Advances in High Energy Physics

Volume 2017 (2017), Article ID 2878949, 6 pages

https://doi.org/10.1155/2017/2878949

## Exact Computations in Topological Abelian Chern-Simons and BF Theories

LAPTh, Université de Savoie, CNRS, Chemin de Bellevue, BP 110, 74941 Annecy-le-Vieux Cedex, France

Correspondence should be addressed to Philippe Mathieu; rf.srnc.htpal@ueihtam.eppilihp

Received 15 May 2017; Accepted 18 June 2017; Published 30 August 2017

Academic Editor: Ralf Hofmann

Copyright © 2017 Philippe Mathieu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We introduce Deligne cohomology that classifies fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

#### 1. Introduction

Consider the following actions:where and are connections. Here, the coupling constant is any real number.

The gauge transformation , where is a function that leaves the actions (1) invariant. Since in the quantum context we consider the complex exponential of the action, the invariance required is less restrictive. Indeed, we can consider an invariance of up to an integer:which implies that is quantized. Studying the gauge invariance properties of the holonomies, which are the observables of Chern-Simons and BF theories, it turns out that the most general gauge transformation is , where is a closed 1-form with integral periods. On a contractible open set this transformation reduces to the classical one since, by Poincaré Lemma, there exists such that . In particular, this is the case when the theory is defined in which is a contractible space. However, this generalized gauge transformation enables defining a theory on any closed (i.e., compact without boundary) -manifold . The classical gauge transformation appears thus to be a particular case of the quantum one.

In this paper we will consider the equivalence classes according to this quantum gauge transformation. These classes classify fibre bundles over endowed with connections and their collection is the so-called first Deligne cohomology group of . We will show that this structure enables performing exact computations in the framework of Chern-Simons and BF theories.

#### 2. Deligne Cohomology

The most general statement we can start from is a collection of local gauge fields in open sets that cover the manifold we are considering. We suppose these open sets and their intersections to be contractible, so that we can in particular use the Poincaré Lemma inside. To define a global field, we need to explain how and stick together in the intersection . This, by definition, is done thanks to a gauge transformation:The antisymmetry of this relation in and implies that , making a constant in that is an integer (since (4) is nothing but the cocycle condition for a fibre bundle):The symmetry in , , and of this last relation implies that

Thus, the generalization of our gauge potential on any closed 3-manifold imposes considering a collection constituted of a family of potentials defined in open sets , a family of functions defined in the double intersections , and a family of integers defined in the triple intersections (all those open sets and intersections being contractible). Elements of those collections are related byThese statements define a Deligne cocycle.

We need now to describe how this collection transforms when we perform a gauge transformation of the :where the family of is a family of functions defined in the . This implies that have to transform according towhere the family consists in integers, mainly because do. Finally, transform thus according to

Hence, the collection where are functions defined in the and are integers defined in the intersections together with the set of rulesgeneralizes the idea of gauge transformation. These rules define the addition of a Deligne coboundary to a Deligne cocycle. The quotient set of Deligne cocycles by Deligne coboundaries is the first Deligne cohomology group .

#### 3. Structure of the Space of Deligne Cohomology Classes

is naturally endowed with a structure of -modulus. It can be described in particular through two exact sequences. The first one iswhere is the quotient of the 1 form by the closed 1 form with integral periods and is the space of cohomology classes of the manifold. This is an abelian group, which can thus be decomposed as a direct sum of a free part and a torsion part . This exact sequence shows that the space of Deligne cohomology classes can be thought as a set of fibres over the discrete net constituted by and inside which we can move thanks to elements of (see Figure 1).