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).
The second exact sequence that enables representing is where is the first cohomology group -valued and is the set of closed forms with integral periods. This exact sequence leads to the representation shown in Figure 2.
Those two exact sequences contain the same information.
4. Operations and Duality on Deligne Cohomology Classes
Given two Deligne cohomology classes and with respective representatives and , we define a Deligne cohomology class with representative:
The integral of a Deligne cohomology class with representative over a cycle is defined bywhere means that the equality is satisfied in , that is, up to an integer. This integral is nothing but a holonomy, that is, a typical observable of Chern-Simons and BF quantum field theories. This definition ensures gauge invariance in the sense described in the introduction.
We can define in the same way the integral over of which provides a generalization of Chern-Simons and BF abelian actions:Let us point out that the first term is nothing but the local classical action, the other terms ensuring the gluing of local expressions up to an integer.
Note thatdefines a bilinear pairing from the space of cycle and the space of Deligne cohomology classes (both considered as -moduli) in as well as
Starting from that remark and for later convenience, we will consider Pontrjagin dual of a group . Considering as a functor, we can show that the following sequences are exact: Moreover, the information of the first two exact sequences is included in those two new ones. The Pontrjagin dual is a generalization to distributional objects. Finally, we see that in the sense of (17).
5. Decomposition of Deligne Cohomology Classes
The structure of Deligne cohomology classes is such that each class can be decomposed as the sum of an origin indexed on the cohomology of (basis of the discrete fibre bundle of Deligne cohomology classes) and a translation taken in :The result of functional integrals over the space of Deligne cohomology classes will not depend on the choice of the origins, but the complexity of the computations will. Thus, our goal is to find convenient origins with algebraic properties that will enable performing computations easily.
Concerning the translations, we can decompose (noncanonically) aswhere denotes the set of closed form. Furthermore being the first Betti number. We will call zero modes the elements . With this decomposition, we obtain
Let us consider generators of the free part of the homology of . Then, by Pontrjagin duality, we can associate to it a unique element . Thus, for a fibre over we will consider as origin the elementNote thatsince it represents a linking number which is necessarily an integer. We impose as a convention the so-called zero regularisation:which is ill-defined as self-linking. Finally, if we decompose as with , then we obtain
Let us consider now a generator of the component of the torsion part of the homology of . This means that is the boundary of no surface, but is. Consider now defined bywhere . Thus, for a fibre over we will consider as origin the elementThis choice has several advantages since we can show thatwhere is the so-called linking form, which is a quadratic form over the torsion of the cohomology. Alsofor any free origin andfor any translation .
6. Chern-Simons and BF Theories
Chern-Simons abelian action is generalized asSince , then has to be quantized here:The partition function is defined as being a normalization that has to cancel the intrinsic divergence of the functional integral. The functional measure we use is thenAssume that this measure verifies the so-called Cameron-Martin property; that is,for a fixed connection and a translation , then, for a translation in associated with a cycle :Using the algebraic properties given before, we can compute exactly the Chern-Simons abelian partition function.
As a convention, for the normalization we choosewhich corresponds to the trivial fibre of Deligne bundle for our theory defined over a manifold . This trivial fibre is the (only) one that constitutes Deligne bundle if we consider a theory over . This choice enables establishing a link with Reshetikhin-Turaev abelian invariant (see [1]). Note that usually the normalization of Reshetikhin-Turaev invariant is chosen to be related to . However if the normalization is done with respect to then one recovers in the abelian case the invariants obtained with convention (40).
This way, we findAnalogous considerations apply to BF abelian theory whose generalized action is( being here also quantized) which leads to a partition function written as
Computations of expectation values of observables can also be performed thanks to this method in both Chern-Simons and BF abelian theories (see [1, 2]).
7. Conclusion
Several correspondences in the nonabelian case, mainly , have been established formally, that is, with manipulations of ill-defined quantities:(1)Chern-Simons partition function is related to Reshetikhin-Turaev topological invariant [3].(2)BF partition function is related to Turaev-Viro topological invariant [4].(3)The square modulus of Chern-Simons partition function is equal to the BF partition function [5].
This is summed up on the following diagram: the only result perfectly rigorously established being the one of Turaev, Reshetikhin, and Viro (see [6–8]).
In the abelian case, we saw that Deligne cohomology approach enables defining rigorously functional integration in the specific case of Chern-Simons and BF theories. Using this tool, we show that the previous diagram is no longer correct and has to be replaced by the following one: where the hypothesis of Turaev are not necessarily satisfied with abelian representations, leading to an inequality in general.
This shows that the abelian theories, contrary to what could be expected, are not a simple trivial subcase of the nonabelian ones. However, we expect to find some traces of this abelian case in the nonabelian one, which is the aim of present works.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.