Re: Two notions of 2-multiplication

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 12:21:05 -0400

"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0409150200.2a351028-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<2qo2teFvrkvqU1-100000@uni-berlin.de>...

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

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

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

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

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

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

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

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

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

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

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

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

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

In

O. Ganor,
Six-dimensional tensionless strings in the large N limit,
hep-th/9605201

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

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

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

A very recent paper addressing this issue is

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

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

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

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

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

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

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

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

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

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

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

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

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

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