The n-Category Café

Sigma-Models and Nonabelian Differential Cohomology

Posted by Urs Schreiber

At the moment I am travelling a bit, until the HIM trimester program on Geometry and Physics begins next month.

I just spent a week in Aarhus, Denmark, where I was working with Jens Fjelstad on our description of rational conformal field theory as a parallel 2-transport with values in cylinders in the 3-category $\mathbf{B} Bimod(C)$, for $C$ a modular tensor category (discussed before here).

Next week I’ll visit Stephan Stolz at the University of Notre Dame, and then the week after Jim Stasheff at UPenn.

I am talking about aspects of what has happened so far, which I am trying to summarize in

U.S.
On $\Sigma$-models and nonabelian differential cohomology
(pdf).

Abstract. A “$\Sigma$-model” can be thought of as a quantum field theory (QFT) which is determined by pulling back $n$-bundles with connection (aka ($n-1$)-gerbes with connection, aka nonabelian differential cocycles) along all possible maps (the “fields”) from a “parameter space” to the given base space.

If formulated suitably, such $\Sigma$-models include gauge theories such as notably (higher) Chern-Simons theory. If the resulting QFT is considered as an “extended” QFT, it should itself be a nonabelian differential cocycle on parameter space whose parallel transport along pieces of parameter space encodes the QFT propagation and correlators.

We are after a conception of nonabelian differential cocycles and their quantization which captures this.

Our main motivation is the quantization of differential Chern-Simons cocycles to extended Chern-Simons QFT and its boundary conformal QFT, reproducing the cocycle structure implicit in [FFRS].

The CFT aspect of this, mentioned in the abstract, is the content of my work with Jens:

Jens Fjelstad & U.S.
Rational CFT is parallel transport
(pdf)

Abstract. From the data of any semisimple modular tensor category $C$ the prescription [Reshetikhin-Turaev] constructs a 3-dimensional TFT by encoding 3-manifolds in terms of string diagrams in $C$. From the additional data of a certain Frobenius algebra object internal to $C$, the presciption [FFRS, FFRS] obtains (the combinatorial aspect of) the corresponding full boundary CFT by decorating triangulations of surfaces with objects and morphisms in $C$.

We show that these decoration prescriptions are “quantum differential cocycles” on the worldvolume for a 3-functorial extended QFT. The boundary CFT arises from a morphism between two chiral copies of the (locally trivialized) TFT 3-functor.

The crucial observation is that all 3-dimensional string diagrams in [FFRS] are Poincaré-dual to cylinders in $\mathbf{B}\mathrm{Bimod}(C)$ which arise as components of a pseudonatural transformation between two 3-functors that factor through $\mathbf{B}\mathbf{B} C \hookrightarrow \mathbf{B}\mathrm{Bimod}(C)$.

This exhibits the “holographic” relation between 3d TFT and 2d CFT as the hom-adjunction in $3\mathrm{Cat}$, which says that a transformation between two 3-functors is itself, in components, a 2-functor.