Re: categorified Yang-Mills equations
From: John Baez (baez_at_galaxy.ucr.edu)
Date: 08/24/04
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Date: Tue, 24 Aug 2004 09:41:17 +0000 (UTC)
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/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.
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