + a e^i (unit vector)

U_{ij}(x + ae_k) = \exp(i a^2 (A_{ij} + a e^k d_k A_{ij})).

Now one can compute the 3-curvature along the (1,1,1)-diagonal

U_{12}(x) U_{13}(x) U_{23}(x) U_{21}(x+ae^3) U_{31}(x+ae^2) U_{32}(x+ae^1)

using the BHC formula. If everything is done right one should get
\exp(i a^3 F_{123}), where

F_{123} ~ d_1 A_{23} + d_2 A_{31} + d_3 A_{12}+ [A_{12}, A_{23}] + [A_{23}, A_{31}] + [A_{31}, A_{12}].

Or something very similar, at least. This is the formula above
for s = (1,1,1), the direction of the relevant diagonal.

So this shows how to get the continuum limit, and also that
A in End(V@V) and F in End(V@V@V).

> 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

There are many lattice models, most of which are as unrelated as
different continuum models. I don't see any relation here. E.g.,
where is the two-form connection?

Urs Schreiber

Sep17-04, 02:27 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:24a23f36.0409170125.48d3ef58-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message\nnews:<2qtrtfF1381kpU1-100000@uni-berlin.de>...\n\n\n> This is the formula above\n> for s = (1,1,1), the direction of the relevant diagonal.\n\n\nIs there a reason for restricting the tangent to the lattice string to\ns=(1,1,1) other than that this gives the result that you seem to expect\nfrom the formula that you proposed?\n\n\n> > BTW, I know of two papers which consider strings on lattices in maybe\n> > roughly a way that you are thinking of. One is\n> >\n> > R. Easther, B. Greene, M. Jackson, Cosmological String Gas on Orbifolds,\n> > hep-th/0204099\n>\n> There are many lattice models, most of which are as unrelated as\n> different continuum models. I don\'t see any relation here. E.g.,\n> where is the two-form connection?\n\n\nThere is no 2-form connection in these papers, but there is a coherent\npicture of strings on the lattice in these papers with strings that are\nallowed to move freely and have various tangents at various points. This is\nwhat one expects to be relavent in a strings-on-the-lattice setup that you\nare apparently thinking about. I am trying to understand the broad\nconcepts of your construction. Currently I am unsure about what you have\nin mind when talking about that degree of freedom called s. Is it supposed\nto be the constant s=(1,1,1) or can it vary?\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:24a23f36.0409170125.48d3ef58-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2qtrtfF1381kpU1-100000@uni-berlin.de>...

> This is the formula above
> for s = (1,1,1), the direction of the relevant diagonal.

Is there a reason for restricting the tangent to the lattice string to
s=(1,1,1) other than that this gives the result that you seem to expect
from the formula that you proposed?

> > 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
>
> There are many lattice models, most of which are as unrelated as
> different continuum models. I don't see any relation here. E.g.,
> where is the two-form connection?

There is no 2-form connection in these papers, but there is a coherent
picture of strings on the lattice in these papers with strings that are
allowed to move freely and have various tangents at various points. This is
what one expects to be relavent in a strings-on-the-lattice setup that you
are apparently thinking about. I am trying to understand the broad
concepts of your construction. Currently I am unsure about what you have
in mind when talking about that degree of freedom called s. Is it supposed
to be the constant s=(1,1,1) or can it vary?

Thomas Larsson

Sep18-04, 07:07 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:<2r0s0oF14h2hcU1-100000@uni-berlin.de>...\n\n> > TL: This is the formula above\n> > for s = (1,1,1), the direction of the relevant diagonal.\n>\n> UrS: Is there a reason for restricting the tangent to the lattice string to\n> s=(1,1,1) other than that this gives the result that you seem to expect\n> from the formula that you proposed?\n\nA cube has eight diagonals, s = (+-1, +-1, +-1). I pick one of those.\n\nThere is a flaw in my argument, though. The cubes diagonal is really\ns = (a,a,a). Since we want s to have a finite limit when a -> 0, we need\nto rescale the non-abelian terms with 1/a. Otherwise we would get a\nresult of the form\n\na^3 F = a^3 dA + a^4 [A,A],\n\nwhich would kill the non-abelian term in the limit. If one sets s = 0\nin my formula for F, only the abelian part remains.\n\n> There is no 2-form connection in these papers, but there is a coherent\n> picture of strings on the lattice in these papers with strings that are\n> allowed to move freely and have various tangents at various points. This is\n> what one expects to be relavent in a strings-on-the-lattice setup that you\n> are apparently thinking about. I am trying to understand the broad\n> concepts of your construction.\n\nPhi^4 theory and general relativity are both continuum field theories.\nWhy should two arbitrary lattice models involving strings be any more\nsimilar than that?\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:<2r0s0oF14h2hcU1-100000@uni-berlin.de>...

> > TL: This is the formula above
> > for s = (1,1,1), the direction of the relevant diagonal.
>
> UrS: Is there a reason for restricting the tangent to the lattice string to
> s=(1,1,1) other than that this gives the result that you seem to expect
> from the formula that you proposed?

A cube has eight diagonals, s = (+-1, +-1, +-1). I pick one of those.

There is a flaw in my argument, though. The cubes diagonal is really
s = (a,a,a). Since we want s to have a finite limit when a -> 0, we need
to rescale the non-abelian terms with 1/a. Otherwise we would get a
result of the form

a^3 F = a^3 dA + a^4 [A,A],

which would kill the non-abelian term in the limit. If one sets s =
in my formula for F, only the abelian part remains.

> There is no 2-form connection in these papers, but there is a coherent
> picture of strings on the lattice in these papers with strings that are
> allowed to move freely and have various tangents at various points. This is
> what one expects to be relavent in a strings-on-the-lattice setup that you
> are apparently thinking about. I am trying to understand the broad
> concepts of your construction.

\Phi^4 theory and general relativity are both continuum field theories.
Why should two arbitrary lattice models involving strings be any more
similar than that?

Thomas Larsson

Sep19-04, 06:12 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> wrote in message news:<24a23f36.0409170125.48d3ef58-100000@posting.google.com>...\n> Let me retract a little. In a given coordinate system, the above\n> equation is completely well-defined - more so than expressions in\n> loop space. Loop space is infinite-dimensional and giving a precise\n> meaning to the space of functionals over it is nasty, especially\n> if you want to field theory which is delicate even for fields\n> over spacetime.\n>\n> But here we deal with ordinary functions of 2n variables (x^i, s^i).\n> In components, the equation above reads\n>\n> F_ijk = d_i A_jk + [s^l A_il, A_jk] + anti-symmetrization,\n>\n> where d_i = d/dx^i. Moreover, we can define a covariant derivative D\n> = d + s.A. Thus F = DA and [D,D] = s.F. I haven\'t seen a notion of a\n> covariant derivative in 2-group theory.\n>\n> In section 4 of http://www.arxiv.org/abs/math-ph/0205017 I also\n> discussed the transformation laws under diffeomorphisms and string\n> gauge transformations. The algebras are written down in (4.6) and (4.7).\n> Note that understanding the transformation properties is the only\n> thing that is missing; once we have these rules and the equations in\n\n\n\nThis is sort of embarassing: I had a look at the continuum section\n4 of math-ph/0205017, and I realized that I remembered my own\ncalculations wrongly. Or rather, I remembered something that I\ntried to do but didn\'t quite work out. Here is how it should be.\n\nRecall that (x,s)-space is a poor man\'s version of string space\n(with the advantage of being finite-dimensional and thus more\ntractable). A 2-connection in ordinary x-space corresponds to a\n1-connection A_i(x,s) in string space. For clarity, I\'ll write out\narguments and indices explicitly. The 1-connection transforms in\nthe usual ways under gauge transformations in (x,s)-space and\ndiffeomorphisms in x-space, extended to (x,s)-space by requiring\nthat s transforms as a vector. This makes the 2-form curvature\n\nF_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\nnice invariant action which generalizes Yang-Mills.\n\nWe should probably also require that\n\ns^i A_i(x,s) = s^i F_ij(x,s) = 0\n\n(A does parallel transport perpendicular to s), and that\n\ns^i D_i phi(x,s) = 0\n\nfor every string field phi (D_i = d_i + A_i(x,s) = covariant\nderivative); this says that if you move the point x in the\nstring\'s direction you still describe the same string.\n\nNow assume that A_i(x,s) = s^j A_ij(x). We then have F_ij(x,s) =\ns^k F_ijk(x), where\n\nF_ijk(x) = d_i A_jk(x) + [A_ij(x), s^m A_km(x)] + anti-symm.\n\nThis is the 3-form curvature of the 2-form connection. However,\nthe condition A_i = s^j A_ij is not invariant under the gauge\ngroup. So we have a well-defined gauge theory, which is a\nnon-abelian generalization of 2-form electromagnetism in a\nparticular gauge. And this is enough, isn\'t it?\n\nThis shows one major benefit of writing papers: you can look up\nthe right answer when you grow older and dumber and have forgotten\nwhat you once did. For details I refer to my paper, written at a\ntime when I evidently was less senile than I am now.\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> wrote in message news:<24a23f36.0409170125.48d3ef58-100000@posting.google.com>...
> Let me retract a little. In a given coordinate system, the above
> equation is completely well-defined - more so than expressions in
> loop space. Loop space is infinite-dimensional and giving a precise
> meaning to the space of functionals over it is nasty, especially
> if you want to field theory which is delicate even for fields
> over spacetime.
>
> But here we deal with ordinary functions of 2n variables (x^i, s^i).
> In components, the equation above reads
>
> F_{ijk} = d_i A_{jk} + [s^l A_{il}, A_{jk}] + anti-symmetrization,
>
> where d_i = d/dx^i. Moreover, we can define a covariant derivative D
> = d + s.A. Thus F = DA and [D,D] = s.F. I haven't seen a notion of a
> covariant derivative in 2-group theory.
>
> In section 4 of http://www.arxiv.org/abs/http://www.arxiv.org/abs/math-ph/0205017 I also
> discussed the transformation laws under diffeomorphisms and string
> gauge transformations. The algebras are written down in (4.6) and (4.7).
> Note that understanding the transformation properties is the only
> thing that is missing; once we have these rules and the equations in

This is sort of embarassing: I had a look at the continuum section
4 of http://www.arxiv.org/abs/math-ph/0205017, and I realized that I remembered my own
calculations wrongly. Or rather, I remembered something that I
tried to do but didn't quite work out. Here is how it should be.

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)[/itex] well-defined, and we can use it to write down a
nice invariant action which generalizes Yang-Mills.

We should probably also require that

[itex]s^i A_i(x,s) = s^i F_{ij}(x,s) =

(A does parallel transport perpendicular to s), and that

s^i D_i \phi(x,s) =

for every string field \phi (D_i = d_i + A_i(x,s) = covariant
derivative); this says that if you move the point x in the
string's direction you still describe the same string.

Now assume that A_i(x,s) = s^j A_{ij}(x). We then have F_{ij}(x,s) =s^k F_{ijk}(x), where

F_{ijk}(x) = d_i A_{jk}(x) + [A_{ij}(x), s^m A_{km}(x)] + anti-symm.

This is the 3-form curvature of the 2-form connection. However,
the condition A_i = s^j A_{ij} is not invariant under the gauge
group. So we have a well-defined gauge theory, which is a
non-abelian generalization of 2-form electromagnetism in a
particular gauge. And this is enough, isn't it?

This shows one major benefit of writing papers: you can look up
the right answer when you grow older and dumber and have forgotten
what you once did. For details I refer to my paper, written at a
time when I evidently was less senile than I am now.

Urs Schreiber

Sep19-04, 07:38 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:24a23f36.0409182307.677df5d7-100000@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\nWhen you go to the continuum limit with this you see that the surface\nholonomy 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)\nis abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986) 63).\n\nYou can then augment your connection by the adjoint action of a target space\n1-form as in hep-th/9710147, hep-th/0207017, hep-th/0407122 (which is\nimplied by gauge transformations on loop space) and find that now the\nsurface holonomy is independent of the path in loop space precisely if the\nnon-abelian 1-form and the 2-form together satisfy a certain condition,\nwhich is precisely the condition that these forms define a "weak\n2-connection", i.e. a functor from the strict 2-groupoid of bigons to a\nsesqui-group. A special case of this is a slightly stromger condition which\nmakes this a strict 2-connection (hep-th/0309173), i.e. a functor to a\nstrict 2-group.\n\nThe objects on loop space can be shown to be well-defined precisely if the\n"background" 1+2 form gauge fields satisfy certain equations of motion. For\nthe case B=0 this is shown in hep-th/0312260 for the non-abelian and in\nhep-th/9909027 and JHEP 04 (2000) 023 for the abelian case. This easily\ngeneralizes to non-abelian and nontrivial B, as discussed in hep-th/0407122.\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:24a23f36.0409182307.677df5d7-100000@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).

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.

The objects on loop space can be shown to be well-defined precisely if the
"background" 1+2 form gauge fields satisfy certain equations of motion. For
the case B=0 this is shown in http://www.arxiv.org/abs/hep-th/0312260 for the non-abelian and in
http://www.arxiv.org/abs/hep-th/9909027 and JHEP 04 (2000) 023 for the abelian case. This easily
generalizes to non-abelian and nontrivial B, as discussed in http://www.arxiv.org/abs/hep-th/0407122.

Urs Schreiber

Sep19-04, 08:57 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:24a23f36.0409182307.677df5d7-100000@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\nWhen you go to the continuum limit with this you see that the surface\nholonomy 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)\nis abelian (first noticed by C. Teitelboim in Phys. Lett. B 167 (1986) 63).\n\nYou can then augment your connection by the adjoint action of a target space\n1-form as in hep-th/9710147, hep-th/0207017, hep-th/0407122 (which is\nimplied by gauge transformations on loop space) and find that now the\nsurface holonomy is independent of the path in loop space precisely if the\nnon-abelian 1-form and the 2-form together satisfy a certain condition,\nwhich is precisely the condition that these forms define a "weak\n2-connection", i.e. a functor from the strict 2-groupoid of bigons to a\nsesqui-group. A special case of this is a slightly stromger condition which\nmakes this a strict 2-connection (hep-th/0309173), i.e. a functor to a\nstrict 2-group.\n\nThe objects on loop space can be shown to be well-defined precisely if the\n"background" 1+2 form gauge fields satisfy certain equations of motion. For\nthe case B=0 this is shown in hep-th/0312260 for the non-abelian and in\nhep-th/9909027 and JHEP 04 (2000) 023 for the abelian case. This easily\ngeneralizes to non-abelian and nontrivial B, as discussed in hep-th/0407122.\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:24a23f36.0409182307.677df5d7-100000@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).

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.

The objects on loop space can be shown to be well-defined precisely if the
"background" 1+2 form gauge fields satisfy certain equations of motion. For
the case B=0 this is shown in http://www.arxiv.org/abs/hep-th/0312260 for the non-abelian and in
http://www.arxiv.org/abs/hep-th/9909027 and JHEP 04 (2000) 023 for the abelian case. This easily
generalizes to non-abelian and nontrivial B, as discussed in http://www.arxiv.org/abs/hep-th/0407122.

Thomas Larsson

Sep23-04, 11:32 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> wrote in message news:<24a23f36.0409220540.2b54d4dc-100000@posting.google.com>...\n\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<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 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>\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\nAha, now I realize what you are saying. Yes of course, my\n2-holonomy does depend on the path in string space, even on the\nlattice. In fact, even a single plaquette has four 2-holonomies\n(two inverse pairs), which I call NW, NE, SW and SE. The\ncontinuum formulation also depends on s; I can write\nA_i(x,s) = B_ij(x) s^j, but this is not a gauge-invariant\nstatement.\n\nHowever, as I pointed out before, we have exactly the same kind\nof dependence in 1-gauge theory. A Wilson line does not only\ndepend on the line, but also on direction. Hence we must specify\nthe in side and the out side of the line. The two directed Wilson\nlines are *not* related by conjugation, but rather by inversion.\nConjugation does not change the determinant (if the endpoints are\nat the same place) but inversion does, unless det = +-1.\n\nSimilarly, my 2-holonomy depends not only on the surface itself,\nbut on the division of the boundary into in and out sides. But\nthis is all, on the lattice at least. So for a surface decorated\nwith an in side, the 2-holonomy is unique. I always talk about\nsuch decorate surfaces, just as I always talk about decorated, or\ndirected, Wilson lines. In this way we may have talked past each\nother.\n\nBut I think you are right when 2-groups define a unique\n2-holonomy in the stronger sense that it depends only on the\nsurface and not the decoration. But this is surely too strong a\ncondition. You said before that moving the reference points on\nthe boundary only gives a conjugation. This is problematic,\nbecause moving the reference points 180 degrees should give\ninversion, and conjugation does not give you inversion. Again,\ncheck the determinants!\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> 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.

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.

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!

Thomas Larsson

Sep24-04, 08:14 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>This thread started with two notions of 2-multiplication. Before\nwe agree to disagree, let me just point out that analogously\nthere are two notions of directed manifolds. A Wilson line is a\ndirected line, which can be generalized to higher dimensions in\ntwo ways.\n\n1. Regard the directed line as an oriented 1D manifold. The\ngeneralization is then an oriented surface, or more generally\nan oriented N-dimensional manifold.\n\n2. Regard the directed line as a 1D manifold whose boundary\nsplits into an in side and an out side. The generalization to N\ndimensions is then an open N-dimensional disk whose boundary\nsplits into an in side and an out side. The generalization of a\nclosed, directed circle is a globe with two poles and a direction\nof rotation. I\'m unsure about more complicated topologies, but it\nworks for sphereical surfaces in higher dimensions as well.\n\nYou, Pfeiffer and presumably Teitelboim always talk about Wilson\nsurfaces in the first sense, whereas I use the second one. There\nis nothing wrong with neither, just two different concepts that\nhappen to coincide in 1D. Both are probably useful, but in in\ndifferent contexts. The second concept allows a p-form\ngeneralization of non-abelian gauge theory with the expected\nnumber of dofs. It seems to me that the first concept does not,\ndue to the consistency conditions.\n\nI thought that diff and gauge covariance, which my continuum\nmodel manifestly has, are enough to construct a reparametrization-\ninvariant surface-ordered integral in the second sense. Perhaps\nnot, but at least I have a p-form version of the local aspects of\nnon-abelian gauge theory in the continuum. Maybe we can agree\nthat the local condition for covariance is that everything\ntransforms in a well-defined way under diffeomorphisms and gauge\ntransformations?\n\nSo we keep talking about different things. With that observation,\nit is probably time to let this thread come to an end.\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>This thread started with two notions of 2-multiplication. Before
we agree to disagree, let me just point out that analogously
there are two notions of directed manifolds. A Wilson line is a
directed line, which can be generalized to higher dimensions in
two ways.

1. Regard the directed line as an oriented 1D manifold. The
generalization is then an oriented surface, or more generally
an oriented N-dimensional manifold.

2. Regard the directed line as a 1D manifold whose boundary
splits into an in side and an out side. The generalization to N
dimensions is then an open N-dimensional disk whose boundary
splits into an in side and an out side. The generalization of a
closed, directed circle is a globe with two poles and a direction
of rotation. I'm unsure about more complicated topologies, but it
works for sphereical surfaces in higher dimensions as well.

You, Pfeiffer and presumably Teitelboim always talk about Wilson
surfaces in the first sense, whereas I use the second one. There
is nothing wrong with neither, just two different concepts that
happen to coincide in 1D. Both are probably useful, but in in
different contexts. The second concept allows a p-form
generalization of non-abelian gauge theory with the expected
number of dofs. It seems to me that the first concept does not,
due to the consistency conditions.

I thought that diff and gauge covariance, which my continuum
model manifestly has, are enough to construct a reparametrization-
invariant surface-ordered integral in the second sense. Perhaps
not, but at least I have a p-form version of the local aspects of
non-abelian gauge theory in the continuum. Maybe we can agree
that the local condition for covariance is that everything
transforms in a well-defined way under diffeomorphisms and gauge
transformations?

So we keep talking about different things. With that observation,
it is probably time to let this thread come to an end.