### What Has Happened So Far

#### Posted by Urs Schreiber

The $n$-Category Café has recently passed beyond $6 \cdot 10^2$ entries, $1.3 \cdot 10^3$ trackbacks and $1\cdot 10^4$ comments. Maybe a good time to look back at what has happened so far.

Our subtitle says “A blog on math, physics and philosophy”. For me, there is one major question sitting at the intersection of these three subjects. It is

The fundamental question of quantum physics:What is a $\Sigma$-model,really?

I have been exclusively talking about this question ever since we started the blog. I started referring to it as the question of the *QFT of the charged $n$-particle* #. I still think this is the more descriptive term, but it was rightly indicated to me that it is not politically advisable for somebody in my position to make up new terminology.

Since it was also pointed out to me ## that it may at times be hard to remember the big picture, let me recall:

The proposed answer to the fundamental question of quantum physics:Pull-push of nonabelian differential cocycles.We are in the setting of general cohomology theory, where generalized/homotopy/ana-morphisms $X \stackrel{\nabla}{\to} \mathbf{B}G$ between “spaces” (usually # presheaves with values in a homotopy category) are “cocycles” encoding higher fiber bundles. And also higher fiber bundles with connection, which are addressed as (nonabelian)

differential cocycles#.Given a (nonabelian, differential) cocycle on $X$, and given another “space” $\Sigma$, there is a canonical way to obtain a cocycle on $\Sigma$: we pull-push $\nabla$ through the correspondence $\array{ && hom(\Sigma,X)\otimes \Sigma \\ & {}^{ev}\swarrow && \searrow^{p_2} \\ X&&&& \Sigma \\ \nabla && \stackrel{\Gamma_\Sigma ev^*(-)}{\mapsto} && \Gamma_\Sigma( ev^* \nabla ) }\,.$

The pullback along $\mathrm{ev}$ (followed by the hom-adjunction) is

transgressionof the cocycle on $X$ to a cocycle on $hom(\Sigma,X)$. The push-forward along $p_2$ is “taking sections” ## #.Usually the push-forward along $p_2$ won’t exist. The chances that it exists increase when the original cocycle is pushed-forward along a

representation$\rho : \mathbf{B} G \to n\mathrm{Vect} \,.$In the context of quantum physics, $X$ is the

target spacein which an “($n-1$)-brane” (= $n$-particle) withworldvolume# of shape $\Sigma$ propagates and is charged # # under abackground field$\rho_* \nabla$. The pull-push $\Gamma_{\Sigma} ev^*(-)$ isquantizationin the extended/localized # sense of Freed ##. $\Gamma_{\Sigma} ev^*(\nabla)$ is theSchrödinger picture # propagation. Applying an endomorphism functor sends it to theHeisenberg picture# of AQFT #. Since quantization sends differential cocycles to differential cocycles, we can iterate. This issecond quantization#.

While following through this program, we ran into one big puzzle, concerning the proper nature of $n$-curvature: it turned out that a differential cocycle “with values in $\mathbf{B}G$” is actually a certain constrained generalized morphism into # $\mathbf{B E}G$. Understanding that funny *shift in dimension* properly used up maybe 50 percent of my time here, and is probably the reason if the effort looked less than coherent at times.

Making recourse to the “rationalized” approximation of $L_\infty$-connections # the pattern was finally understood, and now there are very nice relations emerging # between this question and major programs of my co-bloggers: *higher topos theory* and *geometric representation theory/groupoidification*.

There is one main class of examples which motivates all this effort: quantization of # (higher) Chern-Simons bundles with connection to Chern-Simons QFT ## and its holographic # #boundary theory. Indeed, the realization # that the known modular category theoretic formulation of 2-dimensional CFT # # was in fact secretly a differential cocycle was what originally lead to the *proposed answer* above. This is being worked out with Jens Fjeldstad #.

The hardest part of figuring out the pull-push of a given cocycle is in top dimension. This is no surprise, since there it must reproduce the “path integral”. But first consistency checks in simple toy examples suggest that it does work # # allright.

But with the big picture finally stabilizing, many details need to be worked out further.

Posted at March 27, 2008 10:47 AM UTC
## Re: What has happened so far

Dear Urs,

you posted on Peter Woit´s blog:

I thought that the models that are uniquely said to be truely superstrings. Or at least that is what seems but taking a daily look at hep-th. Some trolle people even troll to say that those are uniquely the true ones. I mean, that’s the impression I take by looking everywhere…

At that time I thought ( in an aesthetical way, because we always think of this theories as pants and strings and sponges…) of the foam models and (perhaps) LQG could be adapted to string theories… But right now, on this post, you pointed to this, and I saw Eric´s post

http://golem.ph.utexas.edu/string/archives/000813.html

In a sense, are you and J. Baez trying to unify “non string theories” and “string theories” ?

Daniel de França