# Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

@article{Carey2004BundleGF, title={Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories}, author={Alan Carey and Stuart Johnson and Michael K. Murray and Danny Stevenson and Bai-Ling Wang}, journal={Communications in Mathematical Physics}, year={2004}, volume={259}, pages={577-613} }

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by…

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## References

SHOWING 1-10 OF 50 REFERENCES

### Some comments on Chern-Simons gauge theory

- Mathematics
- 1989

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe…

### Topological gauge theories and group cohomology

- Mathematics
- 1990

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a…

### BUNDLE GERBES APPLIED TO QUANTUM FIELD THEORY

- Mathematics
- 2000

This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah–Patodi–Singer index…

### Gerbes, Simplicial Forms and Invariants for Families of Foliated Bundles

- Mathematics
- 2003

The notion of smooth Deligne cohomology is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this…

### The Formulation of the Chern-Simons Action for General Compact Lie Groups Using Deligne Cohomology

- Mathematics
- 2001

We formulate the Chern-Simons action for any com- pact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underly- ing 3-manifold to…

### The Universal Gerbe and Local Family Index Theory

- Mathematics
- 2004

The goal of this paper is to apply the universal gerbe developed in [CMi1] and [CMi2] and the local family index theorems to give a unified viewpoint on the known examples of geometrically…

### CLASSICAL CHERN-SIMONS THEORY, PART 2

- Mathematics
- 1992

There is a large mathematical literature on classical mechanics and field theory, especially on the relationship to symplectic geometry. One might think that the classical Chern-Simons theory, which…

### Bundle gerbes

- Mathematics
- 1994

. Just as C × principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral…

### Elliptic Cohomology: The M -theory 3-form and E 8 gauge theory

- Mathematics
- 2007

We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8…

### Gerbes in classical Chern-Simons theory

- Mathematics
- 2001

We construct geometrically a gerbe assigned to a connection on a principal SU(2)-bundle over an oriented closed 1-dimensional manifold. If the connection is given by the restriction of a connection…