<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>Over two years ago John Baez had written in article\nhttp://groups.google.de/groups?selm=aeghes%24j4d%241%40glue.ucr.edu :\n\n> I\'ve finished writing a paper on this stuff, called "Higher Yang-Mills\n> Theory". It will appear on hep-th pretty soon, but you can get it now\n> here:\n\n[ http://xxx.lanl.gov/abs/hep-th/0206130 ]\n\n\n\nI had not completely followed the conversation concerning theories with\n2-form gauge fields back in 2002, but after a major detour it now so happens\nthat I get the impression that I may have something to say about it which\nmight even be new, or might at least be a new viewpoint.\n\nI must say I am coming from physics, and have not even tried to acquaint\nmyself in more than very elementary terms with the vast mathematical\nmachinery associated with 2-categories, 1-gerbes and whatnot.\n\nBut as far as I can see there are open questions in the theory of 2-form\ngauge theories which both physical as well as mathematical reasoning has, to\ndate, failed to answer.\n\nIt was in particular Martin Cederwall who a couple of days ago made me aware\nof the fact that it is an open problem to construct a Lagrangian for "higher\nYang-Mills theory" which is invariant under both the ordinary gauge\ntransformation\n\nA -> U A U^+ U(dU^+)\n(1)\nB -> U B U^+,\n\n(where A is the 1-form connection and B the 2-form connection) as well as\nits 2-form cousin, which is expected to be something like\n\nA -> A\n(2)\nB -> B + d_A L\n\nfor some 1-form gauge parameter L.\n\nI got interested in this business in an attempt to understand deformations\nof loop space differential geometry by means of nonabelian deformation\noperators. In particular, there is a pretty obvious deformation of the\nboundary state of the D9 brane (which is nothing but the constant 0-form on\nloop space) that should somehow describe strings (F- or maybe D-strings) in\nthe background of nonabelian 1- and 2-form fields. In loop space formulation\none can easily read off from the deformed super-Virasoro constraints, which\nare associated to this boundary state deformation, the generalized\n"gauge-covariant" exterior derivative on loop space which describes the\ncorresponding configuration space geometry.\n\nThere was already in 2002 a paper by Christiaan Hofman, hep-th/0207017,\nwhere a form of this covariant derivative on loop space was proposed.\nComparing this proposal with the operator that I derive from the boundary\nstate deformation method one finds a slight but crucial difference. I have\nevidence that this difference might be important for understanding the above\nmentioned open problems.\n\nSince on loop space the target space 2-form field induces an ordinary 1-form\nconnection, it is possible to compute the gauge transformations of this\n1-form connection on loop space in the usual way and then read off what it\nimplies for the target space objects.\n\nI think I successfully checked that the loop space connection associated\nwith B that I derive is consistent in the sense that global gauge\ntransformations on loop space imply the target space gauge transformation\n(1) given above. That this works out correctly crucially depends on the\ndifference between the connection that I am using and that which was\nproposed by Christiaan Hofman before, which is the reason why I think it may\nbe of interest.\n\n(I have talked with Christiaan Hofman about this and he pointed out that the\nconnection he gives might describe a different physical setup, where for\ninstance the B field is in the fundamental of the gauge group, maybe.)\n\nMoreover, when I check what infinitesimal but local loop space gauge\ntransformations imply for the target space fields I find the transformation\n(2) above - but corrected by certain 1-forms on loop space which do not have\na target space interpretation.\n\nNow I am trying to understand what this might mean. It seems to maybe tell\nme that the reason why nobody managed to write down a local field theory\nwhich is invariant under both (1) and (2) is that the full invariance of\n2-form gauge theory is that of 1-form gauge theory on loop space, which can\nin principle only be realized incompletely on a point-particle space.\n\nMaybe in other words this might mean that the boundary string field theory\nfor open strings in a nonabelian 2-form background might have the full\n(1)+(2)+correction term gauge invariance (it certainly should), but that its\nfinite level-truncated effective point particle field theory will never\nhave.\n\nThis does not sound implausible I think. But Martin Cederwall pointed out to\nme that one might have to restrict the set of all admissable gauge\ntransformations on loop space to maybe those which satisfy certain\nconditions. Maybe that is true, but I have not yet figured it out.\n\nTherefore I\'d be grateful if somebody knowledgable could have a look at a\ndraft on the above issues which I have prepared\n\nhttp://www-stud.uni-essen.de/~sb0264/p9.pdf\n\nand maybe make some critical remarks.\n\nI\'d appreciate all responses, ranging from debunking my approach to maybe\ncollaborating on something.\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>Over two years ago John Baez had written in article
http://groups.google.de/groups?selm=aeghes%24j4d%241%40glue.ucr.edu :
> I've finished writing a paper on this stuff, called "Higher Yang-Mills
> Theory". It will appear on hep-th pretty soon, but you can get it now
> here:
I had not completely followed the conversation concerning theories with
2-form gauge fields back in 2002, but after a major detour it now so happens
that I get the impression that I may have something to say about it which
might even be new, or might at least be a new viewpoint.
I must say I am coming from physics, and have not even tried to acquaint
myself in more than very elementary terms with the vast mathematical
machinery associated with 2-categories, 1-gerbes and whatnot.
But as far as I can see there are open questions in the theory of 2-form
gauge theories which both physical as well as mathematical reasoning has, to
date, failed to answer.
It was in particular Martin Cederwall who a couple of days ago made me aware
of the fact that it is an open problem to construct a Lagrangian for "higher
Yang-Mills theory" which is invariant under both the ordinary gauge
transformation
A -> U A U^+ U(dU^+)[/itex]
(1)
B -> U B U^+,
(where A is the 1-form connection and B the 2-form connection) as well as
its 2-form cousin, which is expected to be something like
A -> A
(2)
[itex]B -> B + d_A L
for some 1-form gauge parameter L.
I got interested in this business in an attempt to understand deformations
of loop space differential geometry by means of nonabelian deformation
operators. In particular, there is a pretty obvious deformation of the
boundary state of the D9 brane (which is nothing but the constant 0-form on
loop space) that should somehow describe strings (F- or maybe D-strings) in
the background of nonabelian 1- and 2-form fields. In loop space formulation
one can easily read off from the deformed super-Virasoro constraints, which
are associated to this boundary state deformation, the generalized
"gauge-covariant" exterior derivative on loop space which describes the
corresponding configuration space geometry.
There was already in 2002 a paper by Christiaan Hofman, http://www.arxiv.org/abs/hep-th/0207017,
where a form of this covariant derivative on loop space was proposed.
Comparing this proposal with the operator that I derive from the boundary
state deformation method one finds a slight but crucial difference. I have
evidence that this difference might be important for understanding the above
mentioned open problems.
Since on loop space the target space 2-form field induces an ordinary 1-form
connection, it is possible to compute the gauge transformations of this
1-form connection on loop space in the usual way and then read off what it
implies for the target space objects.
I think I successfully checked that the loop space connection associated
with B that I derive is consistent in the sense that global gauge
transformations on loop space imply the target space gauge transformation
(1) given above. That this works out correctly crucially depends on the
difference between the connection that I am using and that which was
proposed by Christiaan Hofman before, which is the reason why I think it may
be of interest.
(I have talked with Christiaan Hofman about this and he pointed out that the
connection he gives might describe a different physical setup, where for
instance the B field is in the fundamental of the gauge group, maybe.)
Moreover, when I check what infinitesimal but local loop space gauge
transformations imply for the target space fields I find the transformation
(2) above - but corrected by certain 1-forms on loop space which do not have
a target space interpretation.
Now I am trying to understand what this might mean. It seems to maybe tell
me that the reason why nobody managed to write down a local field theory
which is invariant under both (1) and (2) is that the full invariance of
2-form gauge theory is that of 1-form gauge theory on loop space, which can
in principle only be realized incompletely on a point-particle space.
Maybe in other words this might mean that the boundary string field theory
for open strings in a nonabelian 2-form background might have the full
(1)+(2)+correction term gauge invariance (it certainly should), but that its
finite level-truncated effective point particle field theory will never
have.
This does not sound implausible I think. But Martin Cederwall pointed out to
me that one might have to restrict the set of all admissable gauge
transformations on loop space to maybe those which satisfy certain
conditions. Maybe that is true, but I have not yet figured it out.
Therefore I'd be grateful if somebody knowledgable could have a look at a
draft on the above issues which I have prepared
http://www-stud.uni-essen.de/~sb0264/p9.pdf
and maybe make some critical remarks.
I'd appreciate all responses, ranging from debunking my approach to maybe
collaborating on something.
John Baez
Aug24-04, 05:41 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>Almost two months ago Urs Schreiber wrote:\n\n>Over two years ago John Baez had written:\n\n>> I\'ve finished writing a paper on this stuff, called "Higher Yang-Mills\n>> Theory". It will appear on hep-th pretty soon, but you can get it now\n>> here:\n>>\n>>[ http://xxx.lanl.gov/abs/hep-th/0206130 ]\n\n>I had not completely followed the conversation concerning theories with\n>2-form gauge fields back in 2002, but after a major detour it now so happens\n>that I get the impression that I may have something to say about it which\n>might even be new, or might at least be a new viewpoint.\n\nOkay, great! I just saw this post of yours. Our conversation may\nbe slow, but I find it interesting...\n\n>It was in particular Martin Cederwall who a couple of days ago made me aware\n>of the fact that it is an open problem to construct a Lagrangian for "higher\n>Yang-Mills theory" which is invariant under both the ordinary gauge\n>transformation\n>\n> A -> U A U^+ + U(dU^+)\n> (1)\n> B -> U B U^+,\n>\n>(where A is the 1-form connection and B the 2-form connection) as well as\n>its 2-form cousin, which is expected to be something like\n>\n> A -> A\n> (2)\n> B -> B + d_A L\n>\n>for some 1-form gauge parameter L.\n\nRight, this is a famous old puzzle. The reason I never\nbothered trying to publish my paper is that the theory\ndescribed in there is invariant under (1) but not (2).\n\nWorse, I\'m not even sure these categorified gauge theories\n*should* be invariant under transformations of type (2).\nHere by "should" I\'m referring to the intuitions we get\nfrom 2-group theory (as opposed to other more physical\nintuitions).\n\nA 2-group is a special sort of category where the objects describe\n"symmetries" just like the elements of a group, while the morphisms\ndescribe "symmetries of symmetries". In categorified gauge theory\nthere is not a group of gauge transformations, but a 2-group. It\nmay make sense to demand that the Lagrangian in such a theory is\ninvariant under the "symmetries", and I\'m pretty sure equation (1)\nexpresses that idea. But, it\'s far less clear that equation (2)\nexpresses the right sort of "invariance" under the "symmetries of\nsymmetries".\n\nIn other words, I can\'t derive equations (1) and (2) from sensible\nideas on how the 2-group of gauge transformations should "act" on\nconnections in categorified gauge theory.\n\nFor one thing, 2-groups naturally act not on sets but on categories!\nSo, for a sensible setup, I need not just a set of connections, but a\ncategory of them. If I got this straightened out, I should be able to\nturn the crank and see how the 2-group of gauge transformations "acts"\non this category. Then maybe I could figure out what it means for\nequations to be "invariant" under this action. But, my brain always\nmelts down right around this point.\n\n>Moreover, when I check what infinitesimal but local loop space gauge\n>transformations imply for the target space fields I find the transformation\n>(2) above - but corrected by certain 1-forms on loop space which do not have\n>a target space interpretation.\n\nThat\'s interesting!\n\n>Now I am trying to understand what this might mean. It seems to maybe tell\n>me that the reason why nobody managed to write down a local field theory\n>which is invariant under both (1) and (2) is that the full invariance of\n>2-form gauge theory is that of 1-form gauge theory on loop space, which can\n>in principle only be realized incompletely on a point-particle space.\n\nI\'ve thought about this a bit. I\'ve thought about the relation\nbetween connections on loop space LX and 2-connections on X, and\nI think like you I noticed they didn\'t quite match. But it was a\nlong time ago and it\'s blurry in my mind.\n\n>Therefore I\'d be grateful if somebody knowledgeable could have a look at a\n>draft on the above issues which I have prepared\n>\n> http://www-stud.uni-essen.de/~sb0264/p9.pdf\n>\n>and maybe make some critical remarks.\n\nI\'ll take a look! Knowledgeable or not, I\'ll certainly be interested.\n\nBy the way, I\'ve been meaning to talk to Hendryk Pfeiffer about these things\nwhile I\'m Cambridge, but I still haven\'t found the time. I really should\ndo it soon! I\'ll show him this paper of yours, and maybe we can figure\nsomething out.\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>Almost two months ago Urs Schreiber wrote:
>Over two years ago John Baez had written:
>> I've finished writing a paper on this stuff, called "Higher Yang-Mills
>> Theory". It will appear on hep-th pretty soon, but you can get it now
>> here:
>>
>>[ http://xxx.lanl.gov/abs/http://www.arxiv.org/abs/hep-th/0206130 ]
>I had not completely followed the conversation concerning theories with
>2-form gauge fields back in 2002, but after a major detour it now so happens
>that I get the impression that I may have something to say about it which
>might even be new, or might at least be a new viewpoint.
Okay, great! I just saw this post of yours. Our conversation may
be slow, but I find it interesting...
>It was in particular Martin Cederwall who a couple of days ago made me aware
>of the fact that it is an open problem to construct a Lagrangian for "higher
>Yang-Mills theory" which is invariant under both the ordinary gauge
>transformation
>
> A -> U A U^+ + U(dU^+)
> (1)
> B -> U B U^+,
>
>(where A is the 1-form connection and B the 2-form connection) as well as
>its 2-form cousin, which is expected to be something like
>
> A -> A
> (2)
> B -> B + d_A L
>
>for some 1-form gauge parameter L.
Right, this is a famous old puzzle. The reason I never
bothered trying to publish my paper is that the theory
described in there is invariant under (1) but not (2).
Worse, I'm not even sure these categorified gauge theories
*should* be invariant under transformations of type (2).
Here by "should" I'm referring to the intuitions we get
from 2-group theory (as opposed to other more physical
intuitions).
A 2-group is a special sort of category where the objects describe
"symmetries" just like the elements of a group, while the morphisms
describe "symmetries of symmetries". In categorified gauge theory
there is not a group of gauge transformations, but a 2-group. It
may make sense to demand that the Lagrangian in such a theory is
invariant under the "symmetries", and I'm pretty sure equation (1)
expresses that idea. But, it's far less clear that equation (2)
expresses the right sort of "invariance" under the "symmetries of
symmetries".
In other words, I can't derive equations (1) and (2) from sensible
ideas on how the 2-group of gauge transformations should "act" on
connections in categorified gauge theory.
For one thing, 2-groups naturally act not on sets but on categories!
So, for a sensible setup, I need not just a set of connections, but a
category of them. If I got this straightened out, I should be able to
turn the crank and see how the 2-group of gauge transformations "acts"
on this category. Then maybe I could figure out what it means for
equations to be "invariant" under this action. But, my brain always
melts down right around this point.
>Moreover, when I check what infinitesimal but local loop space gauge
>transformations imply for the target space fields I find the transformation
>(2) above - but corrected by certain 1-forms on loop space which do not have
>a target space interpretation.
That's interesting!
>Now I am trying to understand what this might mean. It seems to maybe tell
>me that the reason why nobody managed to write down a local field theory
>which is invariant under both (1) and (2) is that the full invariance of
>2-form gauge theory is that of 1-form gauge theory on loop space, which can
>in principle only be realized incompletely on a point-particle space.
I've thought about this a bit. I've thought about the relation
between connections on loop space LX and 2-connections on X, and
I think like you I noticed they didn't quite match. But it was a
long time ago and it's blurry in my mind.
>Therefore I'd be grateful if somebody knowledgeable could have a look at a
>draft on the above issues which I have prepared
>
> http://www-stud.uni-essen.de/~sb0264/p9.pdf
>
>and maybe make some critical remarks.
I'll take a look! Knowledgeable or not, I'll certainly be interested.
By the way, I've been meaning to talk to Hendryk Pfeiffer about these things
while I'm Cambridge, but I still haven't found the time. I really should
do it soon! I'll show him this paper of yours, and maybe we can figure
something out.
Urs Schreiber
Aug24-04, 07:29 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"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag\nnews:cg2rbp\\$krk\\$1@glue.ucr.edu...\n\n> >Moreover, when I check what infinitesimal but local loop space gauge\n> >transformations imply for the target space fields I find the\ntransformation\n> >(2) above - but corrected by certain 1-forms on loop space which do not\n> >have a target space interpretation.\n>\n> That\'s interesting!\n\nThere is even a stronger version of this statement:\n\nThe annoying extra terms disappear precisely in the case that B = -F, where\nF is the field strength of A, and in this case the connection on loop space is\nflat. (It is gauge equivalent to the trivial connection by adjoining with\nthe holonomy of A around a given loop.)\n\nBut B = -F is also the condition that is implicit in proposition 5 of your\nhep-th/0206130 on the properties of 2-groups, as emphasized by\nGirelli&Pfeiffer. I have mentioned that first in our discussion of Thomas\nLarsson\'s proposal (which is suffering precisely from a lack of this\ncondition) at\n\nhttp://golem.ph.utexas.edu/string/archives/000405.html#c001419 .\n\nAs I wrote in a recent post to spr, a loop space connection of the schematic\nform\n\nint dsigma W_A(sigma) B(sigma) W_A^{-1}(sigma)\n\ncan be shown (http://golem.ph.utexas.edu/string/archives/000416.html)\nto compute precisely the 2-group holonomy as discussed by you,\nGirelly&Pfeiffer, when integraded over a loop in _based_ loop space.\n\nBut there are flat connections on loop space (where flatness ensures\nuniqueness of surface holonomy associated with holonomies of loops in based\nloop space) which are not gauge equivalent to the above sort of connection.\nThese were first discussed by Alvarez, Ferreira and Sanchez-Guillen, as\nreviewed in http://golem.ph.utexas.edu/string/archives/000405.html. They do\nhowever involve 2-form fields which are heavily restricted, namely they must\neither be in an abelian ideal of the algeba of the 1-form, or they must be\ncovariantly constant with respect to that 1-form.\n\nStill, in these cases the 1-form can be arbitrarily non-abelian. So I am\nwondering if by reverse engineering this construction from loop space\nlanguage to algebraic higher-group theory, one can find a previously unnoticed sort\nof "2-group" or something similar which does not have B = -F but has\n"quasi-abelian" B and non-abelian A. I would think the loop space result\nshows that this must be possible, but I haven\'t thought about it in any detail.\nPossibly this is already known in some disguise?\n\n\n> I\'ll take a look! Knowledgeable or not, I\'ll certainly be interested.\n\nGreat, thanks. Please note that the final version, including a discussion of\nthe relation to the work by Alvarez, Ferreira and Sanchez-Guillen is\navailable as http://arxiv.org/abs/hep-th/0407122 .\n\nYou\'ll note that the discussion of the mathy aspects is inferior, but I have\ndiscussed all this in some detail with the above three authors as well as\nwith Jens Fjelstad, a mathematical physicist from Hamburg university, and it\nseems that the ideas are correct.\n\n> By the way, I\'ve been meaning to talk to Hendryk Pfeiffer about these\n> things while I\'m Cambridge, but I still haven\'t found the time. I really should\n> do it soon! I\'ll show him this paper of yours, and maybe we can figure\n> something out.\n\nI had some email exchange with Pfeiffer, who originally responded by private\nmail to my spr post from 2 months ago and made me aware of his work. It was\nkind of funny that I received his email exactly while doing the calculation\nwhich told me from my point of view that B=-F is a consistency condition.\nHis result was a nice check that I am on the right track.\n\nI think using divergence calculation in string theory one can show that the\neffective field theory of the non-abelian A and B field is just described by\nthe YM equations for A together with the above constraint.\n\nThis is of course not higher gauge theory in the original sense. But it _is_\ncorrectly invariant under both first order and second order gauge\ntransformations.\n\nLooking back, I tend to agree with the opinion expressed in the concluding\nsection of Girelli&Pfeiffer\'s paper, namely that the problems to produce an\n"interesting" 2-group version of Yang-Mills is precisely due to the\nconstraint B=-F. Girelli&Pfeiffer express some hopes that this "no-go theorem" can be\ncircumvented by using "large" gauge transformations (if I recall correctly)\nbut it seems to me that in any case this does not yield what one might\noriginally have expected to find.\n\nPerhaps the real use of 2-groups is really rather in the study of integrable\nsystems, maybe.\n\nOr else: No matter what one thinks of string theory, it certainly has some\nvalue as a means to produce (effective) field theories. One big open\nquestion is apparently under what circumstances a non-abelian B-field really appears\nin string theory. There are at least various indications that effective field\ntheories on NS-5 branes play a role. These provide a natural form of\n"categorized gauge theory" in the sense that here the open strings ending on\nD-branes are replaces with open _membranes_ ending on NS branes. The\nenpoints of the open membranes are closed strings on the NS brane, and these carry\nthe information which NS brane they are sitting on, which should give them a\nhigher-order Chan-Paton factor. This makes sense, since precisely such a\nsingle CP factor (a "fundamental index") is really implicit in the boundary\nstates which I used in my paper to derive the loop space connections from.\nChristiaan Hofmann in his paper mentioned more reasons to expect nonabelian\n2-forms to appear in the NS5 brane effective field theories.\n\nSo I think investigation of these field theories on NS5 branes might reveal\nthe proper "target space field theory interpretation" of what I tried to\nderive in the worldsheet theory on loop space. But apparently nobody has\nreally studied this, yet.\n\nThat\'s maybe no wonder, given that even superstring boundary states for\nnon-abelian A fields without any B field present are still under\ninvestigation\n(e.g. http://golem.ph.utexas.edu/string/archives/000417.html).\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>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
news:cg2rbp$krk$1@glue.ucr.edu...
> >Moreover, when I check what infinitesimal but local loop space gauge
> >transformations imply for the target space fields I find the
transformation
> >(2) above - but corrected by certain 1-forms on loop space which do not
> >have a target space interpretation.
>
> That's interesting!
There is even a stronger version of this statement:
The annoying extra terms disappear precisely in the case that B = -F, where
F is the field strength of A, and in this case the connection on loop space is
flat. (It is gauge equivalent to the trivial connection by adjoining with
the holonomy of A around a given loop.)
But B = -F is also the condition that is implicit in proposition 5 of your
http://www.arxiv.org/abs/hep-th/0206130 on the properties of 2-groups, as emphasized by
Girelli&Pfeiffer. I have mentioned that first in our discussion of Thomas
Larsson's proposal (which is suffering precisely from a lack of this
condition) at
can be shown (http://golem.ph.utexas.edu/string/archives/000416.html)
to compute precisely the 2-group holonomy as discussed by you,
Girelly&Pfeiffer, when integraded over a loop in _based_ loop space.
But there are flat connections on loop space (where flatness ensures
uniqueness of surface holonomy associated with holonomies of loops in based
loop space) which are not gauge equivalent to the above sort of connection.
These were first discussed by Alvarez, Ferreira and Sanchez-Guillen, as
reviewed in http://golem.ph.utexas.edu/string/archives/000405.html. They do
however involve 2-form fields which are heavily restricted, namely they must
either be in an abelian ideal of the algeba of the 1-form, or they must be
covariantly constant with respect to that 1-form.
Still, in these cases the 1-form can be arbitrarily non-abelian. So I am
wondering if by reverse engineering this construction from loop space
language to algebraic higher-group theory, one can find a previously unnoticed sort
of "2-group" or something similar which does not have B = -F but has
"quasi-abelian" B and non-abelian A. I would think the loop space result
shows that this must be possible, but I haven't thought about it in any detail.
Possibly this is already known in some disguise?
> I'll take a look! Knowledgeable or not, I'll certainly be interested.
Great, thanks. Please note that the final version, including a discussion of
the relation to the work by Alvarez, Ferreira and Sanchez-Guillen is
available as http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0407122 .
You'll note that the discussion of the mathy aspects is inferior, but I have
discussed all this in some detail with the above three authors as well as
with Jens Fjelstad, a mathematical physicist from Hamburg university, and it
seems that the ideas are correct.
> By the way, I've been meaning to talk to Hendryk Pfeiffer about these
> things while I'm Cambridge, but I still haven't found the time. I really should
> do it soon! I'll show him this paper of yours, and maybe we can figure
> something out.
I had some email exchange with Pfeiffer, who originally responded by private
mail to my spr post from 2 months ago and made me aware of his work. It was
kind of funny that I received his email exactly while doing the calculation
which told me from my point of view that B=-F is a consistency condition.
His result was a nice check that I am on the right track.
I think using divergence calculation in string theory one can show that the
effective field theory of the non-abelian A and B field is just described by
the YM equations for A together with the above constraint.
This is of course not higher gauge theory in the original sense. But it _is_
correctly invariant under both first order and second order gauge
transformations.
Looking back, I tend to agree with the opinion expressed in the concluding
section of Girelli&Pfeiffer's paper, namely that the problems to produce an
"interesting" 2-group version of Yang-Mills is precisely due to the
constraint B=-F. Girelli&Pfeiffer express some hopes that this "no-go theorem" can be
circumvented by using "large" gauge transformations (if I recall correctly)
but it seems to me that in any case this does not yield what one might
originally have expected to find.
Perhaps the real use of 2-groups is really rather in the study of integrable
systems, maybe.
Or else: No matter what one thinks of string theory, it certainly has some
value as a means to produce (effective) field theories. One big open
question is apparently under what circumstances a non-abelian B-field really appears
in string theory. There are at least various indications that effective field
theories on NS-5 branes play a role. These provide a natural form of
"categorized gauge theory" in the sense that here the open strings ending on
D-branes are replaces with open _membranes_ ending on NS branes. The
enpoints of the open membranes are closed strings on the NS brane, and these carry
the information which NS brane they are sitting on, which should give them a
higher-order Chan-Paton factor. This makes sense, since precisely such a
single CP factor (a "fundamental index") is really implicit in the boundary
states which I used in my paper to derive the loop space connections from.
Christiaan Hofmann in his paper mentioned more reasons to expect nonabelian
2-forms to appear in the NS5 brane effective field theories.
So I think investigation of these field theories on NS5 branes might reveal
the proper "target space field theory interpretation" of what I tried to
derive in the worldsheet theory on loop space. But apparently nobody has
really studied this, yet.
That's maybe no wonder, given that even superstring boundary states for
non-abelian A fields without any B field present are still under
investigation
(e.g. http://golem.ph.utexas.edu/string/archives/000417.html).
Urs Schreiber
Aug24-04, 12:29 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>I wrote:\n\n> Still, in these cases the 1-form can be arbitrarily non-abelian. So I am\n> wondering if by reverse engineering this construction from loop space\n> language to algebraic higher-group theory, one can find a previously\n> unnoticed sort of "2-group" or something similar which does not have B = -F\n> but has "quasi-abelian" B and non-abelian A. I would think the loop space result\n> shows that this must be possible, but I haven\'t thought about it in any\n> detail.\n> Possibly this is already known in some disguise?\n\nI have tried to figure it out. Seems to me that there is a notion of \'weak\'\n2-group which does see the surface holonomies for nonvanishing B + F.\n\nI discuss the details here:\n\nhttp://golem.ph.utexas.edu/string/archives/000421.html .\n\nI\'d be glad to know what you think about this.\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>I wrote:
> Still, in these cases the 1-form can be arbitrarily non-abelian. So I am
> wondering if by reverse engineering this construction from loop space
> language to algebraic higher-group theory, one can find a previously
> unnoticed sort of "2-group" or something similar which does not have B = -F
> but has "quasi-abelian" B and non-abelian A. I would think the loop space result
> shows that this must be possible, but I haven't thought about it in any
> detail.
> Possibly this is already known in some disguise?
I have tried to figure it out. Seems to me that there is a notion of 'weak'
2-group which does see the surface holonomies for nonvanishing B + F.
<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>There are lots of things to say, but here\'s one:\n\nIn article <412b269b@news.sentex.net>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>So I am\n>wondering if by reverse engineering this construction from loop space\n>language to algebraic higher-group theory, one can find a previously\n>unnoticed sort of "2-group" or something similar which does not have\n>B = -F but has "quasi-abelian" B and non-abelian A.\n\nA strict 2-group is the same as a crossed module: a group G, a group\nH, an action of G on H and a homomorphism t: H -> G, satisfying some\nequations described in my paper.\n\nTaking Lie algebras we see a strict Lie 2-algebra is the same\nas a differential crossed module: a Lie algebra g, a Lie algebra h,\nan action of g on h and a homomorphism dt: h -> g, satisfying some\nequations.\n\nWhen we think about connections in this context, we see the natural\nequation is not\n\nB = -F\n\n(which makes no sense!) but instead\n\nt(B) = -F\n\nwhere F is a g-valued 2-form and B is an h-valued 2-form.\n\nWe can then consider various special cases.\n\nAt one extreme we can have G trivial and H abelian; then the\nabove equation is vacuous. This is what happens in 2-form\nelectromagnetism.\n\nAt another extreme we can have G = H and t the identity; then\nyou get B = -F. That\'s the case you seem to like best.\n\nBut there are lots of intermediate cases. Maybe you want some\nconcrete examples? I can manufacture examples, but not very\ninteresting ones if G and H are required to be *compact* Lie groups,\nbecause the Lie algebra of these is always semisimple + abelian,\nand the options for homomorphisms dt are severely limited.\n\n(As usual I will cc this to you.)\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>There are lots of things to say, but here's one:
In article <412b269b@news.sentex.net>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>So I am
>wondering if by reverse engineering this construction from loop space
>language to algebraic higher-group theory, one can find a previously
>unnoticed sort of "2-group" or something similar which does not have
>B = -F but has "quasi-abelian" B and non-abelian A.
A strict 2-group is the same as a crossed module: a group G, a group
H, an action of G on H and a homomorphism t: H -> G, satisfying some
equations described in my paper.
Taking Lie algebras we see a strict Lie 2-algebra is the same
as a differential crossed module: a Lie algebra g, a Lie algebra h,
an action of g on h and a homomorphism dt: h -> g, satisfying some
equations.
When we think about connections in this context, we see the natural
equation is not
B = -F
(which makes no sense!) but instead
t(B) = -F
where F is a g-valued 2-form and B is an h-valued 2-form.
We can then consider various special cases.
At one extreme we can have G trivial and H abelian; then the
above equation is vacuous. This is what happens in 2-form
electromagnetism.
At another extreme we can have G = H and t the identity; then
you get B = -F. That's the case you seem to like best.
But there are lots of intermediate cases. Maybe you want some
concrete examples? I can manufacture examples, but not very
interesting ones if G and H are required to be *compact* Lie groups,
because the Lie algebra of these is always semisimple + abelian,
and the options for homomorphisms dt are severely limited.
(As usual I will cc this to you.)
Urs Schreiber
Aug25-04, 04:54 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>Many thanks for your reply!\n\nYou wrote:\n\n> When we think about connections in this context, we see the natural\n> equation is not\n>\n> B = -F\n>\n> (which makes no sense!)\n>\n> but instead\n>\n> t(B) = -F\n>\n> where F is a g-valued 2-form and B is an h-valued 2-form.\n>\n> We can then consider various special cases.\n>\n> At one extreme we can have G trivial and H abelian; then the\n> above equation is vacuous. This is what happens in 2-form\n> electromagnetism.\n>\n> At another extreme we can have G = H and t the identity; then\n> you get B = -F. That\'s the case you seem to like best.\n\nTrue, I like that best at the moment, because it seems to me that a very\ninteresting class of physical applications corresponds to G = H = U(N).\n\nBut I do understand that 2-group theory covers the more general case where G\n\\neq H. I think pretty much everything I wrote so far with G = H in mind\ndirectly translates to this more general case when you think of all my Bs as\nt(B)s. But I will make that more precise and explicit in the future.\n\nMeanwhile, since I have the feeling that I didn\'t express myself well\nenough, let me try to rephrase the question that I am currently concerned with:\n\nAlvarez, Ferreira&Sanchez-Guillen in hep-th/9710147 found two classes of\nconsistent surface holonomies which happen to have G = H but B + F \\neq 0.\n(I am pretty sure that their construction can be straightforwardly generalized\nto G \\neq H in which case it would give consistent surface holonomy for t(B) +\nF \\neq 0. But the case with G=H alone already raises the following questions.)\n\nAs far as I can see, there are only two possible reactions to this result:\n\n1) It contains a flaw and 2-group theory is right that only t(B)+F = 0 gives\nwell defined surface holonomy.\n\n2) It is correct. Then there must be a reason why 2-group theory cannot\nobtain these surface holonomies with t(B) + F nonvanishing.\n\n\nI argued that the latter is the case (namely that there are more solutions\nto the 2-associativity condition than just those with t(B)+F=0), but if I am\nwrong about that please let me know where I went astray.\n\nI would like to know\n\na) if you think there is a 3rd alternative to 1) and 2) above (maybe that\nsomehow Alvarez, Ferreira&Sanchez-Guillen are secretly speaking about a\ndifferent notion of surface holonomy than 2-group theory does or something\nlike that)\n\nb) or else, if you think that their result is flawed, what you think the\nmistake is\n\nc) or finally, if you think their result is correct, how you see it fit\ntogether with the results of 2-group theory.\n\n> But there are lots of intermediate cases. Maybe you want some\n> concrete examples? I can manufacture examples, but not very\n> interesting ones if G and H are required to be *compact* Lie groups,\n> because the Lie algebra of these is always semisimple + abelian,\n> and the options for homomorphisms dt are severely limited.\n\nI am not sure that examples where t is nontrivial are of help for answering\nthe question that I am concerned with. But maybe that\'s precisely my\nproblem.\n\nIn any case, thanks for your help!\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>Many thanks for your reply!
You wrote:
> When we think about connections in this context, we see the natural
> equation is not
>
> B = -F
>
> (which makes no sense!)
>
> but instead
>
> t(B) = -F
>
> where F is a g-valued 2-form and B is an h-valued 2-form.
>
> We can then consider various special cases.
>
> At one extreme we can have G trivial and H abelian; then the
> above equation is vacuous. This is what happens in 2-form
> electromagnetism.
>
> At another extreme we can have G = H and t the identity; then
> you get B = -F. That's the case you seem to like best.
True, I like that best at the moment, because it seems to me that a very
interesting class of physical applications corresponds to G = H = U(N).
But I do understand that 2-group theory covers the more general case where G
\neq H. I think pretty much everything I wrote so far with G = H in mind
directly translates to this more general case when you think of all my Bs as
t(B)s. But I will make that more precise and explicit in the future.
Meanwhile, since I have the feeling that I didn't express myself well
enough, let me try to rephrase the question that I am currently concerned with:
Alvarez, Ferreira&Sanchez-Guillen in http://www.arxiv.org/abs/hep-th/9710147 found two classes of
consistent surface holonomies which happen to have G = H but B + F \neq .
(I am pretty sure that their construction can be straightforwardly generalized
to G \neq H in which case it would give consistent surface holonomy for t(B) +
F \neq . But the case with G=H alone already raises the following questions.)
As far as I can see, there are only two possible reactions to this result:
1) It contains a flaw and 2-group theory is right that only t(B)+F = gives
well defined surface holonomy.
2) It is correct. Then there must be a reason why 2-group theory cannot
obtain these surface holonomies with t(B) + F nonvanishing.
I argued that the latter is the case (namely that there are more solutions
to the 2-associativity condition than just those with t(B)+F=0), but if I am
wrong about that please let me know where I went astray.
I would like to know
a) if you think there is a 3rd alternative to 1) and 2) above (maybe that
somehow Alvarez, Ferreira&Sanchez-Guillen are secretly speaking about a
different notion of surface holonomy than 2-group theory does or something
like that)
b) or else, if you think that their result is flawed, what you think the
mistake is
c) or finally, if you think their result is correct, how you see it fit
together with the results of 2-group theory.
> But there are lots of intermediate cases. Maybe you want some
> concrete examples? I can manufacture examples, but not very
> interesting ones if G and H are required to be *compact* Lie groups,
> because the Lie algebra of these is always semisimple + abelian,
> and the options for homomorphisms dt are severely limited.
I am not sure that examples where t is nontrivial are of help for answering
the question that I am concerned with. But maybe that's precisely my
problem.
In any case, thanks for your help!
John Baez
Aug25-04, 09:12 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>\nIn article <412c53c6\\$1@news.sentex.net>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>John Baez wrote:\n\n>> When we think about connections in this context, we see the natural\n>> equation is not\n>>\n>> B = -F\n>>\n>> (which makes no sense!)\n>>\n>> but instead\n>>\n>> t(B) = -F\n>>\n>> where F is a g-valued 2-form and B is an h-valued 2-form.\n>>\n>> We can then consider various special cases.\n>>\n>> At one extreme we can have G trivial and H abelian; then the\n>> above equation is vacuous. This is what happens in 2-form\n>> electromagnetism.\n>>\n>> At another extreme we can have G = H and t the identity; then\n>> you get B = -F. That\'s the case you seem to like best.\n\n>True, I like that best at the moment, because it seems to me that a very\n>interesting class of physical applications corresponds to G = H = U(N).\n\nBy the way, there are closely related ways to build a 2-group, that\nare easy to mix up. It would be unfortunate to think you were working\nwith one when you were really working with the other! I\'m not accusing\nyou of doing this, but I just think I should warn you of the danger,\nsince I\'m gotten confused myself plenty of times. Here they are:\n\n1) Take G = H, let t: H -> G be the identity homomorphism, and let\nalpha be the action of G on H be via conjugation:\n\nalpha(g)(h) = ghg^{-1}\n\n2) Take G = Aut(H) be the group of all automorphisms of H.\nLet t: H -> G be the map sending any element h to the corresponding\n"inner automorphism" of H, that is, the automorphism given by\nconjugating by h:\n\nt(h)(h\') = hh\'h^{-1}\n\nLet alpha be the obvious action of G = Aut(H) as automorphisms\nof H.\n\nThese two constructions agree when the map t in construction 2)\nis one-to-one and onto. This is the case for SO(3) but not for\nSU(n) or U(n), since these have a nontrivial center.\n\n>From a pure mathematical viewpoint, construction 2) is actually\na lot simpler and more important than construction 1), even though\nit doesn\'t look simpler in the lowbrow way I just described it.\n\n>But I do understand that 2-group theory covers the more general case where G\n>\\neq H. I think pretty much everything I wrote so far with G = H in mind\n>directly translates to this more general case when you think of all my Bs as\n>t(B)s. But I will make that more precise and explicit in the future.\n\nOkay... it may be no big deal for your applications, but it can be...\n\n>Meanwhile, since I have the feeling that I didn\'t express myself well\n>enough, let me try to rephrase the question that I am currently concerned with:\n>\n>Alvarez, Ferreira&Sanchez-Guillen in hep-th/9710147 found two classes of\n>consistent surface holonomies which happen to have G = H but B + F \\neq 0.\n\nOf course it\'s possible to have G = H but t: H -> G not equal to the\nidentity! That would be one possible explanation of their work, since\nthis can give B + dt(F) = 0 but B + F =/= 0.\n\nHowever, instead of guessing, I should just read their paper.\n\n>1) It contains a flaw and 2-group theory is right that only t(B)+F = 0 gives\n>well defined surface holonomy.\n\nI should note that it\'s not "2-group theory" which makes this claim,\nbut a paper by Girelli and Pfeiffer. And I\'m not even sure they claim\nthis in an ironclad way. Personally I\'m very confused about all this\nstuff, so more confusion is actually a good thing - it might disentangle\nsomething.\n\n>2) It is correct. Then there must be a reason why 2-group theory cannot\n>obtain these surface holonomies with t(B) + F nonvanishing.\n\n>I argued that the latter is the case (namely that there are more solutions\n>to the 2-associativity condition than just those with t(B)+F=0), but if I am\n>wrong about that please let me know where I went astray.\n\nI don\'t know what the "2-associativity" condition is.\n\nI guess I\'ll just have to read their stuff. Of course I wouldn\'t mind\na wee summary in plain English....\n\n\n\n\n\n\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>In article <412c53c6$1@news.sentex.net>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>John Baez wrote:
>> When we think about connections in this context, we see the natural
>> equation is not
>>
>> B = -F
>>
>> (which makes no sense!)
>>
>> but instead
>>
>> t(B) = -F
>>
>> where F is a g-valued 2-form and B is an h-valued 2-form.
>>
>> We can then consider various special cases.
>>
>> At one extreme we can have G trivial and H abelian; then the
>> above equation is vacuous. This is what happens in 2-form
>> electromagnetism.
>>
>> At another extreme we can have G = H and t the identity; then
>> you get B = -F. That's the case you seem to like best.
>True, I like that best at the moment, because it seems to me that a very
>interesting class of physical applications corresponds to G = H = U(N).
By the way, there are closely related ways to build a 2-group, that
are easy to mix up. It would be unfortunate to think you were working
with one when you were really working with the other! I'm not accusing
you of doing this, but I just think I should warn you of the danger,
since I'm gotten confused myself plenty of times. Here they are:
1) Take G = H, let t: H -> G be the identity homomorphism, and let
\alpha be the action of G on H be via conjugation:
\alpha(g)(h) = ghg^{-1}
2) Take G = Aut(H) be the group of all automorphisms of H.
Let t: H -> G be the map sending any element h to the corresponding
"inner automorphism" of H, that is, the automorphism given by
conjugating by h:
t(h)(h') = hh'h^{-1}
Let \alpha be the obvious action of G = Aut(H) as automorphisms
of H.
These two constructions agree when the map t in construction 2)
is one-to-one and onto. This is the case for SO(3) but not for
SU(n) or U(n), since these have a nontrivial center.
>From a pure mathematical viewpoint, construction 2) is actually
a lot simpler and more important than construction 1), even though
it doesn't look simpler in the lowbrow way I just described it.
>But I do understand that 2-group theory covers the more general case where G
>\neq H. I think pretty much everything I wrote so far with G = H in mind
>directly translates to this more general case when you think of all my Bs as
>t(B)s. But I will make that more precise and explicit in the future.
Okay... it may be no big deal for your applications, but it can be...
>Meanwhile, since I have the feeling that I didn't express myself well
>enough, let me try to rephrase the question that I am currently concerned with:
>
>Alvarez, Ferreira&Sanchez-Guillen in http://www.arxiv.org/abs/hep-th/9710147 found two classes of
>consistent surface holonomies which happen to have G = H but B + F \neq .
Of course it's possible to have G = H but t: H -> G not equal to the
identity! That would be one possible explanation of their work, since
this can give B + dt(F) = but B + F =/= .
However, instead of guessing, I should just read their paper.
>1) It contains a flaw and 2-group theory is right that only t(B)+F = gives
>well defined surface holonomy.
I should note that it's not "2-group theory" which makes this claim,
but a paper by Girelli and Pfeiffer. And I'm not even sure they claim
this in an ironclad way. Personally I'm very confused about all this
stuff, so more confusion is actually a good thing - it might disentangle
something.
>2) It is correct. Then there must be a reason why 2-group theory cannot
>obtain these surface holonomies with t(B) + F nonvanishing.
>I argued that the latter is the case (namely that there are more solutions
>to the 2-associativity condition than just those with t(B)+F=0), but if I am
>wrong about that please let me know where I went astray.
I don't know what the "2-associativity" condition is.
I guess I'll just have to read their stuff. Of course I wouldn't mind
a wee summary in plain English....
Urs Schreiber
Aug25-04, 11:05 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"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag\nnews:cgi1dm\\$daq\\$1@glue.ucr.edu...\n\n> By the way, there are closely related ways to build a 2-group, that\n> are easy to mix up. It would be unfortunate to think you were working\n> with one when you were really working with the other! I\'m not accusing\n> you of doing this, but I just think I should warn you of the danger,\n> since I\'m gotten confused myself plenty of times. Here they are:\n>\n> 1) Take G = H, let t: H -> G be the identity homomorphism, and let\n> alpha be the action of G on H be via conjugation:\n>\n> alpha(g)(h) = ghg^{-1}\n>\n> 2) Take G = Aut(H) be the group of all automorphisms of H.\n> Let t: H -> G be the map sending any element h to the corresponding\n> "inner automorphism" of H, that is, the automorphism given by\n> conjugating by h:\n>\n> t(h)(h\') = hh\'h^{-1}\n>\n> Let alpha be the obvious action of G = Aut(H) as automorphisms\n> of H.\n\nNow I am a little confused. I think I do understand these two examples. But\naren\'t both just two special cases of the general notion of 2-group as\ndiscussed in your paper?\n\n\n> From a pure mathematical viewpoint, construction 2) is actually\n> a lot simpler and more important than construction 1), even though\n> it doesn\'t look simpler in the lowbrow way I just described it.\n\n\nI think I roughly do appreciate the relation to 2-categories that you\nexplain somewhere:\n\nWe can think of a group H as a category with a single object where group\nelements are morphism with source and target all the same single object.\n\nThe invertible functors between such group categories are the group\nautomorphisms. These can be taken to be the objects of a category called\nAut(H).\n\nThe morphisms of Aut(H) must be morphisms between these functors. Let g be\nsuch an automorphism of H. Then any element h of H maps g to another\nautomorphism g\' by\n\ng \'(x) := h(g(h^-1 x)) .\n\nSo the elements of H correspond to morphisms between the g.\n\nSince Aut(H) of which g are the elements is itself a group and since it\nmorphisms H are a group, too, we are dealing with a 2-category being a\ngroup, i.e. a 2-group.\n\nI hope that\'s roughly correct. (But it doesn\'t affect anything of what I say\nbelow ;-)\n\n\n> >But I do understand that 2-group theory covers the more general case\nwhere G\n> >\\neq H. I think pretty much everything I wrote so far with G = H in mind\n> >directly translates to this more general case when you think of all my Bs\nas\n> >t(B)s. But I will make that more precise and explicit in the future.\n>\n> Okay... it may be no big deal for your applications, but it can be...\n\n\nOk. If it doesn\'t disturb you too much I would be very happy if we could for\nthe moment restrict the discussion to the case where t is the identity and\ntry to clarify the question I am thinking about in this special case first.\nIf and when we think we clarified this special case I believe it will be\neasy to hit everything in sight with a nontrivial t and see what we get in\nthis more general case.\n\n\n> >Meanwhile, since I have the feeling that I didn\'t express myself well\n> >enough, let me try to rephrase the question that I am currently concerned\nwith:\n> >\n> >Alvarez, Ferreira&Sanchez-Guillen in hep-th/9710147 found two classes of\n> >consistent surface holonomies which happen to have G = H but B + F \\neq\n0.\n>\n> Of course it\'s possible to have G = H but t: H -> G not equal to the\n> identity! That would be one possible explanation of their work, since\n> this can give B + dt(F) = 0 but B + F =/= 0.\n\n\nHm. Over at the SCT Amitabha Lahiri made a possibly similar suggestion\n(http://golem.ph.utexas.edu/string/archives/000421.html#c001484). But I am\nnot sure yet if this really helps. See below for more on that.\n\n\n> I don\'t know what the "2-associativity" condition is.\n\n\nI mean the condition that it does not matter whether we first perform the\nhorizontal and then the vertical products, or the other way round. The\nequation in definition 1 of your paper. Ah, let me see, you call it the\n"exchange law". Sorry, somehow I thought it should be called\n2-associativity.\n\n\n> >1) It contains a flaw and 2-group theory is right that only t(B)+F = 0\ngives\n> >well defined surface holonomy.\n>\n> I should note that it\'s not "2-group theory" which makes this claim,\n> but a paper by Girelli and Pfeiffer.\n\n\nOh, good that you are saying this. This is getting to the heart of the\nmatter. From how Girelli&Pfeiffer quote your paper as a proof for their\nclaim I took it that they are referring to your "proposition 5" as implying\nthis. There you show that in order to get from a crossed module with\nelements h and g to a 2-group one has to identify 2-morphisms with pairs\n(h,g) and set\n\nsource(h,g) = g\n\nand\n\ntarget(h,g) = hg\n\n(as I said, I\'ll drop the t that should be appearing here for the time\nbeing, if you allow).\n\nBut this means that if (h,g) is the morphism\n\ng1\n--------->\n| ^\n| h |\n|------- g2\n\nwe have g2 = h g1 and hence\n\nh = g2 g1^{-1} .\n\nThe differential version of this, i.e. the second order term in epsilon when\nyou write\n\nh = 1 + epsilon^2 B\n\ng = 1 + epsilon A\n\nis\n\nB + F_A = 0 .\n\nWhen I read your and Pfeiffer\'s papers I derived the same in a similar but\nsomewhat more direct way:\n\nWhen the 2-associativity law (in definition 1 on p. 8 of your paper) is\nexplicitly written in terms of the definition of the vertical and the\nhorizontal product it is equivalent to\n\n\n(*) f1 g1 f2\' g1^{-1} = g2 f2\' g2^{-1} f1 .\n\n(For more details on what I mean here please see my recent reply to Amitabha\nat the SCT http://golem.ph.utexas.edu/string/archives/000421.html#c001485)\n\nMy very point which I tried to make in the comment starting this discussion\nis that we should try to find all solutions to this equation in order to\nfind all conditions under which we get a consistent surface holonomy from\negde and surface group labels.\n\nOne can easily see that _one_ solution to (*) is\n\nf1 = g2 g1^{-1},\n\nwhich is precisely the solution implying B+F=0 discussed above. So that\'s\nhow this condition comes up.\n\nMy point was that there are in fact _other_ solutions of this equation. For\ninstance let the edge labels be arbitrary and restrict the surface labels to\ntake values in an abelian ideal of the group. Then this equation is\nsatisfied, too. This corresponds to one of the classes of surface holonomies\nthat Alverz et al. discuss in terms of loop space formalism.\n\nMoreover, I pointed out that when we go from the condition (*) to its\ninfinitesimal/differential version by again setting g = 1+ epsilon A etc, we\nobtain an equation which admits still one more solution. Namely flat edge\nholonomies together with covariantly constant surface holonomies. Because it\nis really this infinitesimal version which is needed to compute surface\nholonomies in the continuum, this does give a consistent surface holonomy,\nand indeed it corresponds to the other class of loop space connections\ndiscussed by Alvarez et al.\n\n\n\n> I guess I\'ll just have to read their stuff. Of course I wouldn\'t mind\n> a wee summary in plain English....\n\n\nThere are a lot of ideas in that paper, but for our discussion really only a\ncouple of them are relevant. I have reviewed and briefly discussed them in\nthe second half of section 3.4 of http://arxiv.org/abs/hep-th/0407122\nwhich again is based on this SCT entry:\nhttp://golem.ph.utexas.edu/string/archives/000405.html .\n\nThe idea is to write down a connection \\mathcal{A} on loop space of the form\n\n\\mathcal{A}\n=\nint_0^{2\\pi} W(sigma) B_mn(sigma) W^{-1}(sigma) X\'^m(sigma) dX^n(sigma)\n\nwhere the integral is over the given loop, W is the holonomy of A along that\nloop and X\' is the sigma-derivative of the loop X : (0,2pi) -> M.\n\nThis connection gives us a consistent surface holonomy of surfaces which are\nimages of curves in based loop space in particular if it is flat (since then\nit is independent of the parameterization of the surface in terms of sigma\nand in terms of the foliation by loops. (Flatness is not necessary, but\nsufficient for this to be true.)\n\nAlvarez et al. thought they need to set F_A = 0. When one does that it is\npretty easy to see that there are two cases in which \\mathcal{A} is flat,\nnamely\n\n1) When B is in an abelian ideal of the common algebra in which A and B\ntake values.\n\n2) When B is covariantly constant with respect to A.\n\nThese two conditions dont have B+F=0, obviously (in general).\n\nI have disucssed above and in more detail at the SCT how both these\nconditions are mapped to additional solutions of the "exchange law" that\ndiffer from\n\nh = g2 g1^{-1} .\n\n\nIt turned out that Alvarez et al. were overly restrictive by assuming they\nneed F_A = 0, but this is how they found these 2 classes of solutions. The\nfirst even gives consistent surface holonomy for F_A =/= 0.\n\nBut, while still unaware of the work by Alvarez et al., I derived in\nhep-th/0407122 that there are more loop space connections of the above form\nwhich are flat. Namely those obtained by setting B+F_A = 0. These are\nmoreover precisely those that preserve the above form of the connection\nunder local loop space gauge transformations. This is one sign of many why\nB+F_A = 0 is the most "natural" class of surface holonomies.\n\nIf you instead make a loop space gauge transformation to any of the loop\nspace connections by Alvarez et al. the result is something which can no\nlonger be written in the above form, but which involves in general _several_\n1-forms together with the 2-form. So in a sense the surface holonomies by\nAlvarez et al live in a framework larger that the standard 2-group theory.\n\nBut all this I have already said at the SCT in reply to Amitabha Lahiri,\nwhere you can find it, so I\'ll stop here. I am in the proces of typing some\nLaTeX notes about some of the ideas that we are here discussing in ASCII.\nThat might help the discussion.\n\nMeanwhile you can find pretty-printed versions of the formulas that I am\nconsidering at the SCT. With MSIE it takes just the free download of the\nMathPlayer plugin, with Mozilla just the free download of a certain font to\nread these equations. More details can be found here:\nhttp://golem.ph.utexas.edu/string/archives/000316.html\n\nThanks a lot for your time and comments.\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:cgi1dm$daq$1@glue.ucr.edu...
> By the way, there are closely related ways to build a 2-group, that
> are easy to mix up. It would be unfortunate to think you were working
> with one when you were really working with the other! I'm not accusing
> you of doing this, but I just think I should warn you of the danger,
> since I'm gotten confused myself plenty of times. Here they are:
>
> 1) Take G = H, let t: H -> G be the identity homomorphism, and let
> \alpha be the action of G on H be via conjugation:
>
> \alpha(g)(h) = ghg^{-1}
>
> 2) Take G = Aut(H) be the group of all automorphisms of H.
> Let t: H -> G be the map sending any element h to the corresponding
> "inner automorphism" of H, that is, the automorphism given by
> conjugating by h:
>
> t(h)(h') = hh'h^{-1}
>
> Let \alpha be the obvious action of G = Aut(H) as automorphisms
> of H.
Now I am a little confused. I think I do understand these two examples. But
aren't both just two special cases of the general notion of 2-group as
discussed in your paper?
> From a pure mathematical viewpoint, construction 2) is actually
> a lot simpler and more important than construction 1), even though
> it doesn't look simpler in the lowbrow way I just described it.
I think I roughly do appreciate the relation to 2-categories that you
explain somewhere:
We can think of a group H as a category with a single object where group
elements are morphism with source and target all the same single object.
The invertible functors between such group categories are the group
automorphisms. These can be taken to be the objects of a category called
Aut(H).
The morphisms of Aut(H) must be morphisms between these functors. Let g be
such an automorphism of H. Then any element h of H maps g to another
automorphism g' byg '(x) := h(g(h^-1 x)) .
So the elements of H correspond to morphisms between the g.
Since Aut(H) of which g are the elements is itself a group and since it
morphisms H are a group, too, we are dealing with a 2-category being a
group, i.e. a 2-group.
I hope that's roughly correct. (But it doesn't affect anything of what I say
below ;-)
> >But I do understand that 2-group theory covers the more general case
where G
> >\neq H. I think pretty much everything I wrote so far with G = H in mind
> >directly translates to this more general case when you think of all my Bs
as
> >t(B)s. But I will make that more precise and explicit in the future.
>
> Okay... it may be no big deal for your applications, but it can be...
Ok. If it doesn't disturb you too much I would be very happy if we could for
the moment restrict the discussion to the case where t is the identity and
try to clarify the question I am thinking about in this special case first.
If and when we think we clarified this special case I believe it will be
easy to hit everything in sight with a nontrivial t and see what we get in
this more general case.
> >Meanwhile, since I have the feeling that I didn't express myself well
> >enough, let me try to rephrase the question that I am currently concerned
with:
> >
> >Alvarez, Ferreira&Sanchez-Guillen in http://www.arxiv.org/abs/hep-th/9710147 found two classes of
> >consistent surface holonomies which happen to have G = H but B + F \neq
.
>
> Of course it's possible to have G = H but t: H -> G not equal to the
> identity! That would be one possible explanation of their work, since
> this can give B + dt(F) = but B + F =/= .
Hm. Over at the SCT Amitabha Lahiri made a possibly similar suggestion
(http://golem.ph.utexas.edu/string/archives/000421.html#c001484). But I am
not sure yet if this really helps. See below for more on that.
> I don't know what the "2-associativity" condition is.
I mean the condition that it does not matter whether we first perform the
horizontal and then the vertical products, or the other way round. The
equation in definition 1 of your paper. Ah, let me see, you call it the
"exchange law". Sorry, somehow I thought it should be called
2-associativity.
> >1) It contains a flaw and 2-group theory is right that only t(B)+F =
gives
> >well defined surface holonomy.
>
> I should note that it's not "2-group theory" which makes this claim,
> but a paper by Girelli and Pfeiffer.
Oh, good that you are saying this. This is getting to the heart of the
matter. From how Girelli&Pfeiffer quote your paper as a proof for their
claim I took it that they are referring to your "proposition 5" as implying
this. There you show that in order to get from a crossed module with
elements h and g to a 2-group one has to identify 2-morphisms with pairs
(h,g) and set
source(h,g) = g
and
target(h,g) = hg
(as I said, I'll drop the t that should be appearing here for the time
being, if you allow).
But this means that if (h,g) is the morphism
g1
--------->
| ^
| h |
|------- g2
we have g2 = h g1 and hence
h = g2 g1^{-1} .
The differential version of this, i.e. the second order term in \epsilon when
you write
h = 1 + \epsilon^2 Bg = 1 + \epsilon A
is
B + F_A =[/itex] .
When I read your and Pfeiffer's papers I derived the same in a similar but
somewhat more direct way:
When the 2-associativity law (in definition 1 on p. 8 of your paper) is
explicitly written in terms of the definition of the vertical and the
horizontal product it is equivalent to
(*) f1 g1 f2' g1^{-1} = g2 f2' g2^{-1} f1 .
(For more details on what I mean here please see my recent reply to Amitabha
at the SCT http://golem.ph.utexas.edu/string/archives/000421.html#c001485)
My very point which I tried to make in the comment starting this discussion
is that we should try to find all solutions to this equation in order to
find all conditions under which we get a consistent surface holonomy from
egde and surface group labels.
One can easily see that _one_ solution to (*) is
f1 = g2 g1^{-1},
which is precisely the solution implying B+F=0 discussed above. So that's
how this condition comes up.
My point was that there are in fact _other_ solutions of this equation. For
instance let the edge labels be arbitrary and restrict the surface labels to
take values in an abelian ideal of the group. Then this equation is
satisfied, too. This corresponds to one of the classes of surface holonomies
that Alverz et al. discuss in terms of loop space formalism.
Moreover, I pointed out that when we go from the condition (*) to its
infinitesimal/differential version by again setting g = 1+ \epsilon A etc, we
obtain an equation which admits still one more solution. Namely flat edge
holonomies together with covariantly constant surface holonomies. Because it
is really this infinitesimal version which is needed to compute surface
holonomies in the continuum, this does give a consistent surface holonomy,
and indeed it corresponds to the other class of loop space connections
discussed by Alvarez et al.
> I guess I'll just have to read their stuff. Of course I wouldn't mind
> a wee summary in plain English....
There are a lot of ideas in that paper, but for our discussion really only a
couple of them are relevant. I have reviewed and briefly discussed them in
the second half of section 3.4 of http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0407122
which again is based on this SCT entry:
http://golem.ph.utexas.edu/string/archives/000405.html .
The idea is to write down a connection \mathcal{A} on loop space of the form
where the integral is over the given loop, W is the holonomy of A along that
loop and X' is the \sigma-derivative of the loop X : (0,2pi) -> M.
This connection gives us a consistent surface holonomy of surfaces which are
images of curves in based loop space in particular if it is flat (since then
it is independent of the parameterization of the surface in terms of \sigma
and in terms of the foliation by loops. (Flatness is not necessary, but
sufficient for this to be true.)
Alvarez et al. thought they need to set F_A = . When one does that it is
pretty easy to see that there are two cases in which \mathcal{A} is flat,
namely
1) When B is in an abelian ideal of the common algebra in which A and B
take values.
2) When B is covariantly constant with respect to A.
These two conditions dont have B+F=0, obviously (in general).
I have disucssed above and in more detail at the SCT how both these
conditions are mapped to additional solutions of the "exchange law" that
differ from
h = g2 g1^{-1} .
It turned out that Alvarez et al. were overly restrictive by assuming they
need F_A = 0, but this is how they found these 2 classes of solutions. The
first even gives consistent surface holonomy for F_A =/= .
But, while still unaware of the work by Alvarez et al., I derived in
http://www.arxiv.org/abs/hep-th/0407122 that there are more loop space connections of the above form
which are flat. Namely those obtained by setting B+F_A = . These are
moreover precisely those that preserve the above form of the connection
under local loop space gauge transformations. This is one sign of many why
B+F_A = is the most "natural" class of surface holonomies.
If you instead make a loop space gauge transformation to any of the loop
space connections by Alvarez et al. the result is something which can no
longer be written in the above form, but which involves in general _several_
1-forms together with the 2-form. So in a sense the surface holonomies by
Alvarez et al live in a framework larger that the standard 2-group theory.
But all this I have already said at the SCT in reply to Amitabha Lahiri,
where you can find it, so I'll stop here. I am in the proces of typing some
LaTeX notes about some of the ideas that we are here discussing in ASCII.
That might help the discussion.
Meanwhile you can find pretty-printed versions of the formulas that I am
considering at the SCT. With MSIE it takes just the free download of the
MathPlayer plugin, with Mozilla just the free download of a certain font to
read these equations. More details can be found here:
http://golem.ph.utexas.edu/string/archives/000316.html
Thanks a lot for your time and comments.
Hendryk Pfeiffer
Aug26-04, 07:56 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,\n\napologies that I didn\'t manage to reply earlier.\n\nI had a rather superficial look at the coffee table discussion. The key\nissues that have come up there and in the previous postings here, seem\nto be the following.\n\n1) Since 2-groups are 2-categories, there is one more level of symmetry\naround. Configurations can be related by gauge transformations. But\ngauge transformations can be related by a second level of symmetry.\n\nIn the case of lattice (or triangulation-) higher gauge theory, you can\ncompute how this relationship looks like if you go back to the category\ntheory that works behind the scenes. The second level of transformations\nare called `modifications\'. Have a look at the last section of\nhep-th/0304074 and maybe at some category theory textbook.\n\nThe problem is now the following. Without changing the rules of physics\nI can, of course, search for a Lagrangian that is invariant under the\ngauge transformations. This is what I have done in hep-th/0304074, and\nthe paper with Florian (hep-th/0309173) more or less depends on that\ndecision. But then the second level of symmetry transformations is\ncompletely useless (even though it looks as if they really `want\' to be\nthere).\n\nAs John says, this may be too `naive\' a choice and too restrictive. In\norder to do better, however, I guess you have to change the rules of\n(path integral) physics and to replace the exp(iS) terms (S=action) in\nthe path integral by something more sophisticated.\n\n2) If you just use the strict 2-group which comes from a crossed module\nwith G=H, t=id, and the action of G on H by conjugation, I guess you are\nnot really doing anything non-trivial beyond ordinary non-Abelian gauge\ntheory.\n\nWhy? Have a look at the discrete formulation again. Start with any\nordinary gauge connection, i.e. with elements of G assigned to the\nedges. For each face, you can now choose the label h in H which is just\nthe holonomy of the G-connection around its boundary. So the ordinary\ngauge theory you have started with, completely determines the second\nlevel you put on top of it, and this is always possible.\n\nIf you study this sort of higher gauge theory, the only thing it does is\nto carry some ballast around and (in the differential picture) verify\nthe Bianchi identity again and again. In the differential picture, this\nis the theory with B=-F.\n\nAlso, this sort of theory does not have intersting Wilson surfaces. The\nsurface label (in H) of any surface (Wilson surface) of topology S^2\ngets a label in the kernel of t:H->G which is the trivial group in this\ncase.\n\nIf you want something non-trivial and new, you can go for the\n`automorphism 2-group\' instead. H is any group, G=Aut(H), t:H->G assigns\nto each h in H its inner automorphism, and the action of G on H is just\nby the corresponding automorphism.\n\nA simple example: H=SU(2), G=SO(3). The Wilson surfaces have labels in H\nfrom the kernel of t which is now Z_2. If you consider a B^3 (ball) in\nd=3 space-time dimensions with boundary S^2, and the surface label of\nthis S^2 is the non-trivial element of Z_2, you can ask: if we subdivide\nthe B^3, where does the non-trivial Z_2 element come from? This looks as\nif the theory has a sort of `defects\' built-in.\n\nNotice that you need what we call the `integral\' formulation\n(hep-th/0304074) when you want to talk about consistency of curve and\nsurface holonomies. One obstacle in passing to the differential\nformulation (the one using differential 1- and 2-forms) is the fact that\nkernel t is discrete in the above example (and in many others). Since\ncontinuous Z_2-valued functions are locally constant, you cannot capture\nthe non-trivial Wilson surfaces of the above example by differential\nforms alone !! Some people might say that such a theory is genuinely\nnon-perturbative.\n\n3) Concerning the paper by Alvarez, Ferreira&Sanchez-Guillen you\nmention, I just like to emphasize that having consistent curve and\nsurface holonomies (not just some differential 1- and 2-forms) is a\nrather strong condition.\n\nWe know that strict 2-groups do the job, and presumably this can also be\ndone by weak 2-groups (see math.QA/0307200).\n\nThe challence would then be to take the construction of Alvarez et al,\nto convert this into statements about holonomies, and to figure out\neither which 2-group they use or whether and in which way the result is\nmore general than the construction with 2-groups. I don\'t know the answer.\n\n4)\n\n>> 1) It contains a flaw and 2-group theory is right that only t(B)+F =\n>> 0 gives well defined surface holonomy.\n\n>I should note that it\'s not "2-group theory" which makes this claim,\n>but a paper by Girelli and Pfeiffer. And I\'m not even sure they claim\n>this in an ironclad way. Personally I\'m very confused about all this\n>stuff, so more confusion is actually a good thing - it might\n>disentangle something.\n\nWe actually don\'t claim that at all. The only thing we claim is that if\nyou take the `integral\' formulation of higher gauge theory\n(hep-th/0304074), assume (!!) that all curve and surface labels depend\nsmoothly on the positions of the curves and surfaces, and then\ndifferentiate everything, you get t(B)=-F.\n\nMy remark at the end of (2) above is probably a good reason why you may\nnot want everything to be smooth. This means that things are more\ncomplicated than just having differential 1- and 2-forms around and some\nrelations between them.\n\nBest regards\n\nHendryk\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,
apologies that I didn't manage to reply earlier.
I had a rather superficial look at the coffee table discussion. The key
issues that have come up there and in the previous postings here, seem
to be the following.
1) Since 2-groups are 2-categories, there is one more level of symmetry
around. Configurations can be related by gauge transformations. But
gauge transformations can be related by a second level of symmetry.
In the case of lattice (or triangulation-) higher gauge theory, you can
compute how this relationship looks like if you go back to the category
theory that works behind the scenes. The second level of transformations
are called `modifications'. Have a look at the last section of
http://www.arxiv.org/abs/hep-th/0304074 and maybe at some category theory textbook.
The problem is now the following. Without changing the rules of physics
I can, of course, search for a Lagrangian that is invariant under the
gauge transformations. This is what I have done in http://www.arxiv.org/abs/hep-th/0304074, and
the paper with Florian (http://www.arxiv.org/abs/hep-th/0309173) more or less depends on that
decision. But then the second level of symmetry transformations is
completely useless (even though it looks as if they really `want' to be
there).
As John says, this may be too `naive' a choice and too restrictive. In
order to do better, however, I guess you have to change the rules of
(path integral) physics and to replace the \exp(iS) terms (S=action) in
the path integral by something more sophisticated.
2) If you just use the strict 2-group which comes from a crossed module
with G=H, t=id, and the action of G on H by conjugation, I guess you are
not really doing anything non-trivial beyond ordinary non-Abelian gauge
theory.
Why? Have a look at the discrete formulation again. Start with any
ordinary gauge connection, i.e. with elements of G assigned to the
edges. For each face, you can now choose the label h in H which is just
the holonomy of the G-connection around its boundary. So the ordinary
gauge theory you have started with, completely determines the second
level you put on top of it, and this is always possible.
If you study this sort of higher gauge theory, the only thing it does is
to carry some ballast around and (in the differential picture) verify
the Bianchi identity again and again. In the differential picture, this
is the theory with B=-F.
Also, this sort of theory does not have intersting Wilson surfaces. The
surface label (in H) of any surface (Wilson surface) of topology S^2
gets a label in the kernel of t:H->G which is the trivial group in this
case.
If you want something non-trivial and new, you can go for the
`automorphism 2-group' instead. H is any group, G=Aut(H), t:H->G assigns
to each h in H its inner automorphism, and the action of G on H is just
by the corresponding automorphism.
A simple example: H=SU(2), G=SO(3). The Wilson surfaces have labels in H
from the kernel of t which is now Z_2. If you consider a B^3 (ball) in
d=3 space-time dimensions with boundary S^2, and the surface label of
this S^2 is the non-trivial element of Z_2, you can ask: if we subdivide
the B^3, where does the non-trivial Z_2 element come from? This looks as
if the theory has a sort of `defects' built-in.
Notice that you need what we call the `integral' formulation
(http://www.arxiv.org/abs/hep-th/0304074) when you want to talk about consistency of curve and
surface holonomies. One obstacle in passing to the differential
formulation (the one using differential 1- and 2-forms) is the fact that
kernel t is discrete in the above example (and in many others). Since
continuous Z_2-valued functions are locally constant, you cannot capture
the non-trivial Wilson surfaces of the above example by differential
forms alone !! Some people might say that such a theory is genuinely
non-perturbative.
3) Concerning the paper by Alvarez, Ferreira&Sanchez-Guillen you
mention, I just like to emphasize that having consistent curve and
surface holonomies (not just some differential 1- and 2-forms) is a
rather strong condition.
We know that strict 2-groups do the job, and presumably this can also be
done by weak 2-groups (see math.QA/0307200).
The challence would then be to take the construction of Alvarez et al,
to convert this into statements about holonomies, and to figure out
either which 2-group they use or whether and in which way the result is
more general than the construction with 2-groups. I don't know the answer.
4)
>> 1) It contains a flaw and 2-group theory is right that only t(B)+F =
>> gives well defined surface holonomy.
>I should note that it's not "2-group theory" which makes this claim,
>but a paper by Girelli and Pfeiffer. And I'm not even sure they claim
>this in an ironclad way. Personally I'm very confused about all this
>stuff, so more confusion is actually a good thing - it might
>disentangle something.
We actually don't claim that at all. The only thing we claim is that if
you take the `integral' formulation of higher gauge theory
(http://www.arxiv.org/abs/hep-th/0304074), assume (!!) that all curve and surface labels depend
smoothly on the positions of the curves and surfaces, and then
differentiate everything, you get t(B)=-F.
My remark at the end of (2) above is probably a good reason why you may
not want everything to be smooth. This means that things are more
complicated than just having differential 1- and 2-forms around and some
relations between them.
Best regards
Hendryk
Urs Schreiber
Aug26-04, 09:06 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:cgkind\\$jj8\\$1@pegasus.csx.cam.ac.uk...\n\n> As John says, this may be too `naive\' a choice and too restrictive. In\n> order to do better, however, I guess you have to change the rules of\n> (path integral) physics and to replace the exp(iS) terms (S=action) in\n> the path integral by something more sophisticated.\n\nHm, that sounds pretty drastic.\n\nAnother possibility is to go to loop space. The 2-group holonomies that you\nare considering become holonomies of an ordinary 1-form \\mathcal{A} in loop\nspace. I have shown how the ordinary gauge transformations on loop space\nproduce the two sorts of gauge transformations in 2-group theory. So if you\nwant a theory invariant under these gauge transformations all you have to do\nis write a Lagrangian in terms of gauge-invariant quantities constructed\nfrom \\mathcal{A}. One simple example would be Yang-Mills theory on loop\nspace in terms of this loop space connection. I haven\'t tried to work out\nany consequences of that, but at least you can write it down and it does\nhave the gauge invariances that you are talking about, for _any_ choice of\n1-forms A and 2-forms B. (Even though some gauge transformations will take\nyou out of the class of loop space connections that are expressible in terms\nof a single 2-form and a single 1-form on target space.)\n\nIt is maybe interesting to note that a local field theory on loop space has\ndifferent interactions of loops than ordinary string field theory does.\nString field theory (e.g. open bosonic cubic SFT) is _non-local_ in loop\nspace, because two strings merge by overlapping with one half, each.\nHowever, in a field theory which is _local_ on loop space (like loop space\nYM) two strings interact only when they overlap completely.\n\n\n> 2) If you just use the strict 2-group which comes from a crossed module\n> with G=H, t=id, and the action of G on H by conjugation, I guess you are\n> not really doing anything non-trivial beyond ordinary non-Abelian gauge\n> theory.\n>\n> Why?\n\nYup. I know why. That is, I know what you are getting at here, even though I\nwouldn\'t say there is not anything non-trivial to be found. You are saying\nthat the condition B+F = 0 completely fixed my 2-form when the 1-form is\nchosen. True, but...\n\n\n> If you study this sort of higher gauge theory, the only thing it does is\n> to carry some ballast around and (in the differential picture) verify\n> the Bianchi identity again and again. In the differential picture, this\n> is the theory with B=-F.\n\n\nYes. If you are just interested in gauge theory equations of motion then\nthis is just ordinary YM with the "ballast" that you can now also talk about\nsurface holonomies.\n\nBut when you put a string in this 2-form background with B+F=0 you see that\nit sees quite a different background than just an ordinary gauge field A.\nThe boundary state of the 2-form theory is different from that of the 1-form\ntheory.\n\nSo the background equations of motion are the same in this case as of the\ngauge theory alone, still there is now a notion of surface holonomy and so\nsurfaces can couple to the background field, too. And yes, since B+F=0\nimplies that the connection on loop space is flat, it follows that closed\nsurfaces have trivial surface holonomy. But that\'s fine. Bounded surfaces\nstill have non-trivial coupling.\n\n\n> Also, this sort of theory does not have intersting Wilson surfaces. The\n> surface label (in H) of any surface (Wilson surface) of topology S^2\n> gets a label in the kernel of t:H->G which is the trivial group in this\n> case.\n\n\nTrue. (Still, there are nontrivial holonomies for bounded surfaces.) One of\nthe points that I am discussing here is that even with G=H and t trivial one\ncan have non-trivial Wilson surfaces. The conserved charges in 2+1d\nintegrable models studied by Alvarez et al are examples for that. This is\npossible because they noted that there are consistent surface holonomies\neven for B+F non-vanishing.\n\n\n> If you want something non-trivial and new, you can go for the\n> `automorphism 2-group\' instead. H is any group, G=Aut(H), t:H->G assigns\n> to each h in H its inner automorphism, and the action of G on H is just\n> by the corresponding automorphism.\n>\n> A simple example: H=SU(2), G=SO(3). The Wilson surfaces have labels in H\n> from the kernel of t which is now Z_2. If you consider a B^3 (ball) in\n> d=3 space-time dimensions with boundary S^2, and the surface label of\n> this S^2 is the non-trivial element of Z_2, you can ask: if we subdivide\n> the B^3, where does the non-trivial Z_2 element come from? This looks as\n> if the theory has a sort of `defects\' built-in.\n>\n> Notice that you need what we call the `integral\' formulation\n> (hep-th/0304074) when you want to talk about consistency of curve and\n> surface holonomies. One obstacle in passing to the differential\n> formulation (the one using differential 1- and 2-forms) is the fact that\n> kernel t is discrete in the above example (and in many others). Since\n> continuous Z_2-valued functions are locally constant, you cannot capture\n> the non-trivial Wilson surfaces of the above example by differential\n> forms alone !! Some people might say that such a theory is genuinely\n> non-perturbative.\n\n\nI have seen your discussion of this point in your papers. That\'s a maybe\ninteresting alternative to the high restrictivivty of the continuum theory.\n\n\n> 3) Concerning the paper by Alvarez, Ferreira&Sanchez-Guillen you\n> mention, I just like to emphasize that having consistent curve and\n> surface holonomies (not just some differential 1- and 2-forms) is a\n> rather strong condition.\n\n\nPlease note that nobody is claiming that it suffices to have "just some\ndifferential 1- and 2-forms". I discuss conditions for loop space\nconnections (including those cooked up from one 1-form and one 2-form on\ntarget space) to give consistent surface holonomies in hep-th/0407122.\n\nOne such condition (covering most of the gound but being not the only one)\nis (in based loop space) that the connection on loop space is flat. This\nflatness ensures that the surface holonomy is independent of the foliation\nby loops and hence well defined.\n\nTaking the results by Alvarez et al. and of my paper together, there are\nprecisely three conditions known which make a loop space connection cooked\nup from a 1-form and a 2-from on target space flat.\n\n1) B + F = 0\n2) d_A B = 0, F=0, B in an abelian ideal\n3) \\partial_A B = 0, F=0\n\nThese results are derived with G=H assumed, but I think it is pretty obvious\nthat they generalize to the more general cases with G\\neq H and/or t\nnontrivial. I\'ll write that up as soon as possible.\n\n\n> We know that strict 2-groups do the job, and presumably this can also be\n> done by weak 2-groups (see math.QA/0307200).\n\n\nMaybe people working on weak 2-groups have reproduced conditions 2) and 3)\nabove?\n\n\n> The challence would then be to take the construction of Alvarez et al,\n> to convert this into statements about holonomies, and to figure out\n> either which 2-group they use or whether and in which way the result is\n> more general than the construction with 2-groups. I don\'t know the answer.\n\n\nI think I do know the answer to that. I have discussed in\n\nhttp://golem.ph.utexas.edu/string/archives/000416.html\n\nhow one can translate from the loop space formalism used by Alvarez,\nFerreira, Sanchez-Guillen and myself to 2-group calculations. I show that\nthe (based) loop space holonomies computed with a loop space connection of\nthe form\n\n\\mathcal{A} = int_0^{2\\pi}dsigma (W_A B_mn W_A^{-1} X\'^m dX^n)(sigma)\n\ncompute precisely the same surface holonomy as obtained from 2-group theory\nwith 1-form A and 2-form B. You can easily convince yourself that the\nderivation of this result remains true when you choose H =/= G, replace the\nabove conjugation by W_A with the triangle operation in your paper and so\non.\n\nThat\'s precisely the reason why I started this entire discussion here. The\nabove gives a map between computations of surface holonomies using loop\nspace technology and those using 2-group theory, so both are two ways to\nlook at the same thing. Still, in the literature using loop space formalisms\nmore consistent surface holonomies are known than in 2-group theory. My\nquestion is: What is going on here? I believe the answer is that t(B) +F = 0\nis not the most general solution of the exchange law (which is precisely the\nlaw which ensures consistent surface holonomy in the 2-group approach).\n\n\n> >> 1) It contains a flaw and 2-group theory is right that only t(B)+F =\n> >> 0 gives well defined surface holonomy.\n>\n> >I should note that it\'s not "2-group theory" which makes this claim,\n> >but a paper by Girelli and Pfeiffer. And I\'m not even sure they claim\n> >this in an ironclad way. Personally I\'m very confused about all this\n> >stuff, so more confusion is actually a good thing - it might\n> >disentangle something.\n>\n> We actually don\'t claim that at all. The only thing we claim is that if\n> you take the `integral\' formulation of higher gauge theory\n> (hep-th/0304074), assume (!!) that all curve and surface labels depend\n> smoothly on the positions of the curves and surfaces, and then\n> differentiate everything, you get t(B)=-F.\n\n\nHm. So you do claim that for the case that all forms are sufficiently\ndifferentiable. That is the case I am talking about. I think this is natural\nand is the case that should be fully understood before we consider exotic\nsetups where our fields are not differentiable.\n\nI am claiming that in this case, where the fields are differentiable, you\ndon\'t necessarily get t(B) = -F. You do get that from differentiating\nequation (2.10) in your hep-th/0309173, which says that surface labels are\nthe same as their sorrounding holonomies. Where does this condition come\nfrom? I am claiming that ultimately this condition comes from the exchange\nlaw. It is one solution of the exchange law condition. But there are other\nsolutions of this condition and I believe these other solutions are\nprecisely those found by Alvarez et al in loop space formalism.\n\n\n> My remark at the end of (2) above is probably a good reason why you may\n> not want everything to be smooth. This means that things are more\n> complicated than just having differential 1- and 2-forms around and some\n> relations between them.\n\n\nMaybe. Before considering non-smooth fields I would like to understand the\nsubtleties of the smooth case.\n\nI am very grateful for all your comments, even though I feel that to some\nextent we are still somewhat talking past each other. This is probably due\nto my comments being scattered in various posts here and at the SCT. I\'ll\nsee if I can write up a more bundled discussion.\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:cgkind$jj8$1@pegasus.csx.cam.ac.uk...
> As John says, this may be too `naive' a choice and too restrictive. In
> order to do better, however, I guess you have to change the rules of
> (path integral) physics and to replace the \exp(iS) terms (S=action) in
> the path integral by something more sophisticated.
Hm, that sounds pretty drastic.
Another possibility is to go to loop space. The 2-group holonomies that you
are considering become holonomies of an ordinary 1-form \mathcal{A} in loop
space. I have shown how the ordinary gauge transformations on loop space
produce the two sorts of gauge transformations in 2-group theory. So if you
want a theory invariant under these gauge transformations all you have to do
is write a Lagrangian in terms of gauge-invariant quantities constructed
from \mathcal{A}. One simple example would be Yang-Mills theory on loop
space in terms of this loop space connection. I haven't tried to work out
any consequences of that, but at least you can write it down and it does
have the gauge invariances that you are talking about, for _any_ choice of
1-forms A and 2-forms B. (Even though some gauge transformations will take
you out of the class of loop space connections that are expressible in terms
of a single 2-form and a single 1-form on target space.)
It is maybe interesting to note that a local field theory on loop space has
different interactions of loops than ordinary string field theory does.
String field theory (e.g. open bosonic cubic SFT) is _non-local_ in loop
space, because two strings merge by overlapping with one half, each.
However, in a field theory which is _local_ on loop space (like loop space
YM) two strings interact only when they overlap completely.
> 2) If you just use the strict 2-group which comes from a crossed module
> with G=H, t=id, and the action of G on H by conjugation, I guess you are
> not really doing anything non-trivial beyond ordinary non-Abelian gauge
> theory.
>
> Why?
Yup. I know why. That is, I know what you are getting at here, even though I
wouldn't say there is not anything non-trivial to be found. You are saying
that the condition B+F = completely fixed my 2-form when the 1-form is
chosen. True, but...
> If you study this sort of higher gauge theory, the only thing it does is
> to carry some ballast around and (in the differential picture) verify
> the Bianchi identity again and again. In the differential picture, this
> is the theory with B=-F.
Yes. If you are just interested in gauge theory equations of motion then
this is just ordinary YM with the "ballast" that you can now also talk about
surface holonomies.
But when you put a string in this 2-form background with B+F=0 you see that
it sees quite a different background than just an ordinary gauge field A.
The boundary state of the 2-form theory is different from that of the 1-form
theory.
So the background equations of motion are the same in this case as of the
gauge theory alone, still there is now a notion of surface holonomy and so
surfaces can couple to the background field, too. And yes, since B+F=0
implies that the connection on loop space is flat, it follows that closed
surfaces have trivial surface holonomy. But that's fine. Bounded surfaces
still have non-trivial coupling.
> Also, this sort of theory does not have intersting Wilson surfaces. The
> surface label (in H) of any surface (Wilson surface) of topology S^2
> gets a label in the kernel of t:H->G which is the trivial group in this
> case.
True. (Still, there are nontrivial holonomies for bounded surfaces.) One of
the points that I am discussing here is that even with G=H and t trivial one
can have non-trivial Wilson surfaces. The conserved charges in 2+1d
integrable models studied by Alvarez et al are examples for that. This is
possible because they noted that there are consistent surface holonomies
even for B+F non-vanishing.
> If you want something non-trivial and new, you can go for the
> `automorphism 2-group' instead. H is any group, G=Aut(H), t:H->G assigns
> to each h in H its inner automorphism, and the action of G on H is just
> by the corresponding automorphism.
>
> A simple example: H=SU(2), G=SO(3). The Wilson surfaces have labels in H
> from the kernel of t which is now Z_2. If you consider a B^3 (ball) in
> d=3 space-time dimensions with boundary S^2, and the surface label of
> this S^2 is the non-trivial element of Z_2, you can ask: if we subdivide
> the B^3, where does the non-trivial Z_2 element come from? This looks as
> if the theory has a sort of `defects' built-in.
>
> Notice that you need what we call the `integral' formulation
> (http://www.arxiv.org/abs/hep-th/0304074) when you want to talk about consistency of curve and
> surface holonomies. One obstacle in passing to the differential
> formulation (the one using differential 1- and 2-forms) is the fact that
> kernel t is discrete in the above example (and in many others). Since
> continuous Z_2-valued functions are locally constant, you cannot capture
> the non-trivial Wilson surfaces of the above example by differential
> forms alone !! Some people might say that such a theory is genuinely
> non-perturbative.
I have seen your discussion of this point in your papers. That's a maybe
interesting alternative to the high restrictivivty of the continuum theory.
> 3) Concerning the paper by Alvarez, Ferreira&Sanchez-Guillen you
> mention, I just like to emphasize that having consistent curve and
> surface holonomies (not just some differential 1- and 2-forms) is a
> rather strong condition.
Please note that nobody is claiming that it suffices to have "just some
differential 1- and 2-forms". I discuss conditions for loop space
connections (including those cooked up from one 1-form and one 2-form on
target space) to give consistent surface holonomies in http://www.arxiv.org/abs/hep-th/0407122.
One such condition (covering most of the gound but being not the only one)
is (in based loop space) that the connection on loop space is flat. This
flatness ensures that the surface holonomy is independent of the foliation
by loops and hence well defined.
Taking the results by Alvarez et al. and of my paper together, there are
precisely three conditions known which make a loop space connection cooked
up from a 1-form and a 2-from on target space flat.
1) B + F = 2) d_A B = 0, F=0, B[/itex] in an abelian ideal
3) \partial_A B = 0, F=0
These results are derived with G=H assumed, but I think it is pretty obvious
that they generalize to the more general cases with G\neq H and/or t
nontrivial. I'll write that up as soon as possible.
> We know that strict 2-groups do the job, and presumably this can also be
> done by weak 2-groups (see math.QA/0307200).
Maybe people working on weak 2-groups have reproduced conditions 2) and 3)
above?
> The challence would then be to take the construction of Alvarez et al,
> to convert this into statements about holonomies, and to figure out
> either which 2-group they use or whether and in which way the result is
> more general than the construction with 2-groups. I don't know the answer.
I think I do know the answer to that. I have discussed in
how one can translate from the loop space formalism used by Alvarez,
Ferreira, Sanchez-Guillen and myself to 2-group calculations. I show that
the (based) loop space holonomies computed with a loop space connection of
the form
compute precisely the same surface holonomy as obtained from 2-group theory
with 1-form A and 2-form B. You can easily convince yourself that the
derivation of this result remains true when you choose H =/= G, replace the
above conjugation by W_A with the triangle operation in your paper and so
on.
That's precisely the reason why I started this entire discussion here. The
above gives a map between computations of surface holonomies using loop
space technology and those using 2-group theory, so both are two ways to
look at the same thing. Still, in the literature using loop space formalisms
more consistent surface holonomies are known than in 2-group theory. My
question is: What is going on here? I believe the answer is that t(B) +F =
is not the most general solution of the exchange law (which is precisely the
law which ensures consistent surface holonomy in the 2-group approach).
> >> 1) It contains a flaw and 2-group theory is right that only [itex]t(B)+F =
> >> gives well defined surface holonomy.
>
> >I should note that it's not "2-group theory" which makes this claim,
> >but a paper by Girelli and Pfeiffer. And I'm not even sure they claim
> >this in an ironclad way. Personally I'm very confused about all this
> >stuff, so more confusion is actually a good thing - it might
> >disentangle something.
>
> We actually don't claim that at all. The only thing we claim is that if
> you take the `integral' formulation of higher gauge theory
> (http://www.arxiv.org/abs/hep-th/0304074), assume (!!) that all curve and surface labels depend
> smoothly on the positions of the curves and surfaces, and then
> differentiate everything, you get t(B)=-F.
Hm. So you do claim that for the case that all forms are sufficiently
differentiable. That is the case I am talking about. I think this is natural
and is the case that should be fully understood before we consider exotic
setups where our fields are not differentiable.
I am claiming that in this case, where the fields are differentiable, you
don't necessarily get t(B) = -F. You do get that from differentiating
equation (2.10) in your http://www.arxiv.org/abs/hep-th/0309173, which says that surface labels are
the same as their sorrounding holonomies. Where does this condition come
from? I am claiming that ultimately this condition comes from the exchange
law. It is one solution of the exchange law condition. But there are other
solutions of this condition and I believe these other solutions are
precisely those found by Alvarez et al in loop space formalism.
> My remark at the end of (2) above is probably a good reason why you may
> not want everything to be smooth. This means that things are more
> complicated than just having differential 1- and 2-forms around and some
> relations between them.
Maybe. Before considering non-smooth fields I would like to understand the
subtleties of the smooth case.
I am very grateful for all your comments, even though I feel that to some
extent we are still somewhat talking past each other. This is probably due
to my comments being scattered in various posts here and at the SCT. I'll
see if I can write up a more bundled discussion.