We introduce Deligne cohomology that classifies U1 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 U1 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:(1)SCSNA=2πN∫R3A∧dA=2πN∫R3A∧F,SBFNA,B=2πN∫R3B∧dA=2πN∫R3B∧F,where A and B are U1 connections. Here, the coupling constant N is any real number.

The gauge transformation A→A+dΛ, where Λ 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 S up to an integer:(2)S⟶S+2πn,n∈Z⟹eiS⟶eiSwhich implies that N 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 A→A+ωZ, where ωZ 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 Λ such that ωZ=dΛ. In particular, this is the case when the theory is defined in R3 which is a contractible space. However, this generalized gauge transformation enables defining a theory on any closed (i.e., compact without boundary) 3-manifold M. 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 U1 fibre bundles over M endowed with connections and their collection is the so-called first Deligne cohomology group of M. We will show that this structure enables performing exact computations in the framework of U1 Chern-Simons and BF theories.

2. Deligne Cohomology

The most general statement we can start from is a collection of local gauge fields Aα in open sets Uα that cover the manifold M 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 Aα and Aβ stick together in the intersection Uα∩Uβ. This, by definition, is done thanks to a gauge transformation:(3)Aβ=Aα+dΛαβin Uαβ=Uα∩Uβ.The antisymmetry of this relation in α and β implies that dΛαβ+Λβγ+Λγα=0, making Λαβ+Λβγ+Λγα a constant in Uα∩Uβ∩Uγ that is an integer (since (4) is nothing but the cocycle condition for a U1 fibre bundle):(4)Λαβ+Λβγ+Λγα=nαβγ∈Zin Uαβγ=Uα∩Uβ∩Uγ.The symmetry in α, β, and γ of this last relation implies that(5)nαβγ-nαβδ+nαγδ-nβγδ=0.

Thus, the generalization of our gauge potential on any closed 3-manifold M imposes considering a collection Aα,Λαβ,nαβγ constituted of a family of potentials Aα defined in open sets Uα, a family of functions Λαβ defined in the double intersections Uαβ, and a family of integers defined in the triple intersections Uαβγ (all those open sets and intersections being contractible). Elements of those collections are related by(6)Aβ=Aα+dΛαβin Uαβ.Λαβ+Λβγ+Λγα=nαβγ∈Zin Uαβγnαβγ-nαβδ+nαγδ-nβγδ=0These statements define a Deligne cocycle.

We need now to describe how this collection transforms when we perform a gauge transformation of the Aα:(7)Aα⟶Aα+dqαin Uα,where the family of qα is a family of functions defined in the Uα. This implies that Λαβ have to transform according to(8)Λαβ⟶Λαβ+qα-qβ-mαβin Uαβ,where the family mαβ consists in integers, mainly because nαβγ do. Finally, nαβγ transform thus according to(9)nαβγ⟶nαβγ-mβγ+mαγ-mαβin Uαβγ.

Hence, the collection qα,mαβ where qα are functions defined in the Uα and mαβ are integers defined in the intersections Uαβ together with the set of rules(10)Aα⟶Aα+dqαin UαΛαβ⟶Λαβ+qα-qβ-mαβin Uαβnαβγ⟶nαβγ-mβγ+mαγ-mαβin Uαβγgeneralizes 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 HD1.

3. Structure of the Space of Deligne Cohomology Classes

HD1 is naturally endowed with a structure of Z-modulus. It can be described in particular through two exact sequences. The first one is(11)0⟶Ω1ΩZ1=Ω1Ω01×Ω01ΩZ1⟶HD1Z⟶H2Z=F2⊕T2⟶0,where Ω1/ΩZ1 is the quotient of the 1 form by the closed 1 form with integral periods and H2Z 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 F2=Zb2 and a torsion part T2=Zp1⊕⋯⊕Zpn. 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 H2Z and inside which we can move thanks to elements of Ω1/ΩZ1(see Figure 1).

The second exact sequence that enables representing HD1 is (12)0⟶H1RZ=Ω01ΩZ1×T2⟶HD1Z⟶ΩZ2=Ω1Ω01×F2⟶0,where H1R/Z is the first cohomology group R/Z-valued and ΩZ2 is the set of closed 2 forms with integral periods. This exact sequence leads to the representation shown in Figure 2.

Those two exact sequences contain the same information.(13)

4. Operations and Duality on Deligne Cohomology Classes

Given two Deligne cohomology classes A and B with respective representatives Aα,Λαβ,nαβγ and Bα,Θαβ,mαβγ, we define a Deligne cohomology class A⋆B with representative:(14)Aα∧dBα,ΛαβBβ,nαβγBγ,nαβγΘγρ,nαβγmγρσ.

The integral of a Deligne cohomology class A with representative Aα,Λαβ,nαβγ over a cycle z is defined by(15)∮zA=Z∑α∫zα=Uα∩zAα-∑αβ∫zαβ=Uαβ∩zΛαβ,where =Z means that the equality is satisfied in R/Z, 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 M of A⋆B which provides a generalization of Chern-Simons and BF abelian actions:(16)∫MA⋆B=Z∑α∫UαAα∧dBα-∑αβ∫UαβΛαβBβ+∑αβγ∫UαβγnαβγBγ-∑αβγδ∫UαβγδnαβγΘγρ.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 that(17)Z1×HD1⟶RZz,A⟼∮zAdefines a bilinear pairing from the space Z1 of 1 cycle and the space of Deligne cohomology classes (both considered as Z-moduli) in R/Z as well as(18)HD1×HD1⟶RZA,B⟼∫MA⋆B.

Starting from that remark and for later convenience, we will consider Pontrjagin dual X#=HomX,R/Z of a group X. Considering Hom as a functor, we can show that the following sequences are exact: (19)0⟶ΩZ2#⟶HD1#⟶H1RZ#⟶0,0⟶H2#⟶HD1#⟶Ω1ΩZ1#⟶0.Moreover, the information of the first two exact sequences is included in those two new ones. (20)The Pontrjagin dual is a generalization to distributional objects. Finally, we see that Z1⊂HD1# in the sense of (17).

5. Decomposition of Deligne Cohomology Classes

The structure of Deligne cohomology classes is such that each class A can be decomposed as the sum of an origin indexed on the cohomology of M (basis of the discrete fibre bundle of Deligne cohomology classes) and a translation taken in Ω1/ΩZ1:(21)A=Aa0+ω,a∈H2Z.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) Ω1/ΩZ1 as(22)Ω1ΩZ1≃Ω1Ω01×Ω01ΩZ1,where Ω01 denotes the set of closed 1 form. Furthermore(23)Ω01ΩZ1≃RZb1,b1 being the first Betti number. We will call zero modes the elements ω0∈Ω01/ΩZ1. With this decomposition, we obtain(24)∀ω0∈Ω01ΩZ1,∀ω∈Ω1ΩZ1,∫Mω⋆ω0=Z0.

Let us consider generators za of the free part of the homology of M. Then, by Pontrjagin duality, we can associate to it a unique element ηza∈HD1#. Thus, for a fibre over ∑amaza∈F1≃F2 we will consider as origin the element(25)Am=∑amaηza∈HD1#.Note that(26)∫MAm⋆An=Z0since it represents a linking number which is necessarily an integer. We impose as a convention the so-called zero regularisation:(27)∫MAm⋆Am=Z0which is ill-defined as self-linking. Finally, if we decompose ω0∈Ω01/ΩZ1 as ω0=∑bθbρb with ∮zaρb=δab, then we obtain(28)∫MAm⋆ω0=Zm·θ.

Let us consider now a generator τa of the component Zpa of the torsion part of the homology of M. This means that τa is the boundary of no surface, but paτa is. Consider now ητa∈HD1# defined by(29)ητa=0,mαβpa,nαβγ,where panαβγ=mβγ-mαγ+mαβ. Thus, for a fibre over ∑aκaτa∈T1≃T2 we will consider as origin the element(30)Aκ0=∑aκaητa∈HD1#.This choice has several advantages since we can show that(31)∫MAκ10⋆Aκ20=Z-Qκ1,κ2,where Q is the so-called linking form, which is a quadratic form over the torsion of the cohomology. Also(32)∫MAκ0⋆Am=Z0for any free origin Am and(33)∫MAκ0⋆ω=Z0for any translation ω.

6. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M149"><mml:mi>U</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:mfenced></mml:math></inline-formula> Chern-Simons and BF Theories

Chern-Simons abelian action is generalized as(34)SCSNA=2πN∫MA⋆A.Since ∫MA⋆A∈R/Z, then N has to be quantized here:(35)N∈Z.The partition function is defined as(36)ZCSN=1NCSN∫HD1#eiSCSNADA,NCSN being a normalization that has to cancel the intrinsic divergence of the functional integral. The functional measure we use is then(37)dμCSNA=DAeiSCSNA.Assume that this measure verifies the so-called Cameron-Martin property; that is,(38)dμCSNA+ω=dμCSNAe4iπN∫MA⋆ωe2iπN∫Mω⋆ωfor a fixed connection A and a translation ω, then, for jγ a translation in HD1# associated with a cycle γ:(39)dμCSNA+mjγ2N=dμCSNA.Using the algebraic properties given before, we can compute exactly the Chern-Simons abelian partition function.

As a convention, for the normalization we choose(40)NCSN=∫Ω1/ΩZ1#eiSCSNωDω=∫Ω1/Ω01#eiSCSNωDωwhich corresponds to the trivial fibre of Deligne bundle for our theory defined over a manifold M. This trivial fibre is the (only) one that constitutes Deligne bundle if we consider a theory over S3. 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 S1×S2. However if the normalization is done with respect to S3 then one recovers in the abelian case the invariants obtained with convention (40).

This way, we find(41)ZCSN=∑τA∈T2e-2iπNQτA,τA.Analogous considerations apply to BF abelian theory whose generalized action is(42)SBFNA,B=2πN∫MA⋆B(N being here also quantized) which leads to a partition function written as(43)ZBFN=∑τA∈T2∑τB∈T2e-2iπNQτA,τB=∏i=1ngcdpi,Npi.

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 SU2, have been established formally, that is, with manipulations of ill-defined quantities:

Chern-Simons partition function is related to Reshetikhin-Turaev topological invariant [3].

BF partition function is related to Turaev-Viro topological invariant [4].

The square modulus of Chern-Simons partition function is equal to the BF partition function [5].

This is summed up on the following diagram: (44)ZCSN2⋯CattaneoZBFNWitten⋮⋮Ponzano,Regge…RTN2=TuraevTVNthe 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: (45)ZCSN2≠Thuillier&M.ZBFNGuadagnini&ThuillierThuillier&M.RTN2≠TVN,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.

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