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April 11, 2008

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 BBimod(C)\mathbf{B} Bimod(C), for CC 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

On Σ\Sigma-models and nonabelian differential cohomology

Abstract. A “Σ\Sigma-model” can be thought of as a quantum field theory (QFT) which is determined by pulling back nn-bundles with connection (aka (n1n-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 Σ\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

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

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 BBimod(C)\mathbf{B}\mathrm{Bimod}(C) which arise as components of a pseudonatural transformation between two 3-functors that factor through BBCBBimod(C)\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 3Cat3\mathrm{Cat}, which says that a transformation between two 3-functors is itself, in components, a 2-functor.

Posted at April 11, 2008 6:14 AM UTC

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Re: Sigma-Models and Nonabelian Differential Cohomology

I am currently giving a series of talks “on nonabelian differential cohomology” at Notre Dame, kindly invited by Stephan Stolz.

I have started turning my hand-written notes into a LaTeX-file:

Talk: On nonabelian differential cohomology (pdf, 26 pages)

Abstract. Nonabelian differential nn-cocycles provide the data for an assignment of “quantities” to nn-dimensional “spaces” which is

  • locally controlled by a given “typical quantity”;
  • globally compatible with all possible gluings of volumes.

For n=1n=1 this encompasses the notion of parallel transport in fiber bundles with connection. In general we think of it as parallel nn-transport.

For low nn and/or “sufficiently abelian quantities” this has been modeled by differential characters, (n1)(n-1)-gerbes, (n1)(n-1)-bundle gerbes and nn-bundles with connection. We give a general definition for all nn in terms of descent data for transport nn-functors along the lines of [BS,SWI,SWII,SWIII]. Concrete realizations, notably Chern-Simons nn-cocycles, are obtained by integrating L L_\infty-algebras and their higher Cartan-Ehresmann connections [SSS].

Tomorrow is the third talk, where I should say a bit more about the general definition of nonabelian differential cohomology and then pass to the realizations in terms of L L_\infty-algebras. At that point the notes currently are still a bit incomplete.

Posted by: Urs Schreiber on April 17, 2008 5:41 AM | Permalink | Reply to this
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