3D Topological Models and Heegaard Splitting II: Pontryagin duality and Observables
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3D Topological Models and Heegaard Splitting II:Pontryagin duality and Observables
F. Thuillier
LAPTh, Universit´e Savoie Mont Blanc, CNRS, B.P. 110, F-74941 Annecy Cedex,France
Abstract
In a previous article, a construction of the smooth Deligne-Beilinson cohomologygroups H pD ( M ) on a closed 3-manifold M represented by a Heegaard splitting X L ∪ f X R was presented. Then, a determination of the partition functions of the U (1) Chern-Simons and BF Quantum Field theories was deduced from this construction. In thissecond and concluding article we stay in the context of a Heegaard spitting of M to defineDeligne-Beilinson 1-currents whose equivalent classes form the elements of H D ( M ) ⋆ , thePontryagin dual of H D ( M ). Finally, we use singular fields to first recover the partitionfunctions of the U (1) Chern-Simons and BF quantum field theories, and next to determinethe link invariants defined by these theories. The difference between the use of smoothand singular fields is also discussed. Introduction
In a first article [1] we explained how to recover the smooth Deligne-Beilinson (DB)cohomology groups H pD ( M ) by using a Heegaard splitting X L ∪ f X R of an oriented closedsmooth 3-manifold M . Then, we identified a decomposition of DB 1-cocycles of X L ∪ f X R which turned out to be very efficient in the Chern-Simon (CS) and BF U (1) topologicaltheories. Doing so, we were able to check that the partition functions for these twotheories coincide with those obtain in the so-called “standard” approach as exposedin [2–5]. Moreover, up to a reciprocity formula, the CS and BF partition functions arenothing but respectively the Reshetikhin-Turaev [6] and Turaev-Viro [7] U (1) invariants[2,4,8]. It is remarkable that we can obtain partition functions by performing a functionalintegration on smooth fields (actually on H D ( M )) when it is well-known that in theonly locally finite and translation-invariant Borel measure on the infinite dimensionalBanach set of smooth fields is the trivial measure. In the U (1) case, the use of smoothDB cohomology allows to identify some finite dimensional sub-integrals from the initialfunctional integral. Once the remaining infinite dimensional integral is factorized outand eliminated with the help of a normalization factor, the desired partition functionsemerge. This technique applies to the standard approach as well as to the one based ona Heegaard splitting.In this second and concluding article we study CS and BF observables, and computetheir expectation values still in the context of a Heegaard splitting. The CS and BFobservables are U (1) holonomies and hence are given by exponentiating the line integralof gauge fields over closed curves, i.e. 1-cycles. It turns out that integration along 1-cycles goes to DB cohomology classes as an R / Z -valued linear functional, thus providingan example of the so-called Pontryagin duality. There are two ways to deal with thedetermination of the expectation values of these observables in the functional integrationpoint of view. The first one is to stick with the smooth DB cohomology by replacing1-cycles with regularization whose support is as close to the 1-cycles as we want. Theholonomies are then given as integral of DB products. At the end of the computation,we take the limit which brings the regularized 1-cycles back to their original singularposition. The drawback of this method is that we must assume that functional integrationcommutes with the limit procedure. Another approach is to deal with 1-cycle themselvesand consequently with distributional fields. The drawback of this procedure is that weare faced with the usual problem of giving a meaning to products of distributions whichappear all along the computation. Fortunately as we will see, this problem turns out tobe quite easy to handle.We can wonder about the differences between the standard and Heegaard splittingapproaches to the U (1) CS and BF theories. The most significant difference concernthe degrees of freedom used in the functional integral. In the standard approach amongthe finite dimensional sub-integrals a sum over T ( M ), the first torsion group of the 3-1anifold M whose elements are integers, must be performed. In the Heegaard splittingapproach the equivalent sum must be performed over T ( M ) ⋆ , the Pontryagin dual of T ( M ) whose elements are angles. This difference is related to the following simple,although relatively unexpected, fact. When using a Heegaard splitting to describe DBclasses, the exact sequence which naturally emerges from the construction is the oneending with the set of (generalized) curvatures and not the one ending with the secondcohomology space of M , as in the standard approach. Beside a simple curiosity, let uspoint out that the Heegaard splitting approach could reveal interesting in the non-abeliancase too where there does not seem to be an equivalent of DB cohomology. Hence, theHeegaard construction could help to identify relevant degrees of freedom in the functionalintegral, just as in the abelian case.The notion of 3-handlebody plays a central role in the construction proposed in thisarticle and to our knowledge physicists are not very familiar with the geometry of smooth3-manifolds with boundary, the same remarks holding true for Heegaard splittings. Thisis why we thought it best to give enough details for this article to be as self contained aspossible, even at the expense of its length.The article is structured as follows: • In Section 2 we exhibit some geometrical data provided by a 3-handlebody X andits boundary ∂X ∼ = Σ, concentrating on singular chains, forms and de Rham currents.The main purpose of the first subsection dedicated to the Riemann surface Σ is Property2. The second subsection exposes some geometrical properties of X relying on the factthat this manifold has a boundary. Both subsections end with few comments on DBcohomology. • In section 3 we present a description of chains, forms and de Rham currents of aHeegaard splitting X L ∪ f X R . We end this section by briefly recalling the construction [1]of H D ( X L ∪ f X R ), the first smooth DB cohomology group of X L ∪ f X R . Beside fixingnotations, this section tries to detail some useful points about Heegaard splitting andalso provide an expression for the linking number of two 1-cycles of X L ∪ f X R . • Section 4 is the core of the article and is dedicated to Pontryagin duality. Asalready mentioned, Pontryagin duality is closely related to integration of smooth DBclasses along 1-cycles of X L ∪ f X R . We first recall important facts concerning integrationof 1-forms and smooth DB 1-cocycles along 1-cycles of X L ∪ f X R . This yields the notionof DB 1-cycles which themselves shed light on the construction of DB 1-currents that wepropose right after. The main difference with smooth DB 1-cocycles comes from the lackof gluing condition for DB 1-currents. Then, and still in connection with DB 1-cycles, anequivalence relation is exhibited which infers the notion of generalized DB classes the setof which is H D ( X L ∪ f X R ) ⋆ , the Pontryagin dual of H D ( X L ∪ f X R ). Next, we show that H D ( X L ∪ f X R ) ⋆ is embedded into a canonical exact sequence which is the Pontryagindual of an exact sequence into which H D ( X L ∪ f X R ) is embedded. This finally yields the2entral property of the article, i.e. Property 13, which provides us with a parametrizationof H D ( X L ∪ f X R ) ⋆ . The rest of this section deals with an attempt to generalize the DBproduct to DB 1-currents in order to extend the CS and BF (quantum) Lagrangians tosingular DB classes. To that end, we first verify that smooth DB 1-cocycles can be seenas regular DB 1-currents, just like forms can be seen as regular de Rham currents. Then,a set of fundamental generalized DB products is exhibited and the zero regularizationprocedure is exposed in connection with the notion of linking number, self-linking andframing. As a collateral effect of the construction, an explicit realization of the so-calledcycle map [9] is obtained. • The last section is completely devoted to U (1) CS and BF theories in the contextof a Heegaard splitting. We first recall how the actions are related to DB product and itsgeneralization to singular classes. Then, we discuss the functional measure which is sup-posed to allow us to compute partition functions and expectation values of holonomies.The so-called color periodicity and zero modes properties are exhibited. The next sub-section more specifically deals with partition functions while the last one deals withexpectation values. We eventually recover the expressions of the expectation values of U (1) holonomies as given in [3, 5]. In this first section we recall some geometrical information which will prove useful inthe other sections. As in this article we consider Heegaard splittings of oriented smoothclosed 3-manifolds, let X be an oriented smooth 3-handlebody whose boundary, denotedΣ = ∂X , is a Riemann surface of genus g . We assume that Σ is endowed with theorientation induced by the one of X and we denote by i Σ : Σ → X the canonicalinclusion. The first subsection deals with Σ and the second with X . Each of thesesubsections start by considering chains as they are particularly easy to visualize in twoand three dimensions with the help of simple drawings. Σ Let C p (Σ) denote the set of p -chains on Σ, and let Ω p (Σ) be the vector space of smooth p -forms on Σ. These spaces are respectively endowed with with the boundary operator ∂ and the de Rham exterior derivative d . The space of p -cycles of Σ is denoted by Z p (Σ)while the space of closed p -forms is then denoted by Ω p (Σ). The space of closed p -formswith integral periods by Ω p Z (Σ).The de Rham p -currents of Σ are the elements of the topological dual, Ω p (Σ) ⋄ , of3 p (Σ) [10]. This space is endowed with the boundary operator d † defined by: ∀ J ∈ Ω p (Σ) ⋄ , ∀ α ∈ Ω p (Σ) (cid:0) d † J (cid:1) [[ α ]] = J [[ dα ]] , (2.1)which gives rise to (cid:0) Ω • (Σ) ⋄ , d † (cid:1) , the dual complex of the differential complex (Ω • (Σ) , d ).We denote by Ω p (Σ) ⋄ the space of closed p -currents of Σ and by Ω p (Σ) ⋄ Z the space of closed p -currents which are Z -valued on Ω p Z (Σ). The complex (cid:0) Ω • (Σ) ⋄ , d † (cid:1) yields homologygroups H ⋄ p (Σ) which fulfill Poincar´e duality according to: H ⋄ p (Σ) ∼ = H − pdR (Σ) , (2.2)the de Rham cohomology groups H qdR (Σ) being those of the complex (Ω • (Σ) , d ).To any ω ∈ Ω − p (Σ) we associate the de Rham p -current J ω defined by: ∀ α ∈ Ω p (Σ) , J ω [[ α ]] = I Σ ω ∧ α . (2.3)This yields a canonical injection J : Ω − p (Σ) → Ω p (Σ) ⋄ . The elements of Im J arecommonly referred to as regular de Rham p -currents of Σ. Integration by parts impliesthat: ∀ ω ∈ Ω − p (Σ) , d † J ω = ( − ) p +1 J dω . (2.4)In particular, for ω ∈ Ω − p (Σ) and α ∈ Ω p (Σ), the 2-form ω ∧ α yields the regular2-current J ω ∧ α . Moreover, we can introduce the “evaluation”:( J ω ∧ J α ) [[1]] = J ω ∧ α [[1]] = I Σ ω ∧ α (2.5)Under certain conditions [10], it is possible to give a meaning to the evaluation ( j ∧ j ) [[1]]even if j and j are singular, i.e. not regular. For instance, when the singular supportof j (resp. j ) does not intersect the singular support of d † j (resp. d † j ), then regular-izations of j and j allow to define properly ( j ∧ j ) [[1]]. In particular, (cid:0) j ∧ d † j (cid:1) [[1]]is well defined when the singular supports of d † j and d † j do not intersect. Anotherexample where ( j ∧ j ) [[1]] is well defined is when j (or j ) is regular since in this caseits singular support is empty.To any c Σ ∈ C p (Σ) we associate the de Rham p -current j c Σ defined by: ∀ α ∈ Ω p (Σ) , j c Σ [[ α ]] = Z c Σ α . (2.6)This generates a homomorphism j : C p (Σ) → Ω p (Σ) ⋄ . As in this article we mainly dealwith integration properties along chains, we decide to identify p -chains which define the4ame de Rham p -current [10]. Under this identification, the homomorphism j becomesinjective. Relation (2.6) can be written as: ∀ α ∈ Ω p (Σ) , j c Σ [[ α ]] = ( j c Σ ∧ α ) [[1]] = ( j c Σ ∧ J α ) [[1]] . (2.7)Indeed, ( j c Σ ∧ J α ) [[1]] as a meaning because the singular support of J α is empty. Moreover,for any c ∈ C p (Σ) we have: j ∂c = d † j c , (2.8)This gives rise to an injection of the complex ( C • (Σ) , ∂ ) into the complex (cid:0) Ω • (Σ) ⋄ , d † (cid:1) .Under this injection, Z p (Σ) is mapped into a subset of Ω p (Σ) ⋄ Z .Let us consider a set of longitude and meridian 1-cycles of Σ, λ Σ a and µ Σ a ( a = 1 , · · · , g )respectively, the Kronecker indexes of which are: λ Σ a ⊙ µ Σ b = δ ab , (2.9)all other indexes being zero. The operator ⊙ is the combination of the transverse intersec-tion operator ⋔ with the degree operator ♯ : C (Σ) → Z which is a non-trivial extensionof ∂ to the set C (Σ) of 0-chains of Σ such that ♯ ◦ ∂ = 0. Typically, ( ± x ) ♯ := ± x seen as a positively oriented 0-chain of Σ. Note that ⋔ is only defined forchains which are in transverse position, hence its name. If j λ Σ a and j µ Σ b are the de Rham1-currents of λ Σ a and µ Σ b respectively, we have: (cid:16) j λ Σ a ∧ j µ Σ b (cid:17) [[1]] = λ Σ a ⊙ µ Σ b = δ ab , (2.10)As an example, let us assume that Σ = S × S endowed with angular coordinates ( ϕ , ϕ ),the orientation being specified by the volume form dϕ ∧ dϕ . To get λ Σ ⊙ µ Σ = 1 wecan chose λ Σ and µ Σ such that j λ Σ = δ ( ϕ ) dϕ and j µ Σ = δ ( ϕ ) dϕ since then we have( j λ ∧ j µ ) [[1]] = ( δ ( ϕ , ϕ ) dϕ ∧ dϕ ) [[1]] = 1 as expected.Relations (2.9) induce an inner product in Z g according to: h ~m, ~n i = g X a =1 m a λ Σ a ! ⊙ g X b =1 n b µ Σ b ! = g X a,b =1 m a δ ab n b , (2.11)Let us stress out that in this definition the longitude 1-cycles appear to the left and themeridian 1-cycles to the right of the operator ⊙ . Hence, this inner product is nothingbut the Euclidean inner product on the free Z -module Z g which is the restriction to Z g of the Euclidean inner product in R g . This last inner product will also be denoted h , i .5he longitude and meridian 1-cycles generate the first homology group of Σ, and wehave: H (Σ) ∼ = Z H (Σ) ∼ = Z g H (Σ) ∼ = Z . (2.12)In anticipation of the Heegaard construction which will be considered in the nextsection, let Σ L and Σ R be two copies of the oriented closed surface Σ and let f : Σ L → Σ R be an orientation reversing diffeomorphism whose action on longitude and meridian 1-cycles is given by: f ( λ Σ L a ) = g P b =1 r ab λ Σ R b + g P b =1 s ab µ Σ R b + ∂ϕ Σ R a f ( µ Σ L a ) = g P b =1 p ab λ Σ R b + g P b =1 q ab µ Σ R b + ∂ψ Σ R a . (2.13)where ϕ Σ R a and ψ Σ R a are 2-chains of Σ R with a = 1 , · · · , g . This diffeomorphism inducesan isomorphism ˆ f : H (Σ L ) → H (Σ R ) which on its turn induces an automorphism of Z g as H (Σ) ∼ = Z g . The matrix of this automorphism is of the form: M ˆ f = (cid:18) R PS Q (cid:19) , (2.14) P , Q , R and S being integer ( g × g ) matrices defined by the integers p ab , q ab , r ab and s ab ,respectively. Since f reverses orientation, it fulfills: f ( λ Σ L a ) ⊙ Σ R f ( µ Σ L b ) = − ( λ Σ L a ⊙ Σ L µ Σ L b ) , (2.15)or in terms of de Rham 1-currents: (cid:16) f ( j λ Σ La ) ∧ f ( j µ Σ Lb ) (cid:17) [1] = − (cid:16) j λ Σ La ∧ j µ Σ Lb (cid:17) [1] . (2.16)These relations imply that det M ˆ f = − M − f = M ˆ f − is: (cid:18) − Q † P † S † − R † (cid:19) , (2.17)which yields the following set of important relations [1, 11]: Q † P = P † Q , P S † − RQ † = 1 , S † R = R † S ,RP † = P R † , P † S − Q † R = 1 , SQ † = QS † , , (2.18)6he matrices P † , Q † , R † and S † being the transpose of respectively P , Q , R and S , withrespect to inner product (2.11).Each of the above matrices can be seen as the matrix of an endomorphism of Z g . Theendomorphisms associated with P and P † are embedded in the following exact sequences:0 → ker P ⊂ −→ Z g P −→ Z g [ ] −→ coker P −→ → ker P † ⊂ −→ Z g P † −→ Z g [ ] † −−→ coker P † −→ , (2.19)where coker P = Z g / Im P and coker P † = Z g / Im P † . We then have the following prop-erty which stems from relations (2.18): Property 1. ~n ∈ Im P ⇔ Q † ~n ∈ Im P † , (2.20)and an obvious corollary: Corollary 1. [ ~n ] = 0 ⇔ Q † [ ~n ] = [ Q † ~n ] † = 0 . (2.21)Indeed, if ~n = P ~m , then Q † ~n = Q † P ~m = P † Q ~m . Conversely, if Q † ~n = P † ~r then RQ † ~n = RP † ~r = P R † ~r on the one hand, and since P S † − RQ † = 1, RQ † ~n = − ~n + P S † ~n on the second hand. By combining these two relations we get ~n = P ( S † ~n − R † ~r ) whichends up establishing the property. Since Q † ( ~n + P ~m ) = Q † ~n + Q † P ~m = Q † ~n + P † Q ~m , wecan extend Q † into a homomorphism Q † : coker P → coker P † by setting Q † [ ~n ] = [ Q † ~n ] † ,[ ] and [ ] † denoting the passage to coker P and coker P † , respectively. This achievesthe proof of the above property and of its corollary.Since it is a finitely generated abelian group, coker P can be decomposed accordingto coker P = F P ⊕ T P , where F P is the free sector of coker P and is homomorphic to Z b for some b ∈ N , whereas F P is the torsion sector of coker P and is homomorphic to Z p ⊕· · ·⊕ Z p N with p i dividing p i +1 . If ~n ∈ Z g represents a torsion element of coker P thenthere exists p ∈ N ∗ such that p~n ∈ Im P , and hence ~n = ˆ P ~m/p for some ~m ∈ Z g , withˆ P the canonical extension of P to R g . Conversely, if ~m/p ∈ Q g is such that ˆ P ~m/p ∈ Z g ,then [ ˆ P ~m/p ] ∈ T P . As P † plays with respect to f − the role that P plays with respect to f , we straightforwardly conclude that coker P † = F P † ⊕ T P † ∼ = Z b ⊕ Z p ⊕ · · · ⊕ Z p N , andthat ~n represents and element of T P † if and only if there exist ( p, ~m ) ∈ N ∗ × Z g such that ~n = ˆ P † ~m/p . All these results can alternatively be obtained with the help of the Smithnormal form of P , which then allows to easily check that ker P ∼ = Z b ∼ = ker P † . For latterconvenience, we introduce the following notations: H P = coker P = F P ⊕ T P , H P † = coker P † = F P † ⊕ T P † , (2.22)7ith F P ∼ = F P † ∼ = Z b and T P ∼ = T P † ∼ = Z p ⊕ · · · ⊕ Z p N .Pontryagin duality playing a central role in this article, let us have a closer look at( Z g ) ⋆ = Hom( Z g , R / Z ), the Pontryagin dual of Z g . It is isomorphic to ( R / Z ) g since any ϑ ∈ ( Z g ) ⋆ can be generated with the help of some θ ∈ ( R / Z ) g according to: ∀ ~n ∈ Z g , ϑ ( ~n ) = h θ , ~n i = D ~θ, ~n E , (2.23)where ~θ ∈ R g is a representative of θ and denotes the restriction to R / Z . The right-hand side of the second equality above obviously depends on θ and not on the particularrepresentative ~θ . From now on, θ will refer to the homomorphism ϑ = h θ , ·i it defines,thus yielding the identification of ( Z g ) ⋆ with ( R / Z ) g .The homomorphism P induces a dual homomorphism P ⋆ : ( R / Z ) g → ( R / Z ) g definedas follows: ∀ ~n ∈ Z g , ( P ⋆ θ )( ~n ) = θ ( P ~n ) . (2.24)In term of representatives the above relation takes the form: ∀ ~n ∈ Z g , ( P ⋆ θ )( ~n ) = D ˆ P † ~θ, ~n E , (2.25)which shows that on representatives P ⋆ is nothing but ˆ P † . We similarly define P † ⋆ by: ∀ ~n ∈ Z g , ( P † ⋆ θ )( ~n ) = D ˆ P ~θ, ~n E , (2.26)which shows that on representatives P † ⋆ is nothing but ˆ P . Then, we have the followingexact sequences associated with P ⋆ and P † ⋆ :0 → ker P ⋆ → ( R / Z ) g P ⋆ −−→ ( R / Z ) g −→ coker P ⋆ → → ker( P † ) ⋆ → ( R / Z ) g P † ⋆ −−→ ( R / Z ) g −→ coker P † ⋆ → . (2.27)The action of θ ∈ ( R / Z ) g goes to coker P † = Z g / Im P † if and only if θ is zero onIm P † , which means that θ ( P † ~m ) = ( P † ⋆ θ )( ~m ) = 0 ∈ R / Z for any ~m ∈ Z g , and henceif and only if θ ∈ ker P † ⋆ . This shows that ker P † ⋆ ∼ = (coker P † ) ⋆ . Similarly, we haveker P ⋆ ∼ = (coker P ) ⋆ so that the exact sequences (2.27) turn out to be the Pontryagindual of the exact sequences (2.19). Thus, we can write:ker P † ⋆ = H ⋆P † = F ⋆P † ⊕ T ⋆P † ∼ = (coker P † ) ⋆ ∼ = ( R / Z ) b ⊕ Z ⋆p ⊕ · · · ⊕ Z ⋆p N . (2.28)If θ is an element of H ⋆P † then for any of its representative ~θ there exists ~L ∈ Z g suchthat P ~θ = ~L . For θ fixed the set of such ~L is given as ~L + P~l with ~l spanning Z g . Let us8ick one ~L θ for each θ ∈ ker P † ⋆ . Now, let θ be an element of F ⋆P † . On the first hand, if ~θ is a representative of θ then there exists ~l ∈ Z g such that P ~θ = ~L θ + P~l . On the secondhand, since P is an integer matrix and θ ∈ F ⋆P † ∼ = ( R / Z ) b , this last equation is fulfilled ifand only if ˆ P ~θ Z = 0, which means that θ admits a representative ~θ f ∈ ker ˆ P . This impliesthat θ ∈ F ⋆P † if and only if we can pick ~L θ = ~ ~L θ ] = 0. Letus now assume that θ ∈ T ⋆P † . Since T ⋆P † ∼ = Z ⋆p ⊕ · · · ⊕ Z ⋆p N there exists ( p, ~m ) ∈ Z × Z g such that ~m/p is a representative of θ , and then ~L θ fulfills p~L θ = P ( ~m + p~l ) for some ~l ∈ Z g . In other words [ ~L θ ] ∈ T P .Let us gather and complete the above results in the following property: Property 2.
Let us introduce the notations: H ⋆P = ker P ⋆ = F ⋆P ⊕ T ⋆P , H ⋆P † = ker P † ⋆ = F ⋆P † ⊕ T ⋆P † . (2.29)
1) We have the following set of abelian group isomorphisms: H ⋆P ∼ = (coker P ) ⋆ , H ⋆P † ∼ = (coker P † ) ⋆ H ⋆P ∼ = ( R / Z ) b ⊕ Z ⋆p ⊕ · · · ⊕ Z ⋆p N ∼ = H ⋆P † . (2.30)
2) Let ~n be an element of Z g . Then: [ ~n ] ∈ T P ⇔ (cid:18) ∃ ( p, ~m ) ∈ N ∗ × Z g , ~n = ˆ P ~mp (cid:19) , (2.31) and similarly for T P † .3) Let θ = θ f + θ τ be an element of H ⋆P † = F ⋆P † ⊕ T ⋆P † . Then θ has a representativeof the form ~θ = ~θ f + ~θ τ such that: • ˆ P ~θ f = ~ , (2.32) • ∃ ( p, ~m ) ∈ N ∗ × Z g , ~θ τ = ~mp , (2.33)ˆ P ~θ = ˆ
P ~θ τ = ˆ P ~m/p ∈ Z g then representing an element of T P .4) Let θ and θ ′ be two elements of H ⋆P † with representatives ~θ = ~θ f + ~θ τ and ~θ ′ = ~θ ′ f + ~θ ′ τ .Then, the expression: Γ( ~θ, ~θ ′ ) = D ˆ P ~θ, ˆ Q~θ ′ E = D ˆ P ~θ τ , ˆ Q~θ ′ τ E = Γ( ~θ τ , ~θ ′ τ ) , (2.34) induces an R / Z -valued symmetric bilinear pairing on T ⋆P † which is referred to as the linking form of T ⋆P † . The linking form of T P is then driven by setting: γ ( ~n τ , ~n ′ τ ) = Γ( ~θ τ , ~θ ′ τ ) , (2.35) with ~n τ = P ~θ τ and ~n ′ τ = P ~θ ′ τ .
9o prove the last point of the above property let us first note that P † Q~θ ′ = Q † P ~θ ′ = Q † P ~θ ′ τ Z = 0 because P † Q = Q † P and P ~θ ′ = P ~θ ′ τ ∈ Z g . This shows that Q~θ ′ representsan element of H ⋆P and yields the following sequence of equalities: D ˆ P ~θ, ˆ Q~θ ′ E = D ˆ P ~θ τ , ˆ Q~θ ′ E = D ~θ τ , ˆ P † ˆ Q~θ ′ E = D ~θ τ , ˆ Q † ˆ P ~θ ′ E = D ~θ τ , ˆ Q † ˆ P ~θ ′ τ E = D ˆ P ~θ τ , ˆ Q~θ ′ τ E . . (2.36)Finally, symmetry and bilinearity being obvious, the relation: D P ( ~θ τ + ~l ) , Q ( ~θ ′ τ + ~l ′ ) E Z = D P ~θ τ , Q~θ ′ τ E , (2.37)shows that Γ induces an R / Z -valued symmetric bilinear pairing on T ⋆P † , as claimed.A DB 1-cocycle of Σ is a way to represent a U (1) principal bundle with U (1)-connection on Σ with the help of local quantities like 1-forms, functions and integers.By identified these principal bundles with connections under the action of U (1) isomor-phisms, we obtain an equivalence relation and hence classes of DB 1-cocycles of Σ, theset of which is denoted H D (Σ). This is a Z -module which stands in the following exactsequence: 0 → Ω (Σ)Ω Z (Σ) → H D (Σ) → H (Σ) → , (2.38)The Pontryagin dual of H D (Σ), H D (Σ) ⋆ := Hom( H D (Σ) , R / Z ), is obtained by taking thePontryagin dual of the above sequence, which yields:0 → H (Σ) ⋆ ∼ = H (Σ , R / Z ) → H D (Σ) ⋆ δ ⋆ − −−→ (cid:18) Ω (Σ)Ω Z (Σ) (cid:19) ⋆ ∼ = Ω (Σ) ⋄ Z → . (2.39)The construction giving rise to H D (Σ) can be generalized thus giving rise to a family of Z -modules H pD (Σ). As H D (Σ) is involved in the construction of H D ( X L ∪ X R ), let recallthat this particular space stands in the exact sequence:0 → H (Σ , R / Z ) → H D (Σ) ¯ d −→ Ω Z (Σ) → , (2.40)where ¯ d : H D (Σ R ) → Ω Z (Σ R ) is a linear morphism that we refer to as the curvature operator of H D (Σ). The DB product which pairs H D (Σ) and H D (Σ) yields a conicalinjection H D (Σ) J −→ H D (Σ) ⋆ . This is consistent with the fact that Ω Z (Σ) is the subsetof regular elements of Ω (Σ) ⋄ Z . We refer to [9, 12] for details concerning all the groups H pD (Σ). 10 .2 Data for X Let C p ( X ) denotes the abelian group of singular p -chains of X. For each p the embedding i Σ : Σ → ∂X induces a chain map i Σ : C p (Σ) → C p ( X ) thanks to which me can identify C p (Σ) as a subgroup of C p ( X ). The quotient C p ( X, ∂X ) := C p ( X ) /i Σ ( C p (Σ)) is anabelian group usually referred to as the group of relative singular p -chains of ( X, ∂X ).The subgroup i Σ ( C p (Σ)) can be seen as the free abelian group generated by standard p -simplices whose image are in ∂X [13]. This induces a decomposition rule of any p -chain c of X according to: c = c ◦ + c Σ , (2.41)where c Σ ∈ i Σ ( C p (Σ)) and c ◦ = c − c Σ corresponds to a unique element of C p ( X, ∂X ).The above decomposition is equivalent to the direct sum decomposition [13] C p ( X ) ∼ = C p (Σ) ⊕ C p ( X, ∂X ) , (2.42)or to the statement that the exact sequence:0 → C p (Σ) → C p ( X ) → C p ( X, ∂X ) → , (2.43)splits. We refer to c ◦ as the relative component of c and to c Σ as its Σ -component ,and with an irrelevant abuse we identify C p ( X, ∂X ) (resp. C p (Σ)) with the subgroup ofrelative components (resp. Σ-components) of p -chains of X .We provide each C p ( X ) with the standard boundary operator ∂ thus getting the sin-gular complex ( C • ( X ) , ∂ ). With respect to the decomposition rule (2.41), the boundaryoperator ∂ has the following behavior: ∂c = ∂c ◦ + ∂c Σ = ( ∂c ◦ ) ◦ + ( ∂c ◦ ) Σ + ∂c Σ (2.44)which entails a decomposition of ∂ itself according to: ∂ = ∂ ◦◦ + ∂ Σ ◦ + ∂ ΣΣ , (2.45)with: ∂ ◦◦ c = ∂ ◦◦ c ◦ = ( ∂c ◦ ) ◦ ∂ Σ ◦ c = ∂ Σ ◦ c ◦ = ( ∂c ◦ ) Σ ∂ ΣΣ c = ∂ ΣΣ c Σ = ∂c Σ . (2.46)The nilpotency relation ∂∂ = 0 then implies: ∂ ◦◦ ◦ ∂ ◦◦ = 0 ∂ ΣΣ ◦ ∂ Σ ◦ + ∂ Σ ◦ ◦ ∂ ◦◦ = 0 ∂ ΣΣ ◦ ∂ ΣΣ = 0 . (2.47)11 p -chain c of X which fulfills: ∂ ◦◦ c = ( ∂c ◦ ) ◦ = 0 . (2.48)is called a relative p -cycle of ( X, ∂X ). This also means that ∂c ∈ C p − (Σ). The subgroupof relative p -cycles of ( X, ∂X ) is denoted Z p ( X, ∂X ). A p -chain c of X which fulfills : c ◦ = ∂ ◦◦ u = ( ∂u ◦ ) ◦ . (2.49)is called a relative p -boundary of ( X, ∂X ). This also means that c = ∂u + v Σ . Thesubgroup of relative p -boundaries of ( X, ∂X ) is denoted B p ( X, ∂X ). The p -th relativehomology group of ( X, ∂X ) is then: H p ( X, ∂X ) = Z p ( X, ∂X ) B p ( X, ∂X ) . (2.50)These groups are the homology groups of the complex ( C • ( X, ∂X ) , ∂ ◦◦ ). Of course, westill have the homology groups H p ( X ) associated with the complex ( C • ( X ) , ∂ ). Moreover,since i Σ is a chain map, for each p , we have an injection i Σ : H p (Σ) → H p ( X ). It is quiteobvious that any p -cycle of X is a relative p -cycle of ( X, ∂X ), the converse being untrue.Nevertheless, any relative p -cycle of ( X, ∂X ) defines a ( p − ∂X . Indeed, let c be relative p -cycle of ( X, ∂X ). Then ∂ ◦◦ c = 0 and from decomposition (2.45) and thelast of the relations (2.47) we deduce that: ∂∂ Σ ◦ c = ∂ ΣΣ ∂ Σ ◦ c = − ∂ Σ ◦ ∂ ◦◦ c = 0 (2.51)In other words, if c is a relative p -cycle of ( X, ∂X ) then ∂ Σ ◦ c is a ( p − ∂X . Moreover, if c is a relative p -boundary of ( X, ∂X ), then c = ∂u + v Σ and hence ∂ Σ ◦ c = − ∂ ΣΣ ( ∂ Σ ◦ u ◦ ). So, if c is a relative p -boundary of ( X, ∂X ) then ∂ Σ ◦ c is a boundary ofΣ = ∂X . From all this we deduce that the operator ∂ Σ ◦ gives rise to a connecting homo-morphism H p ( X, ∂X ) → H p − (Σ), so that we obtain the so-called long exact sequencein homology for X [13]: · · · → H p (Σ) → H p ( X ) → H p ( X, ∂X ) → H p − (Σ) → · · · . (2.52)Let us extend the above construction to de Rham currents. Ω p ( X ) denotes the spaceof smooth p -forms of X , and let Ω p ( X, ∂X ) denotes the subspace of smooth p -forms of X that vanish on ∂X , i.e. the kernel of the pullback i ∗ Σ : Ω p ( X ) → Ω p (Σ). These spaceshave topological duals which are respectively denoted by Ω p ( X ) ⋄ and Ω p ( X, ∂X ) ⋄ . Then,for each p , there is an exact short sequence:0 → Ω p (Σ) ⋄ i Σ −→ Ω p ( X ) ⋄ π ⋄◦ −→ Ω p ( X, ∂X ) ⋄ → , (2.53)12hich is the dual of the short exact sequence:0 → Ω p ( X, ∂X ) π ◦ −→ Ω p ( X ) i ∗ Σ −→ Ω p (Σ) → . (2.54)The first sequence allows to identify Ω p ( X, ∂X ) ⋄ with the quotient Ω p ( X ) ⋄ /i Σ (Ω p (Σ) ⋄ ).The space i Σ (Ω p (Σ) ⋄ ) is then the space of de Rham p -currents of X with support in ∂X ,an element of which is generically denoted by J Σ . By fixing a base for this subspace andby completing this base into a base of Ω p ( X ) ⋄ we generate a decomposition rule of theform: J = J ◦ + J Σ , (2.55)for all J ∈ Ω p ( X ) ⋄ . We will refer to J ◦ as the relative component of J and to J Σ asits Σ- component , as with an inconsequential abuse we can see Ω p ( X, ∂X ) ⋄ and Ω p (Σ) ⋄ as subspaces of Ω p ( X ) ⋄ . We endow Ω p ( X ) ⋄ with the boundary operator d † defined by:( d † J ) [[ α ]] = J [[ dα ]] , (2.56)for all J ∈ Ω p ( X ) ⋄ and all α ∈ Ω p − ( X ). We have the decomposition: d † = d †◦◦ + d † Σ ◦ + d † ΣΣ , (2.57)which stems from the relation: d † J = d †◦◦ J ◦ + d † Σ ◦ J ◦ + d † ΣΣ J Σ . (2.58)The nilpotency property d † d † = 0 is then equivalent to: d †◦◦ d †◦◦ = 0 d † ΣΣ d † Σ ◦ + d † Σ ◦ d †◦◦ = 0 d † ΣΣ d † ΣΣ = 0 . (2.59)Hence, we have three complexes arising from the construction, the complex (cid:0) Ω • ( X ) ⋄ , d † (cid:1) ,the relative complex (cid:0) Ω • ( X, ∂X ) ⋄ , d †◦◦ (cid:1) and the complex (cid:16) i Σ (Ω • (Σ) ⋄ ) , d † ΣΣ (cid:17) which is canon-ically isomorphic to the complex (cid:16) Ω • (Σ) ⋄ , d † Σ (cid:17) . The first two complexes generate familiesof homology groups, H ⋄ p ( X ) and H ⋄ p ( X, ∂X ), and the operator d † Σ ◦ induces a connectinghomomorphism H ⋄ p ( X, ∂X ) → H ⋄ p − ( X ). All this yields the so-called long exact sequencein currents homology: · · · → H ⋄ p (Σ) → H ⋄ p ( X ) → H ⋄ p ( X, ∂X ) → H ⋄ p − (Σ) → · · · H ⋄ ( X, ∂X ) → . (2.60)13lthough they look similar, the modules occurring in the above long sequence are overreal numbers [10] while those appearing in the long sequence (2.52) are over integers.To any ω ∈ Ω − p ( X ) we associate the de Rham p -current J ω defined by: ∀ α ∈ Ω p ( X ) , J ω [[ α ]] = Z X ω ∧ α , (2.61)which defines an injective homomorphism J : Ω − p ( X ) → Ω p ( X ) ⋄ . The elements of Im J are called regular de Rham p -currents of X . If we apply decomposition (2.55) to a regular p -current J ω , then for support reasons J Σ ω = 0. Hence, we have: ∀ ω ∈ Ω − p ( X ) , J ω = J ◦ ω . (2.62)The boundary of a regular p -current J ω is then: d † J ω [[ α ]] = J ω [[ dα ]] = Z X ω ∧ dα . (2.63)A simple integration by parts yields: d † J ω [[ α ]] = ( − ) p Z X dω ∧ α + ( − ) p +1 I Σ ω Σ ∧ α Σ , (2.64)which implies that: d † J ω [[ α ]] = ( − ) p J dω + ( − ) p +1 J ω Σ . (2.65)Recalling that J ω = J ◦ ω and comparing this relation with (2.59), we deduce that: ( d †◦◦ J ω = d †◦◦ J ◦ ω = ( − ) p J dω d † Σ ◦ J ω = d † Σ ◦ J ◦ ω = ( − ) p +1 J ω Σ . (2.66)Hence, unlike J ω the current d † J ω has a Σ-component. We gather this into a propertythat will prove helpful in subsection 4.3. Property 3.
Let J ω be a regular p -current of X . Then J ω = J ◦ ω and: d † J ω = d †◦◦ J ω + d † Σ ◦ J ω = ( − ) p J dω + ( − ) p +1 J ω Σ . (2.67)The presence of a relative sign between the two terms forming the second equality abovewill be clarified a little further. From all this, we deduce that there is a Poincar´e-Lefschetz14uality between these homology groups and the corresponding de Rham cohomologygroups according to: H ⋄ p ( X ) ∼ = H − pdR ( X, ∂X ) , H ⋄ p ( X, ∂X ) ∼ = H − pdR ( X ) . (2.68)As usual, a p -chain c of X defines a de Rham p -current j c according to: ∀ ω ∈ Ω p ( X ) , j c [[ ω ]] = Z c ω . (2.69)As with Σ, p -chains of X which have the same integrals on all p -forms define the samede Rham p -current and hence are considered as equivalent [10]. Furthermore, if c is achain of X with decomposition c = c ◦ + c Σ and de Rham current j c , then we have: j c = j ◦ c + j Σ c . (2.70)with j c ◦ = j ◦ c and j c Σ = j Σ c . Moreover, we have: d † j c = j ∂c , (2.71)with: d †◦◦ j c = j ∂ ◦◦ c , d † Σ ◦ j c = j ∂ Σ ◦ c , d † ΣΣ j c = j ∂ ΣΣ c . (2.72)All this explains the similitude between relations (2.41), (2.45) and (2.47) on the onehand, and relations (2.55), (2.57) and (2.59) on the other hand.Because Σ is the oriented boundary of X , half of the first homology group of thelatter is trivialized. Indeed, each meridian 1-cycle µ Σ a can be seen as the boundary of ameridian disk D a of X . So, we fix once and for all a set of meridian disks D a such that: ∂D a = − µ Σ a , (2.73)for each a = 1 , · · · , g . Hence, we have: ∂ ◦◦ D a = 0 ∂ Σ ◦ D a = − µ Σ a ∂ ΣΣ D a = 0 , (2.74)the first relation implying that each D a is a relative 2-cycle: D a = D ◦ a .Unlike meridian cycles, the longitude 1-cycles λ Σ a aren’t trivial in X . Nevertheless,we can introduce a set of annuli A Σ a in X such that: ∂A a = λ a − λ Σ a . (2.75)15he 1-cycles λ a generate H ( X ) ≃ Z g and are usually referred to as the core 1-cycles of X . The above relation can also be written as: ∂ ◦◦ A a = λ a ∂ Σ ◦ A a = − λ Σ a ∂ ΣΣ A a = 0 . (2.76)Note that relation (2.75) implies that each non-trivial 1-cycle λ a is actually a relativeboundary. This yields: H ( X ) = Z g , H ( X, ∂X ) = 0 . (2.77)For their part, the meridian disks are generators of H ( X, ∂X ) so that we have: H ( X ) = 0 , H ( X, ∂X ) = Z g . (2.78)We complete this with the following, quite trivial, relations: ( H ( X ) = Z , H ( X, ∂X ) = 0 H ( X ) = 0 , H ( X, ∂X ) = Z . (2.79)As the orientation of Σ is inherited from the one of X , relations (2.9) induce thefollowing ones in X : λ a ⊙ D b = δ ab , (2.80)which define the Kronecker indexes of the 1-chains λ a with the 2-chains D b . Theserelations imply that: ∂ Σ ◦ ( D a ⋔ A b ) = − ( ∂ Σ ◦ D a ) ⋔ ( ∂ Σ ◦ A b ) , (2.81)and more generaly: ∂ Σ ◦ ( c ⋔ ˜ c ) = − ∂ Σ ◦ c ⋔ ∂ Σ ◦ ˜ c , (2.82)where c and ˜ c are relative 2-chains of X the intersection of which is well-defined. Ofcourse, we assume that c and ˜ c are such that all the intersections appearing in (2.82)have a meaning. The above Kronecker indexes induces a pairing between H ( X ) and H ( X, ∂X ) and the same Euclidean inner product on Z g as (2.11). In terms of de Rhamcurrents, we have: d †◦◦ j D a = 0 d † Σ ◦ j D a = − j µ Σ a d † ΣΣ j D a = 0 , d †◦◦ j A a = j λ a d † Σ ◦ j A a = − j λ Σ a d † ΣΣ j A a = 0 , (2.83)16elations (2.80) thus reading:( j λ a ∧ j D b ) [[1]] = λ a ⊙ D b = δ ab . (2.84)Finally, still in terms of currents, relation (2.82) takes the form: d † Σ ◦ ( J ∧ ˜ J ) = − d † Σ ◦ J ∧ d † Σ ◦ ˜ J , (2.85)when the exterior product have a meaning. Note that we can recover (2.82) or (2.85) byconsidering two regular 2-currents J ω and J ˜ ω obtained by regularizing c and ˜ c (or J and ˜ J ) and then by writing the following sequence of equalities: d † Σ ◦ ( J ω ∧ J ˜ ω ) = d † Σ ◦ J ω ∧ ˜ ω = − J ω Σ ∧ ˜ ω Σ = − J ω Σ ∧ J ˜ ω Σ = − d † Σ ◦ J ω ∧ d † Σ ◦ J ˜ ω , (2.86)with the help of Property 3.Let us consider the case where ∂X = S × S , with ω A and ω D two regularizations ofthe chain currents j A and j D , respectively. It is no hard to check that ω Σ A , ω Σ D and dω A are regularizations of the chain currents j λ Σ , j µ Σ and j λ , respectively, whereas dω D = 0.Then, by using Property 3, we deduce that: ( d † J ω A = d †◦◦ J ω A + d † Σ ◦ J ω A = J dω A − J ω Σ A d † J ω D = d †◦◦ J ω D + d † Σ ◦ J ω D = − J ω Σ A , (2.87)relations which agree with those of (2.83). Note that ω Σ D is a regularization of j − ∂D and not of j ∂D , which shed lights on the presence of the relative sign in the formula ofProperty (2.83).A U (1) connection on X L ∪ f X R necessarily gives rise to two connections, one on X L and one on X R , that glue on Σ, via f , up to a gauge transformation. Since a U (1)connection is equivalent to a DB 1-cocycle, the set of classes of U (1) connections of X is nothing but the Deligne-Beilinson Z -module H D ( X ) which is embedded into the exactsequence [14]: 0 → Ω ( X )Ω Z ( X ) → H D ( X ) → H ( X ) → . (2.88)From Poincar´e-Lefschetz duality we have H ( X ) ∼ = H ( X, ∂X ) = 0 and hence: H D ( X ) ∼ = Ω ( X ) / Ω Z ( X ) . (2.89)This already suggests to construct DB 1-cocycles of X L ∪ f X R by starting with 1-formson X L and X R that glue on Σ up to a gauge transformation the set of which is Ω Z (Σ),this last space appearing in exact sequence (2.40).17ince X has a boundary, it is also possible to consider the DB space H D ( X, ∂X )which is embedded into the exact sequence:0 → H ( X, ∂X, R / Z ) → H D ( X, ∂X ) → Ω Z ( X, ∂X ) → . (2.90)At the level of Pontryagin dual spaces, there is a natural injection [14] H D ( X ) → H D ( X, ∂X ) ⋆ which suggests to consider this last space in the construction of H D ( X L ∪ f X R ) ⋆ . We then have the following exact sequence:0 → Ω Z ( X, ∂X ) ⋆ ¯ d ∗ −→ H D ( X, ∂X ) ⋆ → H ( X ) = 0 → , (2.91)which implies that H D ( X, ∂X ) ⋆ ∼ = Ω Z ( X, ∂X ) ⋆ . Finally, the following property showsthat H D ( X, ∂X ) ⋆ can be seen as a quotient space, just like H D ( X ). Property 4.
Let Ω ( X, ∂X ) ⋄ Z be the space of relative -currents that are Z -valued on Ω Z ( X, ∂X ) ,the space of closed -forms with integral periods of X that vanish on ∂X . Then, we have: H D ( X, ∂X ) ⋆ ∼ = Ω Z ( X, ∂X ) ⋆ ∼ = Ω ( X, ∂X ) ⋄ Ω ( X, ∂X ) ⋄ Z . (2.92)The first isomorphism above is a simple consequence of the exact sequence (2.91). Thesecond is quite obvious since the elements of Ω ( X, ∂X ) ⋄ Z are by definition the relativelyclosed de Rham currents of ( X, ∂X ) which are Z -valued on Ω Z ( X, ∂X ). We refer thereader to [14] for details. X L ∪ f X R We consider here the following Heegaard construction. Let X L and X R be two copies ofthe handlebody X and consider an orientation reversing diffeomorphism f : Σ L → Σ R ,where Σ L = ∂X L and Σ R = ∂X L . Then, met us consider the oriented smooth 3-manifold X L ∪ f X R obtained by gluing X L and X R along their diffeomorphic boundaries. It iswell-known that any oriented smooth closed 3-manifold can be constructed this way. Inthis second section we explain how to represent chains, forms and currents on X L ∪ f X R with the help of chains, forms and currents on X L and X R . As before, we start withchains, then we consider currents and we finish with smooth DB cohomology groups,dedicating one subsection to each of these topics. An expression of the linking numberis also provided in the second subsection. 18 .1 Chains and cycles We define a p -chain of X L ∪ f X R as a couple ( c L , c R ) of p chains on X L and X R . If wedecompose these chains according to (2.41) we obtain the representation ( c ◦ L + c Σ L , c ◦ R + c Σ R ).However, such a representation is degenerated. Indeed, let us consider a couple of theform ( u Σ , f ( u Σ )). When gluing X L and X R along their boundary via the homeomorphism f , then u Σ obviously “compensates” f ( u Σ ) so that ( u Σ , f ( u Σ )) actually represents thezero p -chain of X L ∪ f X R . Hence, we must consider the equivalence relation:( c L , c R ) ∼ f ( c L + u Σ , c R + f ( u Σ )) , (3.93)for any p -chain u Σ of Σ, the set of p -chains of X L ∪ f X R being then defined as the quotient:( C p ( X L ) × C p ( X R )) / ∼ f . (3.94)This set is denoted by C p ( X L ∪ f X R ). To avoid the use of equivalence classes of chains,we can consider representatives of the form:( c ◦ L , c ◦ R + c Σ R ) , (3.95)called right representatives . Obviously, any p -chain of X L ∪ f X R has a unique rightrepresentative.The boundary of ( c L , c R ) is trivially defined by ∂ ( c L , c R ) = ( ∂c L , ∂c R ). Since f inducesa chain map between C p (Σ L ) and C p (Σ R ), this boundary operator is compatible with theequivalence relation ∼ f and hence goes to classes. On right representatives ∂ is given as: ∂ ( c ◦ L , c ◦ R + c Σ R ) = (cid:0) ∂ ◦◦ c ◦ L , ∂ ◦◦ c ◦ R + ∂ Σ ◦ c ◦ R − f (cid:0) ∂ Σ ◦ c ◦ L (cid:1) + ∂ ΣΣ c Σ R (cid:1) . (3.96)A p -chain of X L ∪ f X R with right representative ( z ◦ L , z ◦ R + z Σ R ) is a p -cycle if: ( ∂ ◦◦ z ◦ L = 0 = ∂ ◦◦ z ◦ R ∂ Σ ◦ z ◦ R + ∂ ΣΣ z Σ R = f (cid:0) ∂ Σ ◦ z ◦ L (cid:1) . (3.97)The first line in the above relations means that z ◦ L and z ◦ R are relative p -cycles, whereasthe second line implies that z Σ R is not necessarily a cycle of Σ R . The set of p -cycles of X L ∪ f X R is denoted by Z p ( X L ∪ f X R ).A p -cycle of X L ∪ f X R with right representative ( b ◦ L , b ◦ R + b Σ R ) is a p -boundary if thereexists a ( p + 1)-chain with right representative ( c ◦ L , c ◦ R + c Σ R ) such that: b ◦ L = ∂ ◦◦ c ◦ L b ◦ R = ∂ ◦◦ c ◦ R b Σ R = ∂ Σ ◦ c ◦ R − f (cid:0) ∂ Σ ◦ c ◦ L (cid:1) + ∂ ΣΣ c Σ R . (3.98)19he first two conditions simply means that b ◦ L and b ◦ R must be relative p -boundaries.According to the relations (2.47), it is quite obvious that a p -boundary is a p -cycle. Theset of p -boundaries of X L ∪ f X R is denoted by B p ( X L ∪ f X R ), the p -th homology groupof X L ∪ f X R thus being: H p ( X L ∪ f X R ) = Z p ( X L ∪ f X R ) B p ( X L ∪ f X R ) . (3.99)As an exercise, let us compute H ( X L ∪ f X R ). As already mentioned, the meridiandisks D a form a set of generators for H ( X, ∂X ). Hence, a couple of the form: g X a =1 m aL D La + ∂ ◦◦ χ L , g X a =1 m aR D Ra + ∂ ◦◦ χ R ! , (3.100)is a generator of H ( X L ∪ f X R ) is and only if the two components are gluing properlyalong Σ, via the gluing mapping f , in order to form a 2-cycle of X L ∪ f X R . This meansthat they fulfill: g X a =1 ( P ~m L ) a λ Σ R a + g X a =1 ( Q ~m L − ~m R ) a µ Σ R a = − ∂ ΣΣ g X a =1 m aL ψ Σ R a + ∂ Σ ◦ χ R − f ( ∂ Σ ◦ χ L ) ! . (3.101)and hence that: ( P ~m L = 0 ~m R = Q ~m L . (3.102)This yields the following property: Property 5.
Any -cycle of X L ∪ f X R have a right representative of the form: g X a =1 m a D La , g X a =1 ( Q ~m ) a D Ra − g X a =1 m a ψ Σ R a ! + ∂V . (3.103) with ~m ∈ ker P and V = ( V ◦ L , V ◦ R ) a -chain of X L ∪ f X R . Hence: H ( X L ∪ f X R ) ∼ = ker P , (3.104)By using the same kind of arguments and exact sequence (2.19) we can show that:20 roperty 6.
Any -cycle of X L ∪ f X R have a right representative of the form: , X a n a λ Ra ! + (cid:0) ∂ ◦◦ c ◦ L , ∂ ◦◦ c ◦ R + ∂ Σ ◦ c ◦ R − f ( ∂ Σ ◦ c ◦ L ) + ∂ ΣΣ c Σ R (cid:1) , (3.105) with ~n ∈ Z g representing an element of coker P . This cycle is homologically trivial if andonly if Q † ~n ∈ Im P † . Hence: H ( X L ∪ f X R ) ∼ = coker P , (3.106)A generator of H ( X L ∪ f X R ) is necessarily a combination of the longitudes of X R (or equivalently of X L ), let say z R~n = (0 , P a n a λ Ra ). Obviously, z R~n is homologous tothe 1-cycle z Σ R ~n = P a n a λ Σ R a , and if ~n = P ~m then this last cycle is the boundaryof (cid:16)P ga =1 m a j D La , P ga =1 ( Q ~m ) a j D Ra − P ga =1 m a j ψ Σ Ra (cid:17) . Hence, z R~n is a boundary when ~n ∈ Im P . Conversely, if z R~n is a boundary then is is easy to check that ~n ∈ Im P .The rest of the proof stems from Property 1.As a last check of the consistency of our definition of chains of X L ∪ f X R , let usconsider ( X L , X R ) which is nothing but X L ∪ f X R itself and as such a 3-cycle. Then, wehave ∂ ( X L , X R ) = ( ∂X L , ∂X R ) = (Σ L , Σ R ) = (0 , Σ R − f (Σ L )) = (0 , A differential p -form of X L ∪ f X R is a couple ( ω L , ω R ) ∈ Ω p ( X L ) × Ω p ( X R ) that fulfillsthe gluing condition: ω Σ R = f ∗ ω Σ L . (3.107)The de Rham differential of a p -form ( ω L , ω R ) of X L ∪ f X R is simply the ( p + 1)-form( dω L , dω R ). Hence, a p -form ( ω L , ω R ) is closed if ω L and ω R are closed. If ( ω L , ω R ) isexact then ω L and ω R are also exact. Conversely, if ω L = dη L and ω R = dη R , then( ω L , ω R ) is exact if and only if ( η L , η R ) is a ( p − X L ∪ f X R , that is to say ifand only if it fulfills (3.107).As among the closed forms of X L ∪ f X R the closed 2-forms with integral periods playa particular role in this article, we describe them in the following property. Property 7.
A closed -form with integral periods of X L ∪ f X R is of the form: α = g X a =1 N a j ∞ D La , g X a =1 ( Q ~N ) a j ∞ D Ra ! + d ( φ L , φ R ) , (3.108)21 ith ~N ∈ ker P and ( φ R , φ L ) ∈ Ω ( X L ) × Ω ( X R ) such that: φ Σ R − f ∗ φ Σ L = g X a =1 N a j ∞ ψ Ra . (3.109) The de Rham cohomology class of α is ~N ∈ ker P so that ker P ∼ = H dR ( X L ∪ f X R ) . As meridian disks are generator of H ( X, ∂X ) and H ( X ) ∼ = H ( X, ∂X ) by Poincar´e-Lefshetz duality, up to an exact term, every closed 1-form of X is a combination of the1-forms j ∞ D a . These forms are regularizations of the chain currents j D a . By restriction toΣ they define regularization j ∞ µ a of the chain currents j µ a . If we consider regularizations j ∞ λ a and j ∞ ψ Ra of j λ a and j ψ Ra , respecitvely, then relations (2.13) as well as the gluingcondition (3.107) quite straightforwardly yield Property 7. Note that the coefficientsof j ∞ D La and j ∞ D Ra must be integers fort the closed form α to have integral periods. The factthat ker P ∼ = H dR ( X L ∪ f X R ) is consistent with the following sequence of isomorphisms H dR ( X L ∪ f X R ) ∼ = H ( X L ∪ f X R ) ∼ = H ( X L ∪ f X R ) ∼ = ker P which stem from Poincar´e-Lefschetz duality.In analogy with p -chains, the set of de Rham p -current of X L ∪ f X R is the quotient:(Ω p ( X L ) ⋄ × Ω p ( X L ) ⋄ ) / ∼ f , (3.110)where the equivalence relation ∼ f is defined by ( J Σ , f ( J Σ )) ∼ f
0. Every de Rham p -current of X L ∪ f X R admits a right representative ( J ◦ L , J ◦ R + J Σ R ), the boundary operator d † acting on them according to: d † ( J ◦ L , J ◦ R + J Σ R ) = (cid:16) d †◦◦ J ◦ L , d †◦◦ J ◦ R + d † Σ ◦ J ◦ R − f (cid:0) d † Σ ◦ J ◦ L (cid:1) + d † ΣΣ J Σ R (cid:17) . (3.111)A de Rham p -current with right representative ( J ◦ L , J ◦ R + J Σ R ) is closed if and only if: ( d †◦◦ J ◦ L = 0 = d †◦◦ J ◦ R d † ΣΣ J Σ R + d † Σ ◦ J ◦ R − f (cid:0) d † Σ ◦ J ◦ L (cid:1) = 0 , (3.112)and it is exact if and only if there exists a de Rham ( p +1)-current with right representative( S ◦ L , S ◦ R + S Σ R ) such that: J ◦ L = ∂ ◦◦ S ◦ L J ◦ R = ∂ ◦◦ S ◦ R J Σ R = ∂ Σ ◦ S ◦ R − f (cid:0) ∂ Σ ◦ S ◦ L (cid:1) + ∂ ΣΣ S Σ R . (3.113)When the de Rham current is the current of a chain, aka a chain current, the aboverelations are nothing but relations (3.97) and (3.98). In the case of chain current, wedenote by j c or C the de Rham current of a chain c .22et z = ∂c and ˜ z = ∂ ˜ c be two homologically trivial 1-cycles of a smooth closed 3-manifold M whose supports do not intersect and such that z and ˜ c on the one hand, and˜ z and c on the second hand, are in transverse position in M . The linking number of z and ˜ z is the integer defined as: lk ( z, ˜ z ) = c ⊙ ˜ z = ˜ c ⊙ z = lk (˜ z, z ) . (3.114)In order to express the above expression of the linking number with respect to the rightrepresentatives c = ( c ◦ L , c ◦ R + c Σ R ) and ˜ z = ( z ◦ L , z ◦ R + z Σ R ), all we have to do is identify thepointwise oriented intersections between the various components of c and ˜ z , which yields: lk ( z, ˜ z ) = c ◦ L ⊙ ˜ z ◦ L + c ◦ R ⊙ ˜ z ◦ R + ∂ Σ ◦ c ◦ R ⊙ ˜ z Σ R + c Σ R ⊙ (cid:0) ∂ Σ ◦ ˜ z ◦ R + ∂ ΣΣ ˜ z Σ R (cid:1) . (3.115)Let us point out that the first two terms in the above expression correspond to intersec-tions in the 3D interior of X L and X R while the last two terms yield intersections in the2D boundary Σ R . Taking into account the second condition of (3.97) as well as the firsttwo of (3.98) we obtain: lk ( z, ˜ z ) = c ◦ L ⊙ ∂ ◦◦ ˜ c ◦ L + c ◦ R ⊙ ∂ ◦◦ ˜ c ◦ R + ∂ Σ ◦ c ◦ R ⊙ ˜ z Σ R + c Σ R ⊙ f ( ∂ Σ ◦ ˜ z ◦ L ) . (3.116)Finally, by using the third relation of (2.47), we deduce that: lk ( z, ˜ z ) = c ◦ L ⊙ ∂ ◦◦ ˜ c ◦ L + c ◦ R ⊙ ∂ ◦◦ ˜ c ◦ R + ∂ Σ ◦ c ◦ R ⊙ ˜ z Σ R − ( ∂ ΣΣ c Σ R ) ⊙ f ( ∂ Σ ◦ ˜ c ◦ L ) , (3.117)with ˜ z Σ R = ∂ Σ ◦ ˜ c ◦ R − f ( ∂ Σ ◦ ˜ c ◦ L ) + ∂ ΣΣ ˜ c Σ R . This expression can be written in term of de Rhamcurrents as: lk ( z, ˜ z ) = (cid:0) j c ◦ L ∧ d †◦◦ j ˜ c ◦ L (cid:1) [[1 L ]] + (cid:0) j c ◦ R ∧ d †◦◦ j ˜ c ◦ R (cid:1) [[1 R ]] ++ (cid:16) d † Σ ◦ j c ◦ R ∧ ( δ ⋆ − ˜ C ) Σ R (cid:17) [[1 Σ R ]] − (cid:16) d † ΣΣ j c Σ R ∧ f ( d † Σ ◦ j ˜ c ◦ L ) (cid:17) [[1 Σ R ]] , (3.118)with ( δ ⋆ − ˜ C ) Σ R = d † Σ ◦ j ˜ c ◦ R − f ( d † Σ ◦ j ˜ c ◦ L )+ d † ΣΣ j ˜ c Σ R = j ˜ z Σ R . Some remarks must be made concerningthis expression of the linking number. Normally, we must take c (resp. ˜ c ) in transverseposition with ˜ z (resp. z ). So, if ˜ z Σ R = 0 (resp. z Σ R = 0) we may have to chose c (resp. ˜ c )such that c Σ R = 0 (resp. ˜ c Σ R = 0). Thanks to the long exact sequence in homology (2.52)such a choice is always possible. The expression of the linking number then reduces to thefirst three terms of the right hand side of (3.118) with the convention that c = ( c ◦ L , c ◦ R ).However, the fourth term in the right hand side of the above expression of the linkingnumber is well-defined when the two currents that make up it are in transverse position,thus making (3.118) an acceptable generalized expression of the linking number. Lastbut not least, although this is not obvious when looking at (3.118), the linking numberfulfills lk ( z, ˜ z ) = lk (˜ z, z ). 23 .3 Smooth DB classes A Deligne-Beilinson 1-cocycle of X L ∪ f X R is a triple ( A L , A R , Λ Σ R ) ∈ Ω p ( X L ) × Ω p ( X R ) × H D (Σ R ) that fulfills the DB-gluing condition: A Σ R − f ∗ A Σ L = ¯ d Λ Σ R . (3.119)Two DB 1-cocycles are said to be DB equivalent if their difference is of the form: g X a =1 m aL j ∞ D La + dq L , g X a =1 m aR j ∞ D Ra + dq R , ξ ∞ ~m L , ~m R + q Σ R − f ∗ q Σ L ! , (3.120)where ξ ∞ ~m L , ~m R ∈ H D (Σ R ) is such that:¯ dξ ∞ ~m L , ~m R = − g X a =1 ( P ~m L ) a j ∞ λ Σ Ra + g X a =1 ( ~m R − Q ~m L ) a j ∞ µ Σ R − dj ∞ ψ Σ Ra , (3.121)and q Σ R − f ∗ q Σ L denotes the restriction to R / Z of q Σ R − f ∗ q Σ L . This choice for ambiguitiescomes from the fact that H D ( X ) ∼ = Ω ( X ) / Ω Z ( X ) so that the elements of Ω Z ( X ) generateambiguities. It turn out that this last space is generated by j ∞ D La , and when the gluingcondition (3.119) is taken into account, we obtain the form (3.120) for the DB ambiguities.The set of DB classes of X L ∪ f X R thus obtained is a Z -module which can be embeddedinto the exact sequence [1]:0 → H ( X L ∪ f X R , R / Z ) → H D ( X L ∪ f X R ) ¯ d −→ Ω Z ( X L ∪ f X R ) → . (3.122)The curvature of A is defined as ¯ dA = ( dA L , dA R ). It is a 2-form of X L ∪ f X R since dA Σ R = d ( f ∗ A Σ L + ¯ d Λ Σ R ) = f ∗ ( dA Σ L ). This 2-form is obviously closed. Moreover, it hasintegral periods as a consequence of Property 5.A DB 3-cocycle is a triple ( G L , G R , υ Σ R ) ∈ Ω ( X L ) × Ω ( X R ) × H D (Σ R ). A DB 3-cocycle of the form ( dF L , dF R , F Σ R − f ∗ ( F Σ L )) is then a DB ambiguity, with : Ω (Σ R ) → H D (Σ R ) denoting the injection associated with the exact sequence 0 → Ω Z (Σ) → Ω (Σ) → H D (Σ) →
0. The set H D ( X L ∪ f X R ) of DB classes thus induced appearsin the exact sequence:0 → H ( X L ∪ f X R , R / Z ) → H D ( X L ∪ f X R ) ¯ d −→ Ω Z ( X L ∪ f X R ) = 0 → , (3.123)from which we deduce that H D ( X L ∪ f X R ) ∼ = H ( X L ∪ f X R , R / Z ) = R / Z .There is a pairing in H D , the DB product ⋆ , which is defined at the level of DBcocyles as follows:( A L , A R , Λ Σ R ) ⋆ ( B L , B R , Π Σ R ) = (cid:16) A L ∧ dB L , A R ∧ dB R , B Σ R ∧ ¯ d Λ Σ R (cid:17) . (3.124)It is obvious that ( A L , A R , Λ Σ R ) ⋆ ( B L , B R , Π Σ R ) is a 3-cocycles of X L ∪ f X R . The abovepairing is commutative and goes to DB classes [1].24 Pontryagin duality on X L ∪ f X R With all the geometrical preliminaries having been done, let us now concentrate onthe core of the article, Pontryagin duality. The construction starts with integration ofDB classes along cycles which shows that 1-cycles are elements of H D ( X L ∪ f X R ) ⋆ :=Hom( H D ( X L ∪ f X R ) , R / Z ), the Pontryagin dual of H D ( X L ∪ f X R ). This yields DB1-cycles which on their turn give rise to DB 1-currents. -cycles If ω = ( ω L , ω R ) is a p -form of X L ∪ f X R and c = ( c L , c R ) is a p -chain of X L ∪ f X R , thenthe integral of ω along c is defined as: Z c ω = Z c L ω L + Z c R ω R . (4.125)This definition is consistent since the integral of ω = ( ω L , ω R ) along any ( u Σ , f ( u Σ )) iszero, these last couples representing the zero p -chain of X L ∪ f X R . Indeed, the gluingcondition ω Σ R = f ( ω Σ L ) fulfilled by ( ω L , ω R ) straightforwardly yields this results. Withrespect to the right representative ( c ◦ L , c ◦ R + c Σ R ) of c the above integral takes the form: Z c ω = Z c ◦ L ω L + Z c ◦ R ω R + Z c Σ R ω Σ R . (4.126)Let ( z ◦ L , z ◦ R + z Σ R ) be the right representative of a 1-cycle z of X L ∪ f X R . Since H ( X, ∂X ) = 0, we deduce that there exist relative 2-chains c ◦ L and c ◦ R such that z ◦ L = ∂ ◦◦ c ◦ L and z ◦ R = ∂ ◦◦ c ◦ R . Moreover, since ∂ ◦◦ c ◦ = ∂c ◦ − ∂ Σ ◦ c ◦ the integral of ω = ( ω L , ω R )along z can be written as: I z ω = Z c ◦ L dω L + Z c ◦ R dω R − Z ∂ Σ ◦ c ◦ L ω Σ L − Z ∂ Σ ◦ c ◦ R ω Σ R + Z z Σ R ω Σ R . (4.127)By taking into account the gluing condition for ( ω L , ω R ), we find that: I z ω = Z c ◦ L dω L + Z c ◦ R dω R + I z Σ R ω Σ R , (4.128)where z Σ R is the 1-cycle of Σ R such that: z Σ R = ∂ Σ ◦ c ◦ R − f ( ∂ Σ ◦ c ◦ L ) + z Σ R . (4.129)Note that ∂ ΣΣ z Σ R = 0 is a consequence of (2.47), (3.97) and of the fact that z ◦ L = ∂ ◦◦ c ◦ L and z ◦ R = ∂ ◦◦ c ◦ R . All this suggests the following property:25 roperty 8. With respect to integration of -forms, any -cycle z of X L ∪ f X R can be representedby a triple: ( c ◦ L , c ◦ R , z Σ R ) ∈ C ( X L , ∂X L ) × C ( X R , ∂X R ) × Z (Σ R ) , (4.130) so that: • the right representative of z is: (cid:0) ∂ ◦◦ c ◦ L , ∂ ◦◦ c ◦ R + ∂ Σ ◦ c ◦ R − f ( ∂ Σ ◦ c ◦ L ) + z Σ R (cid:1) , (4.131) • for any ω ∈ Ω( X L ∪ f X R ) we have: I z ω = Z c ◦ L dω L + Z c ◦ R dω R + I z Σ R ω Σ R , (4.132) A triple c = ( c ◦ L , c ◦ R , z Σ R ) has above will be referred to as a DB -cycle of X L ∪ f X R . To prove this property we must identify ambiguities in the DB 1-cycles which representa given 1-cycle and show that the integrals along these ambiguities are zero. In otherwords we must identify the DB 1-cycles which represent the trivial (i.e. zero) 1-cycle of X L ∪ f X R . It is quite easy to check that these DB 1-cycles are of the form: (cid:0) z ◦ L , z ◦ R , f ( ∂ Σ ◦ z ◦ L ) − ∂ Σ ◦ z ◦ R (cid:1) , (4.133)where z ◦ L and z ◦ R are relative 2-cycles. Finally, it is also easy to check that a DB 1-cycle( c ◦ L , c ◦ R , z Σ R ) represents a 1-boundary of X L ∪ f X R if and only if z Σ R is a 1-boundary ofΣ R .Following the same logic, let us try to extend the above to DB cocycles and classes.We already know that in this case integration is an R / Z -valued linear functional. Definition 1.
The integral of a DB -cocycle A = ( A L , A R , Λ Σ R ) along a -cycles z = ( z L , z R ) isdefined as: I z A := Z z L A L + Z z R A R − I ( ∂z R ) Σ Λ Σ R , (4.134) with ( ∂z R ) Σ = ∂ Σ ◦ z ◦ R + ∂ ΣΣ z Σ R . The third term in the right-hand side of (4.134) is meaningful because ∂ (cid:0) ( ∂z R ) Σ (cid:1) = 0,this term ensuring that the integral is defined modulo integers. Indeed, the integral of A along any representative ( u Σ L , f ( u Σ L )) of the zero 1-cycle of X L ∪ f X R is: Z f ( u Σ L ) ¯ d Λ Σ R − I ∂ ΣΣ f ( u Σ ) Λ Σ R . (4.135)26ince Λ Σ R is a DB class, this number is an integer which not necessarily vanishes unlikethe case of forms.The next step would be to check that the integral along z of a DB ambiguity like(3.120) is an integer so that integration along 1-cycles extends to DB classes as an R / Z -valued functional. We leave it as an exercise and will return to it in a moment. Like for1-forms we can use the fact that z ◦ L = ∂ ◦◦ c ◦ L and z ◦ R = ∂ ◦◦ c ◦ R for some 2-chains c ◦ L and c ◦ R in order to rewrite the integral of a DB cocycle along a z . After some algebraic juggles,we find that: Z c ◦ L dA L + Z c ◦ R dA R + I z Σ R A Σ R . (4.136)where we also use the fact that the curvature ¯ d Λ Σ has integral periods. Of course, theabove expression must be considered in R / Z . Then, like for forms the integral of a DB 1-cocycle along a 1-cycle z can be written with the help of a DB 1-cycle ( c ◦ L , c ◦ R , z Σ R ). More-over, by applying the above integral formula to a DB 1-cycle (cid:0) z ◦ L , z ◦ R , f (cid:0) ∂ Σ ◦ z ◦ L (cid:1) − ∂ Σ ◦ z ◦ R (cid:1) ,known to represent the zero 1-cycle, we obtain: Z z ◦ L dA L + Z z ◦ R dA R + I f ( ∂ Σ ◦ z ◦ L ) − ∂ Σ ◦ z ◦ R A Σ R = Z f ( ∂ Σ ◦ z ◦ L ) ¯ d Λ Σ R Z = 0 , (4.137)which correctly defines integration along a 1-cycles as a R / Z -valued linear functional onthe Z -module of DB 1-cocycles. Let us recall that: 1) ∂ ◦◦ z ◦ L = 0 = ∂ ◦◦ z ◦ R by construction;2) f ( ∂ Σ ◦ z ◦ L ) is a cycle because ∂ ΣΣ f ( ∂ Σ ◦ z ◦ L ) = − f ( ∂ Σ ◦ ∂ ◦◦ z ◦ L ) = − f (0); 3) ¯ d Λ Σ R has integralperiods. It is then a simple exercise to check that the integral of a DB ambiguity alonga 1-cycle is an integer. Property 8 thus extends to DB cocycles and classes as follows. Property 9.
Let z be a -cycle of X L ∪ f X R with c = ( c ◦ L , c ◦ R , z Σ R ) a DB -cycle representing z ,and let A = ( A L , A R , Λ Σ R ) be a DB -cocycle of X L ∪ f X R .1) The evaluation of c on A , defined as: c [[ A ]] := Z c ◦ L dA L + Z c ◦ R dA R + I z Σ R A Σ R , (4.138) is equal, modulo integers, to the integral of A along z . In other words, we have: c [[ A ]] Z = I z A . (4.139)
2) With respect to integration of -forms and evaluation of DB -cocycles, two DB -cycles ( c ◦ L , c ◦ R , z Σ R ) and (˜ c ◦ , ˜ c ◦ R , ˜ z Σ R ) represent the same -cycle if and only if they differby a DB -cycle of the form: (cid:0) z ◦ L , z ◦ R , f ( ∂ Σ ◦ z ◦ L ) − ∂ Σ ◦ z ◦ R (cid:1) , (4.140)27 here z ◦ L and z ◦ R are relative -cycles. Indeed, the evaluation of A on such DB -cycleas given by (4.138) is zero in R / Z and therefore correctly represents the integral of A along the trivial (i.e. zero) -cycle of X L ∪ f X R .3) With respect to integration of -forms and evaluation of DB -cocycles, a DB -cycle ( c ◦ L , c ◦ R , z Σ R ) represents a -boundary if and only if z Σ R = ∂ ΣΣ c Σ R , the correspondingboundary being then ∂ (cid:0) c ◦ L , c ◦ R + c Σ R (cid:1) . This gives rise to an equivalence relation on DB -cycles representing -cycles of X L ∪ f X R .4)Integration of DB -cocycles along -cycles goes to classes. This allows to see any cycle of X L ∪ f X R as an R / Z -valued functional over H D ( X L ∪ f X R ) , which yields thewell known inclusion: Z ( X L ∪ f X R ) ⊂ H D ( X L ∪ f X R ) ⋆ . (4.141) Then, evaluation of DB -cycles of X L ∪ f X R on DB -cocycles of X L ∪ f X R also goesto classes, for both DB -cycles and DB -cocycles. For the sake of completeness, let us precise that the evaluation of a DB 3-cocycle G = ( G L , G R , υ Σ R ) along X L ∪ f X R is defined as: I X L ∪ f X R G := Z X L G L + Z X R G R + I Σ R υ Σ R . (4.142)A simple computation shows that the integral along X L ∪ f X R of an ambiguity is zeromodulo integer. The above definition is very similar to (4.134) as X L and X R are relative3-cycles and ( ∂X R ) Σ = Σ R . Furthermore, the 3-cycles of X L ∪ f X R are just ( nX L , nX R ). To make contact with the previous subsection, let us consider a DB 1-cycle ( c ◦ L , c ◦ R , z Σ R )representing a 1-cycle z of X L ∪ f X R . Its first two components, c ◦ L and c ◦ R , obviouslydefine relative 2-currents, i.e. elements of Ω ( X, ∂X ) ⋄ , j c ◦ L and j c ◦ R . The third component, z Σ R , defines an element of Ω (Σ) ⋄ Z . Thanks to the exact sequence (2.39) we can see z Σ R (or its de Rham current) as the generalized curvature of some ς Σ R ∈ H D (Σ) ⋆ and replacethe original triple ( c ◦ L , c ◦ R , z Σ R ) by the triple (cid:0) j c ◦ L , j c ◦ R , ς Σ R (cid:1) . Let us note that the sameexact sequence tells us that ς Σ R is not unique. To confirm this idea, let us recall thatintegration provides an injection Z (Σ) → H D (Σ) ⋆ , so that ς Σ R is just the image of the1-cycle z Σ R under this injection. However, ς Σ R must not be confused with the de Rhamcurrent of z Σ R as this current is actually the generalized curvature δ ⋆ − ς Σ R of ς Σ R .All these remarks suggest the following definition. Definition 2. ) A DB -current of X L ∪ f X R is a triple: J = ( J ◦ L , J ◦ R , ς Σ R ) ∈ Ω ( X L , ∂X L ) ⋄ × Ω ( X R , ∂X R ) ⋄ × H D (Σ R ) ⋆ . (4.143)
2) The generalized curvature of a DB -current J = ( J ◦ L , J ◦ R , ς Σ R ) is the elementof Ω ( X L ∪ f X R ) ⋄ Z defined as: δ ⋆ − J = ( d † ◦ ◦ J ◦ L , d † ◦ ◦ J ◦ R + d † Σ ◦ J ◦ R − f ( d † Σ ◦ J ◦ L ) + δ ∗− ς Σ R ) , (4.144) δ ⋆ − ς Σ R ∈ Ω (Σ) ⋄ Z being the generalized curvature of ς Σ R ∈ H D (Σ R ) ⋆ . A DB -currentwith zero genralized curvature is said to be flat , in analogy with the smooth case.3) A DB -current ambiguity is a DB -current of the form: j = ( j ◦ L , j ◦ R , ξ Σ R ) , (4.145) with j ◦ L ∈ Ω ( X L , ∂X L ) ⋄ Z , j ◦ R ∈ Ω ( X R , ∂X R ) ⋄ Z and ξ Σ R ∈ H D (Σ) ⋆ such that: δ ⋆ − ξ Σ R = f ( d † Σ ◦ j ◦ L ) − d † Σ ◦ j ◦ R . (4.146)
4) Two DB -currents of X L ∪ f X R that differ by a DB ambiguity are said to be DBequivalent, and the equivalent class of a DB -current of X L ∪ f X R is simply called a generalized DB class of X L ∪ f X R . The set of generalized DB classes of X L ∪ f X R isthe Pontryagin dual, H D ( X L ∪ f X R ) ⋆ , of H D ( X L ∪ f X R ) .5) Let J = ( J ◦ L , J ◦ R , ς R Σ ) a DB -current and A = ( A L , A R , Λ R Σ ) a DB -cocycle. Theevaluation of J on A is defined as: J [[ A ]] := J ◦ L [[ dA L ]] + J ◦ R [[ dA R ]] + (cid:0) δ ⋆ − ς R Σ (cid:1) (cid:2)(cid:2) A Σ R (cid:3)(cid:3) . (4.147) As an R / Z -valued linear functional this evaluation goes to DB classes, for both DB cur-rents and DB cocycles, thus realizing Pontryagin duality. By construction, the de Rham 1-current δ ⋆ − J fulfills relations (3.112) and hence is closed.Moreover, it belongs to Ω ( X L ∪ f X R ) ⋄ Z . Indeed, for any α ∈ Ω Z ( X L ∪ f X R ) decomposedaccording to Property 7 we have: δ ⋆ − J [[ α ]] = g X a =1 ( Q ~N ) a ( δ ⋆ − ς Σ R ) hh j ∞ µ Σ Ra ii , (4.148)and since δ ⋆ − ς Σ R ∈ Ω (Σ R ) ⋄ Z , the above sum is an integer. We leave it as an exercise tocheck the similarity between the above formula and the integral of the curvature along a2-cycle of X L ∪ f X R . Let note that: δ ⋆ − J = (cid:0) , δ ∗− ς Σ R (cid:1) + d † ( J ◦ L , J ◦ R ) , (4.149)29hich provides another way to see why generalized curvature are closed and Z -valued onΩ Z ( X L ∪ f X R ). The choice for DB ambiguities is based on Property 4. Moreover, becauseof the constraint (4.146) they must fulfill, DB ambiguities are flat: δ ⋆ − j = 0. Hence,the generalized curvature of a generalized DB class J is well-defined as the generalizedcurvature of any of its representatives J : δ ⋆ − J = δ ⋆ − J .By comparing evaluations formulas (4.138) and (4.147) we deduce that a 1-cycle z represented by a DB 1-cycle c = ( c ◦ L , c ◦ R , z Σ R ) should define a DB 1-current of X L ∪ f X R andhence a generalized DB class. For this purpose, let us chose a DB class ς Σ R ∈ H D (Σ R ) ⋆ such that δ ⋆ − ς Σ R = j z Σ R . Then, J z = ( j c ◦ L , j c ◦ R , ς Σ R ) is a DB 1-current whose generalizedcurvature is j z , and we have: c [[ A ]] = J z [[ A ]] , (4.150)for any DB 1-cocycle A of X L ∪ f X R . The generalized DB class of J z is said to becanonically associated with z and as such usually identified with z itself. Let us pointout that the de Rham current j z of a 1-cycle z is the curvature of many inequivalent classesof DB 1-currents, J z being one of these classes. So, we have the following property [9]: Property 10.
There is a canonical injection: Z ( X L ∪ f X R ) → H D ( X L ∪ f X R ) ⋆ , (4.151) usually referred to as a cycle map . The generalized DB class canonically associated witha -cycle z is denoted by J z . A DB -current in the class of J z is said to be associated with z . If z is a homologically trivial 1-cycle so that there exists a 2-chain c = (cid:0) c ◦ L , c ◦ R + c Σ R (cid:1) suchthat z = ∂c , then the generalized DB class canonically associated with z is the class ofthe DB 1-current: J z = J ∂c = (cid:16) j c ◦ L , j c ◦ R , j c Σ R (cid:17) , (4.152)where j c Σ R ∈ H D (Σ) ⋆ is defined by: ∀ a Σ ∈ H D (Σ) , j c Σ R [[ a Σ ]] := j c Σ R (cid:2)(cid:2) ¯ da Σ (cid:3)(cid:3) . (4.153)This last relation obviously extend to all de Rham 2-currents of Σ, thus defining acanonical mapping : Ω (Σ) ⋄ → H D (Σ) ⋆ . The kernel of this mapping is obviouslyΩ (Σ) ⋄ Z so that : Ω (Σ) ⋄ / Ω (Σ) ⋄ Z → H D (Σ) ⋆ is injective.Let us complete these first remarks concerning DB 1-currents and generalized DBclasses with the following property about DB ambiguities which is a consequence ofProperty 5. 30 roperty 11. Let (cid:0) j ◦ L , j ◦ R , ξ Σ R (cid:1) be a DB -current ambiguity. We have: j ◦ L = g X a =1 m aL j D La + d † ◦ ◦ q ◦ L and j ◦ R = g X a =1 m aR j D Ra + d † ◦ ◦ q ◦ L , (4.154) where m aL , m aR ∈ Z , q ◦ L ∈ Ω ( X L , ∂X L ) ⋄ and q ◦ R ∈ Ω ( X R , ∂X R ) ⋄ , and such that: δ ⋆ − ξ Σ R = − g X a =1 ( P ~m L ) a j λ Σ Ra − g X a =1 ( Q ~m L − ~m R ) a j µ Σ Ra + d † Σ d † Σ ◦ q ◦ R − f ( d † Σ ◦ q ◦ L ) − g X a =1 ~m aL j ψ Σ Ra ! . (4.155)Now that we have identified the objects to work with, their ambiguities and the classesthus obtained, we can state the main result of this article: Property 12.
The set H D ( X L ∪ f X R ) ⋆ is a Z -module which can be embedded into the following exactsequence: → H ( X L ∪ f X R , R / Z ) → H D ( X L ∪ f X R ) ⋆ → Ω ( X L ∪ f X R ) ⋄ Z → . (4.156)To prove the above property we start by recalling that a DB ambiguity j = ( j ◦ L , j ◦ R , ξ Σ R )is flat. Hence, the curvature operation defined on DB 1-cocycles goes to DB classesthus yielding the morphism δ ⋆ − : H D ( X L ∪ f X R ) ⋆ → Ω ( X L ∪ f X R ) ⋄ Z . This morphism issurjective. Indeed, let F be an element of Ω ( X L ∪ f X R ) ⋄ Z . Since H ( X ) ≃ H ( X, ∂X )is generated by the longitudes λ a of X , we can always write F as (cid:0) , P a n a j λ Ra (cid:1) + d † S with S = (cid:0) S ◦ L , S ◦ R + S Σ R (cid:1) . The triple (cid:16) S ◦ L , S ◦ R + P a n a j A Ra , ς Σ R + S Σ R (cid:17) such that δ ⋆ − ς Σ R = P a n a j λ Σ Ra + d † ΣΣ S Σ R is then a DB 1-current whose generalized curvature is F . The existenceof ς Σ R ∈ H D (Σ) ⋆ is ensured by the exactness of the sequence (2.39) and the fact that j λ Σ Ra and d † ΣΣ S Σ R both belong to Ω (Σ) ⋄ Z . Hence, we obtain the exact sequence: H D ( X L ∪ f X R ) ⋆ → Ω ( X L ∪ f X R ) ⋄ Z → . (4.157)Let us now consider a flat DB 1-current J = ( J ◦ L , J ◦ R , ς Σ R ) which is not in the trivialclass. The curvature of J is then vanishing and we have: d † ◦ ◦ J ◦ L = 0 = d † ◦ ◦ J ◦ R , (4.158)31hich means that J ◦ L and J ◦ R are relatively closed. Since the meridian disks generate H ( X, ∂X ) the relatively closed currents J ◦ L and J ◦ R are necessarily of the form: J ◦ L = g X a =1 x aL j D a + d † ◦ ◦ q ◦ L and J ◦ R = g X a =1 x aR j D a + d † ◦ ◦ q ◦ R , (4.159)the coefficients x aL and x aR being real numbers. Furthermore, the flatness of J implies that δ ⋆ − ς Σ R = f (cid:0) d † Σ ◦ J ◦ L (cid:1) − d † Σ ◦ J ◦ R with δ ⋆ − ς Σ R ∈ Ω (Σ) ⋄ Z , and since this last space is generatedby the median and longitude cycles of Σ, we deduce that there exist ~L, ~M ∈ Z g suchthat: g X a =1 ( P ~x L ) a j λ Σ Ra + g X a =1 ( Q~x L − ~x R ) a j µ Σ Ra = g X a =1 L a Σ j λ Σ Ra + g X a =1 M a Σ j µ Σ Ra , (4.160)and hence that: ( P ~x L = ~LQ~x L − ~x R = ~M . (4.161)We can add to our initial DB 1-current any DB 1-current ambiguity whose generic formis given in Property 11 which induces the shifts ~x L → ~x L + ~m L , ~x R → ~x R + ~m R and ~L → ~L + P ~m L . Hence, at the level of DB classes, we must consider ~θ L , ~θ R ∈ ( R / Z ) g instead of ~x L , ~x R ∈ R g . Note that ~L ∈ ( Z g / Im P ) = coker P . The first of the aboveconstraint can be written in term of the Pontryagin dual, P ⋆ , of P as: P ⋆ ~θ L = ~ , (4.162)which means that ~θ L ∈ ker P ⋆ . Furthermore, since on the one hand ker P ⋆ = (coker P ) ⋆ and on the second hand coker P = H ( X L ∪ f X R ), we conclude that ~θ L ∈ H ( X L ∪ f X R , ( R / Z )) = H ( X L ∪ f X R ) ⋆ , which means that the set of DB classes of a flat DB1-currents is isomorphic to H ( X L ∪ f X R , ( R / Z )). This achieves the demonstration ofProperty 12. Note that the constraints (4.161) are the same as those that yield theextension to the left of the exact sequence (3.122) as explained in [1]. Moreover, thesecond constraints in (4.161) plays no role in the proof of Property 12. It just determines ~θ R in term of ~θ L . In the particular case of the lens space L ( p, q ) with Heegaard splittingsuch that ∂X = S × S we obtain θ L = L/p and θ R = qL/p . A flat DB 1-current of L ( p, q ) is then of the form (cid:16) Lp j D L , qLp j D R , − ζ L − Lp j ψ Σ R (cid:17) with ζ L ∈ H D ( S × S ) ⋆ suchthat δ ⋆ − ζ L = Lj λ Σ R .Thanks to exact sequence (4.156) we can see the Z -module H D ( X L ∪ f X R ) ⋆ as adiscrete fiber space over the space of generalized curvatures Ω ( X L ∪ f X R ) ⋄ Z , the group32 ( X L ∪ f X R , R / Z ) acting on fibers as a group of translation. As a closed de Rham 1-current, any generalized curvature defines an element in F ( X L ∪ f X R ). Hence, if we havechosen a set of de Rham 1-currents, j k ( k = 1 , · · · , b ), which generate F ( X L ∪ f X R ),then any generalized curvature F can be written as: F = b X k =1 m k j k + d † S , (4.163)with S ∈ Ω ( X L ∪ f X R ) ⋄ . To make this decomposition unique we must consider the classof S in the quotient space Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ . This defines an isomorphismbetween Ω ( X L ∪ f X R ) ⋄ Z and F ( X L ∪ f X R ) ⊕ (Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ ). Nowextending what was done for 2-chains to 2-currents, we associate to the 2-current S = (cid:0) S ◦ L , S ◦ R + S Σ R (cid:1) the DB 1-current S = (cid:16) S ◦ L , S ◦ R , S Σ R (cid:17) which fulfills δ ⋆ − S = d † S . However,the class of S in Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ must not be confused with the DB classof S . Indeed, if we add to S a closed 2-current T , then the the DB 1-current associatewith S + T is equivalent to S if and only if T ∈ Ω ( X L ∪ f X R ) ⋄ Z . In other words, if wesee the DB class of S as an element of Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ Z , then the DB classof T , where T ∈ Ω ( X L ∪ f X R ) ⋄ , is an element of Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ Z . TheDB 1-currents S and S + T are generically inequivalent although they have the samegeneralized curvature: d † S . To end this remark, let us point out that the situation isquite difference when S is the current of a 2-chain. Indeed, we saw that the DB class ofthe current of a 2-chain c is the class of DB 1-current J c defined in (4.152). The integralnature of the 2 − chains is then a strong constraint on the associated DB classes, thusmaking this association into a canonical one for these particular 2-currents. This is thecycle map.Let us put all these results together into the following main property which extendsthe decomposition property of smooth DB classes discussed in [1]. Property 13.
For each n f ∈ F ( X L ∪ f X R ) we pick a representative ~n f ∈ Z g , with ~n f = ~ repre-senting n = . Similarly, for each θ ∈ H ⋆P † ∼ = H ( X L ∪ f X R ) ⋆ we pick a representative ~θ = ~θ f + ~θ τ as in Property 2, with ~θ = ~ representing θ = .Any J ∈ H D ( X L ∪ f X R ) ⋆ is represented in a unique way by a DB 1-current of theform: J ( ~n f ,~θ,S ) = J ~n f + J ~θ + S , (4.164)33 here: J ~n f := , g X a =1 n af j A Ra , ζ ~n f ! J ~θ := g X a =1 θ a j D La , g X a =1 ( Q~θ ) a j D Ra , − ζ P ~θ − g X a =1 θ a j ψ Σ Ra ! , (4.165) with: • J ( ~ ,~ , is the representative of J = 0 ; • S ∈ Ω ( X L ∪ f X R ) ⋄ parametrizes an element of Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ Z whose curvature is d † S , with S = (cid:0) S ◦ L , S ◦ R + S Σ R (cid:1) ; • ζ ~n ∈ H D (Σ R ) ⋆ such that: δ ⋆ − ζ ~n = g X a =1 n a j λ Σ Ra . (4.166) Hence, we can consider ( ~n f , ~θ, S ) as a parametrization of H D ( X L ∪ f X R ) ⋆ . Moreover, wehave the following generalized curvatures: δ ⋆ − J ~n f = , g X a =1 n af j λ Ra ! δ ⋆ − J ~θ = 0 δ ⋆ − S = d † Σ ◦ S ◦ R − f (cid:0) d † Σ ◦ S ◦ L (cid:1) + d † Σ S Σ R . (4.167)With respect to exact sequence (4.156), δ ⋆ − J ( ~n f ,~θ,S ) is the base point of the fiber onwhich the DB class of J ( ~n f ,~θ,S ) stands. The DB class of J ~θ τ plays the same role asthe torsion origins introduced in the standard construction [3], and the DB class of J ~θ f is a (free) translation along the corresponding fiber. To clarify the role played by therepresentatives S we can refer to the discussion preceding Property 13. Hence, let us fixsome u ∈ Ω ( X L ∪ f X R ) ⋄ / Ω ( X L ∪ f X R ) ⋄ . We consider two cases: − if u admits a representative which is the 2-current of a 2-chain c = (cid:0) c ◦ L , c ◦ R + c Σ R (cid:1) then we decide to represent u by j c = (cid:16) j c ◦ L , j c ◦ R + j c Σ R (cid:17) , and the DB 1-current weassociate to j c is C = (cid:16) j c ◦ L , j c ◦ R , j c Σ R (cid:17) . Then, the set of DB classes whose generalizedcurvature is d † j c = j ∂c is generated by the DB 1-currents C + J ~θ f when [ ~θ f ] runsthrough ker ˆ P . 34 if there is no current of 2-chains representing u , then we select randomly a rep-resentative 2-current S = (cid:0) S ◦ L , S ◦ R + S Σ R (cid:1) and the DB 1-current we associate to S is S = (cid:16) S ◦ L , S ◦ R , S Σ R (cid:17) . The set of DB classes whose generalized curvature is d † S isgenerated by the DB 1-currents S + J ~θ f when ~θ f runs through ker P .It should be noted that by clarifying the role played by S we have at the same time shedlight on the role played by the free translations J ~θ f . As an exercise the reader can checkthat up to an ambiguity we have J P ~m = C with δ ⋆ − J P ~m = d † j c = j ∂c for some 2-chain c .This is consistent with the fact that P ~m is zero in H ( X L ∪ f X R ) ∼ = coker P and justifythe choice made in Property 13 to represent the trivial class in H ( X L ∪ f X R ) by ~n = ~ H D ( X L ∪ f X R ) ⋆ with the following property Property 14.
For J ( ~n f ,~θ,S ) a DB 1-current, let us write ǫ ⋆ J ( ~n f ,~θ,S ) the class of ~n f + P ~θ ∈ Z g in coker P ∼ = H ( X L ∪ f X R ) . At the level of generalized DB classes, this defines a surjectivehomomorphism: ǫ ⋆ : H D ( X L ∪ f X R ) ⋆ → H ( X L ∪ f X R ) , (4.168) which can be extended to the left to form the exact sequence: → Ω Z ( X L ∪ f X R ) ⋆ → H D ( X L ∪ f X R ) ⋆ → H ( X L ∪ f X R ) → . (4.169)This property, the detailed proof of which is left as an exercise, is a consequence ofdecomposition (4.164) and of the discussion which follows Property 13.We would like to end this section by looking closer at the DB 1-currents associatedwith 1-cycles of X L ∪ f X R . Let us first point out that if z is a 1-cycle and J is a DB1-current whose generalized curvature is j z then the generalized curvature of J + J ~θ is j z too. Hence, with respect to decomposition (4.164) the identification of the DB 1-currentswhich are associated with a given 1-cycle of X L ∪ f X R , in the sense of Property 10, isnot totally obvious. However, as this identification will be essential in the context ofChern-Simons and BF Quantum Field theories, we must exhibit it properly.Let n be an element of coker P and let ~n ∈ Z g be a representative of n . Accordingto Property 2, we can write ~n = ~n f + ~n τ = ~n f + P ~θ τ with ~θ τ = ~m/p for some p ∈ Z and ~m ∈ Z g . Let us introduce the DB 1-current: C ~θ τ = g X a =1 θ aτ j D La , g X a =1 ( P ~θ τ ) a j A Ra + ( Q~θ τ ) a j D Ra , − g X a =1 θ aτ j ψ Σ Ra ! . (4.170)35ccording to decomposition (4.164), this DB 1-current is of type S and not of type J ~θ ,the de Rham 2-current associated with C ~θ τ being: C ~θ τ = g X a =1 θ aτ j D La , g X a =1 ( P ~θ τ ) a j A Ra + ( Q~θ τ ) a j D Ra − θ aτ j ψ Σ Ra ! , (4.171)and we have: δ ⋆ − C ~θ τ = , g X a =1 n aτ j λ Ra ! = d † C ~θ τ ǫ ⋆ C ~θ τ = 0 . (4.172)The first equality indicates that the generalized curvature of C ~θ τ is the de Rham 1-currentof the 1-cycle z ~n τ = (cid:0) , P ga =1 n aτ j λ Ra (cid:1) and the boundary of the de Rham 2-current C ~θ τ which implies that z ~n τ is a torsion cycle. As a matter of fact, although C ~θ τ is not a chaincurrent, the 2-current pC ~θ τ is the de Rham current of the 2-chain: c ~m = g X a =1 m a D La , g X a =1 ( P ~m ) a A Ra + ( Q ~m ) a D Ra − m a ψ Σ R a ! , (4.173)which fulfills ∂c ~m = p z ~n τ . This also explains why we decided to write C ~θ τ instead of S ~θ τ ,the nature of the index ~θ τ preventing a possible confusion with a chain current.Let us now consider the DB 1-current: J ( ~n f , − ~θ τ ,C ~θτ ) = J ~n f − J ~θ τ + C ~θ τ . (4.174)Then, for any DB 1-cocycle A = ( A L , A R , Λ Σ R ) we have: J ( ~n f , − ~θ τ ,C ~θτ ) [[ A ]] = I z ~n A Σ R , (4.175)with: z ~n = , g X a =1 n a λ Ra ! . (4.176)Hence, the DB 1- current J ( ~n f , − ~θ τ ,C ~θτ ) is associated with the 1-cycle z R~n . Note that: δ ⋆ − J ( ~n f , − ~θ τ ,C ~θτ ) = j z ~n ǫ ⋆ J ( ~n f , − ~θ τ ,C ~θτ ) = n . (4.177)36ny 1-cycle in the homology class n can be written as: z ~n = z ~n + ∂c , (4.178)for some 2-chain c = ( c ◦ L , c ◦ R + c Σ R ). Consequently, the DB 1-current: J z ~n = J ( ~n f , − ~θ τ ,C ~θτ ) + C , (4.179)where C = ( j c ◦ L , j c ◦ R , j c Σ R ), is associated with the 1-cycle z ~n , and as such fulfills: J z ~n [[ A ]] = I z~n A , (4.180)for any DB 1-cocycle A = ( A L , A R , Λ Σ R ). We gather this into the following property. Property 15.
Let n be an element of coker P ∼ = H ( X L ∪ f X R ) .Let ~n = ~n f + P ~θ τ , with ~θ τ = ~m/p , be a representative of n in Z g .Let z ~n be a -cycle whose de Rham current is: j z ~n = , g X a =1 n a j λ Ra ! + d † C . (4.181)
The generalized DB class canonically associated with z n is then the class of the DB -current: J z ~n = J ~n f − J ~θ τ + C ~θ τ + C = J ( ~n f , − ~θ τ ,C ~θτ + C ) , (4.182) where C is the DB -current associated with the chain -current C . Let us point out that once a representative of type J ( ~n f ,~θ,S ) has been chosen in eachgeneralized DB class, the DB 1-current (4.182) is the unique one associated with the1-cycle z ~n . Moreover, still with this choice made, J z ~n is the only DB 1-current withgeneralized curvature j z ~n and homology class the class of ~n . These two constraints thusprovide another way to specify the DB 1-current associated with a given 1-cycle of X L ∪ f X R . If z = ∂c , then we have J z = J ∂c = C = (cid:16) j c ◦ L , j c ◦ R , j c Σ R (cid:17) = J ( ~ ,~ ,C ) , with C = j c . In this subsection we shall see how the DB product between DB 1-cocycles can be extendfirstly to a product between DB 1-currents and DB 1-cocycles, and secondly, with some37recautions, to a product between DB 1-currents. The construction is very similar towhat happen for de Rham currents and forms with respect to the exterior product: theexterior product of two forms is well-defined and so is the exterior product of a de Rhamcurrent with a form. The product of two de Rham currents whose total dimension is themanifold’s dimension may be defined when these currents fulfill certain conditions. Ourinterest in trying to extend the DB product to generalized DB classes is twofold. Onthe one hand, the Chern-Simons and BF actions are both defined using the DB product.On the second hand, since Chern-Simons and BF observables all rely on 1-cycles whichare singular DB classes, it seems natural to consider H D ( X L ∪ f X R ) ⋆ , which contains Z ( X L ∪ f X R ) ⋆ , as the space of fields for these two theories. These considerations bringus to wonder whether it is possible to extend the standard BD product to generalizedDB classes.As a first step, let us associate a DB 1-current to a DB 1-cocycle so that we cannaturally pair singular and regular DB classes. Let A = ( A L , A R , Λ R Σ ) be a DB 1-cocycleof X L ∪ f X R . We set: J A := Z X L ∪ f X R A ⋆ • . (4.183)This defines a functional which acts linearly (and continuously) on any BD 1-cocycle B = ( B L , B R , Π R Σ ) according to: J A [[ B ]] = Z X L A L ∧ dB L + Z X R A R ∧ dB R + I Σ R ¯ d Λ R Σ ∧ B Σ R . (4.184)To check that J A is a DB 1-current we need to identify its components according toDefinition 2. The injection Ω ( X ) J −→ Ω ( X, ∂X ) ⋄ , Property 3 and the comparison ofexpressions (4.184) and (4.138) bring us to set: ( A ◦ L = J A L A ◦ R = J A R , (4.185)with: ( d †◦◦ A ◦ L = J dA L d †◦◦ A ◦ R = J dA R , ( d † Σ ◦ A ◦ L = − J A Σ L d † Σ ◦ A ◦ R = − J A Σ R (4.186)After noticing that: f ( J A Σ R ) = J f ∗ A Σ R , (4.187)38e conclude that: d † Σ ◦ A ◦ R − f ( d † Σ ◦ A ◦ L ) = − J A Σ R + J f ∗ A Σ L = − J A Σ R − f ∗ A Σ L = − J ¯ d Λ Σ R . (4.188)The de Rham 1-current J ¯ d Λ Σ R is obviously an element of Ω (Σ) ⋄ Z . Exact sequence (2.39)ensures us that there exists ξ Σ R ∈ H D (Σ) ⋆ such that δ ⋆ − ξ Σ R = J ¯ d Λ Σ R . More precisely,thanks to the canonical injection H D (Σ) J −→ H D (Σ) ⋆ already mentioned in the end ofsubsection 2.1 we can set: ξ Σ R = J Λ Σ R , (4.189)so that: δ ⋆ − ξ Σ R = J ¯ d Λ R Σ . (4.190)Hence, we can see J A as the DB 1-current ( A ◦ L , A ◦ R , ξ Σ R ), with: J A [[ B ]] = A ◦ L [[ dB L ]] + A ◦ R [[ dB R ]] + ( δ ⋆ − ξ Σ R ) (cid:2)(cid:2) B Σ R (cid:3)(cid:3) , (4.191)which perfectly matches evaluation formula (4.147). Since evaluation goes to classes, wehave the following property [14]. Property 16.
There is a canonical injective homomorphism: H D ( X L ∪ f X R ) J −→ H D ( X L ∪ f X R ) ⋆ , (4.192) also called the canonical inclusion of H D ( X L ∪ f X R ) in H D ( X L ∪ f X R ) ⋆ . By analogy withde Rham currents, the elements of Im J are said to be regular , and a DB -current issaid to be regular if it is of the form J A for some DB -cocycle A . Let us note that in a regular class there is always at least one regular representative.However not all representatives are regular. Conversely, in a singular class there is noregular representative.Thanks to the previous discussion, we see that evaluation of DB 1-current on DB1-cocycles can be considered as a first extension of the DB product if we set:( J ⋆ J B ) [[1]] := J [[ B ]] , (4.193)for any DB 1-current J and any regular DB 1-current J B . In particular, for regular DB1-currents J A and J B we find that:( J A ⋆ J B ) [[1]] = Z X L ∪ f X R A ⋆ B , (4.194)39s it must be. This extended DB product goes to classes provided that evaluations aretaken in R / Z .Before going on, let us comment the use of the 1 which appears in the above evalua-tions. The DB product A ⋆ B defines a DB 3-cocycles of X L ∪ f X R whose DB class is anelement of H D ( X L ∪ f X R ). If we follow the logic of de Rham currents, we should havewritten: ( J ⋆ J B ) [[1]] := J [[ B ⋆ . (4.195)By comparing this relation with (4.193) we deduce that the 1 appearing here behaveslike an identity for the star product with B . The DB class of 1 cannot be an elementof H D ( X L ∪ f X R ) as otherwise the product B ⋆ H D ( X L ∪ f X R ), andsince J ∈ H D ( X L ∪ f X R ) ⋆ = Hom( H D ( X L ∪ f X R ) , R / Z ) the evaluation J [[ B ⋆ ⋆ : H pD ( M ) × H qD ( M ) → H p + q +1 D ( M ) [12]. Hence, B ⋆ H D ( X L ∪ f X R ) if and only if1 ∈ H − D ( X L ∪ f X R ), and since by convention H − D ( X L ∪ f X R ) = Z [12], we conclude thatthe DB class 1 appearing in the DB product B ⋆ H − D ( X L ∪ f X R ).To go one step further, let us compare more carefully regular and singular DB 1-currents. Let J A = ( A ◦ L , A ◦ R , ξ Σ R ) be a regular DB 1-current and let J = ( J ◦ L , J ◦ R , ς Σ R )be a singular one. Their generalized curvatures are: ( δ ⋆ − J A = (cid:0) d †◦◦ A ◦ L , d †◦◦ A ◦ R + d † Σ ◦ A ◦ R − f ( d † Σ ◦ A ◦ L ) + δ ⋆ − ξ Σ R (cid:1) δ ⋆ − J = (cid:0) d †◦◦ J ◦ L , d †◦◦ J ◦ R + d † Σ ◦ J ◦ R − f ( d † Σ ◦ J ◦ L ) + δ ⋆ − ς Σ R (cid:1) . (4.196)Furthermore, since the generalized curvature of ξ Σ R is: δ ⋆ − ξ Σ R = J ¯ d Λ Σ R = f ( d † Σ ◦ A ◦ L ) − d † Σ ◦ A ◦ R , (4.197)we deduce that: δ ⋆ − J A = (cid:0) d †◦◦ A ◦ L , d †◦◦ A ◦ R (cid:1) = J ¯ dA . (4.198)Hence, if ( δ ⋆ − J ) Σ R denotes the Σ R component of the generalized curvature of a DB 1-current J , the regular current J A fulfills:( δ ⋆ − J A ) Σ R = d † Σ ◦ A ◦ R − f ( d † Σ ◦ A ◦ L ) + δ ⋆ − ξ Σ R = 0 . (4.199)Let J [[ A ]] be given as in (4.147). Integrating by parts this evaluation yields: J [[ A ]] = ( d † J ◦ L ) [[ A L ]] + ( d † J ◦ R ) [[ A R ]] + ( δ ⋆ − ς Σ R ) (cid:2)(cid:2) A Σ R (cid:3)(cid:3) . (4.200)40ince d † = d †◦◦ + d † Σ ◦ , we can rewrite this evaluation as:( d †◦◦ J ◦ L ) [[ A L ]] + ( d †◦◦ J ◦ R ) [[ A R ]] + ( d † Σ ◦ J ◦ L ) (cid:2)(cid:2) A Σ L (cid:3)(cid:3) + ( d † Σ ◦ J ◦ R + δ ⋆ − ς Σ R ) (cid:2)(cid:2) A Σ R (cid:3)(cid:3) , (4.201)and as:( d † Σ ◦ J ◦ L ) (cid:2)(cid:2) A Σ L (cid:3)(cid:3) = − (cid:0) f ( d † Σ ◦ J ◦ L ) (cid:1) (cid:2)(cid:2) f ∗ A Σ L (cid:3)(cid:3) = − (cid:0) f ( d † Σ ◦ J ◦ L ) (cid:1) (cid:2)(cid:2) A Σ R − ¯ d Λ Σ R (cid:3)(cid:3) . (4.202)we find that: J [[ A ]] = ( d †◦◦ J ◦ L ) [[ A L ]] + ( d †◦◦ J ◦ R ) [[ A R ]] ++( d † Σ ◦ J ◦ R − f ( d † Σ ◦ J ◦ L ) + δ ⋆ − ς Σ R ) (cid:2)(cid:2) A Σ R (cid:3)(cid:3) + f ( d † Σ ◦ J ◦ L ) (cid:2)(cid:2) ¯ d Λ Σ R (cid:3)(cid:3) . (4.203)Note the similitude with formula (4.134). By introducing the Σ R component of J andreplacing evaluation by formal exterior product we get: J [[ A ]] = ( d †◦◦ J ◦ L ∧ J A L ) [[1 L ]] + ( d †◦◦ J ◦ R ∧ J A R ) [[1 R ]] ++ (cid:16) ( δ ⋆ − J ) Σ R ∧ J A Σ R (cid:17) [[1 Σ R ]] + (cid:16) f ( d † Σ ◦ J ◦ L ) ∧ J ¯ d Λ Σ R (cid:17) [[1 Σ R ]] , (4.204)and by using relations (4.186) and (4.190) we obtain: J [[ A ]] = ( d †◦◦ J ◦ L ∧ A ◦ L ) [[1 L ]] + ( d †◦◦ J ◦ R ∧ A ◦ R ) [[1 R ]] + − ( (cid:0) δ ⋆ − J (cid:1) Σ R ∧ d † Σ ◦ A ◦ R ) [[1 Σ R ]] + ( f ( d † Σ ◦ J ◦ L ) ∧ δ ⋆ − ξ Σ R ) [[1 Σ R ]] , (4.205)and finally: J [[ A ]] = ( A ◦ L ∧ d †◦◦ J ◦ L ) [[1 L ]] + ( A ◦ R ∧ d †◦◦ J ◦ R ) [[1 R ]] ++( d † Σ ◦ A ◦ R ∧ (cid:0) δ ⋆ − J (cid:1) Σ R ) [[1 Σ R ]] − ( δ ⋆ − ξ Σ R ∧ f ( d † Σ ◦ J ◦ L )) [[1 Σ R ]] . (4.206)All the terms in this last expression are well defined because the currents A ◦ L , A ◦ R , d † Σ ◦ A ◦ R and δ ⋆ − ξ Σ R are regular [10]. By comparing the above expression with the one of R A ⋆ B and remembering that we started with an integration by parts, it seems more logical toconsider that the former defines ( J A ⋆ J ) [[1]] rather then ( J ⋆ J A ) [[1]] so that:( J A ⋆ J B ) [[1]] = ( A ◦ L ∧ d †◦◦ B ◦ L ) [[1 L ]] + ( A ◦ R ∧ d †◦◦ B ◦ R ) [[1 R ]] ++ ( d † Σ ◦ A ◦ R ∧ ( δ ⋆ − J B ) Σ R ) [[1 Σ R ]] + ( δ ⋆ − ξ Σ R ∧ f ( d † Σ ◦ B ◦ L ) [[1 Σ R ]]= ( A ◦ L ∧ d †◦◦ B ◦ L ) [[1 L ]] + ( A ◦ R ∧ d †◦◦ B ◦ R ) [[1 R ]] + − ( J ¯ d Λ R Σ ∧ d † Σ ◦ B ◦ R ) [[1 Σ R ]] , (4.207)which agrees with (4.184). Note that we use the fact that ( δ ⋆ − J B ) Σ R = 0.All this yields the first generalization of the DB product according to the followingproperty. 41 roperty 17.
1) A DB -current ( A ◦ L , A ◦ R , ξ Σ R ) is regular if and only if there exist a DB -cocycle ( A L , A R , Λ Σ R ) such that: A ◦ L = J A L , A ◦ R = J A R , ξ Σ R = J Λ Σ R , (4.208) with H D (Σ) J −→ H D (Σ) ⋆ the canonical injection. The generalized curvature of a regularDB -current J A is the image of the curvature of A under the injection Ω Z ( X L ∪ f X R ) → Ω ( X L ∪ f X R ) ⋄ Z and as such is a regular de Rham current.2) Let J = ( J ◦ L , J ◦ R , ς Σ R ) and J A = ( A ◦ L , A ◦ R , ξ Σ R ) be two DB -currents, the firstbeing singular and the second regular. The generalized DB product of J A with J is: ( J A ⋆ J ) [[1]] = ( A ◦ L ∧ d †◦◦ J ◦ L ) [[1 L ]] + ( A ◦ R ∧ d †◦◦ J ◦ R ) [[1 R ]] ++( d † Σ ◦ A ◦ R ∧ (cid:0) δ ⋆ − J (cid:1) Σ R ) [[1 Σ R ]] − ( δ ⋆ − ξ Σ R ∧ f ( d † Σ ◦ J ◦ L )) [[1 Σ R ]] , (4.209) and fulfills: ( J A ⋆ J ) [[1]] Z = J [[ A ]] = ( J ⋆ J A ) [[1]] , (4.210) with A the DB -cocycle defining J A . In particular, if J z is associated with a -cycle z we have: ( J z ⋆ J A ) [[1]] = J z [[ A ]] = I z A . (4.211)From the first of the above properties, we deduce that a generalized DB class is regularif and only if it is a smooth Pontryagin dual in the sense of [14]. In other words, ageneralized DB class is regular if and only if its generalized curvature is regular.To further extend the generalized DB product the first idea is to replace the regularDB 1-current J A in formula (4.209) with a general DB 1-current K which yields:( J ⋆ K ) [[1]] = ( J ◦ L ∧ d †◦◦ K ◦ L ) [[1 L ]] + ( J ◦ R ∧ d †◦◦ K ◦ R ) [[1 R ]] ++ ( d † Σ ◦ J ◦ R ∧ ( δ ⋆ − K ) Σ R ) [[1 Σ R ]] − ( δ ⋆ − ς Σ R ∧ f ( d † Σ ◦ K ◦ L )) [[1 Σ R ]] . (4.212)Before wondering whether this expression is meaningful, let us see if it gives rise toa commutative product or not. To this end, we consider the difference ∆( J , K ) =( J ⋆ K ) [[1]] − ( K ⋆ J ) [[1]]. We have:∆( J , K ) = − (cid:0) d †◦◦ ( J ◦ L ∧ K ◦ L ) (cid:1) [[1 L ]] − (cid:0) d †◦◦ ( J ◦ R ∧ K ◦ R ) (cid:1) [[1 L ]]+ ( d † Σ ◦ J ◦ R ∧ ( δ ⋆ − K ) Σ R ) [[1 Σ R ]] − ( d † Σ ◦ K ◦ R ∧ ( δ ⋆ − J ) Σ R ) [[1 Σ R ]] − ( δ ⋆ − ς Σ R ∧ f ( d † Σ ◦ K ◦ L )) [[1 Σ R ]] + ( δ ⋆ − κ Σ R ∧ f ( d † Σ ◦ J ◦ L )) [[1 Σ R ]] , (4.213)42hich after some algebraic juggles yields:∆( J , K ) = (cid:0) ( δ ⋆ − J ) Σ R ) ∧ ( δ ⋆ − K ) Σ R ) (cid:1) [[1 Σ R ]] − (cid:0) δ ⋆ − ς Σ R ∧ δ ⋆ − κ Σ R (cid:1) [[1 Σ R ]] . (4.214)Since δ ⋆ − ς Σ R and δ ⋆ − κ Σ R are generalized curvature (on Σ R ) they are closed which impliesthat the quantity (cid:0) δ ⋆ − ς Σ R ∧ δ ⋆ − κ Σ R (cid:1) [[1 Σ R ]] is firstly meaningful [10] and secondly an inte-ger. So, the term (cid:0) δ ⋆ − ς Σ R ∧ δ ⋆ − κ Σ R (cid:1) [[1 Σ R ]] is not an issue if commutativity is consideredmodulo integers, i.e. in R / Z . Hence, it is the first term in the right-hand side of theabove equation which breaks commutativity. To ensure commutativity, we must add tothe original expression of ( J ⋆ K ) [[1]] the term: − (cid:0) ( δ ⋆ − J ) Σ R ∧ ( δ ⋆ − K ) Σ R (cid:1) [[1 Σ R ]] = 12 (cid:0) ( δ ⋆ − K ) Σ R ∧ ( δ ⋆ − J ) Σ R (cid:1) [[1 Σ R ]] . (4.215)We can now set the following definition: Definition 3.
Let J = ( J ◦ L , J ◦ R , ς Σ R ) and K = ( K ◦ L , K ◦ R , κ Σ R ) be two DB -currents of X L ∪ f X R .When it exists, their generalized DB product is: ( J ⋆ K ) [[1]] = ( J ◦ L ∧ d †◦◦ K ◦ L ) [[1 L ]] + ( J ◦ R ∧ d †◦◦ K ◦ R ) [[1 R ]] ++ (∆ Σ R J ∧ ( δ ⋆ − K ) Σ R [[1 Σ R ]] − ( δ ⋆ − ς Σ R ∧ f ( d † Σ ◦ K ◦ L )) [[1 Σ R ]] , (4.216) with: ∆ Σ R J = d † Σ ◦ J ◦ R −
12 ( δ ⋆ − J ) Σ R = 12 (cid:0) d † Σ ◦ J ◦ R + f ( d † Σ ◦ J ◦ L ) − δ ⋆ − ς Σ R (cid:1) . (4.217)To see the consistency of this definition we must check that the generalized DB productwith a DB 1-current ambiguity is zero (modulo integers). We leave this as an exercise.Of course we find that ( J A ⋆ J B ) [[1]] = R M A ⋆ B is well-defined. If one of the twoDB 1-currents is regular all the evaluations appearing in the above definition are well-defined [10], the additional contribution (4.215) vanishes and as expected we have:( J ⋆ J A ) [[1]] = ( J A ⋆ J ) [[1]] = J [[ A ]] . (4.218)When both DB 1-currents are singular some or all of the evaluations appearing in thedefinition of ( J ⋆ K ) [[1]] may be ill-defined. Fortunately, the situation is not as bad asit seems. To see this, let us give some fundamental generalized DB products relying on43ecomposition (4.164). After some algebraic juggles, we find: • (cid:16) J ~n f ⋆ J ~n ′ f (cid:17) [[1]] Z = 0 , • (cid:0) J ~n f ⋆ J ~θ (cid:1) [[1]] = D Q~θ, ~n f E , • (cid:0) J ~n f ⋆ S (cid:1) [[1]] = g X a =1 (cid:0) n af j λ Ra ∧ S ◦ R (cid:1) [[1 Σ R ]] , • (cid:0) J ~θ ⋆ J ~θ ′ (cid:1) [[1]] = − D P ~θ, Q~θ ′ E = − D P ~θ τ , Q~θ ′ τ E , • (cid:0) J ~θ ⋆ S (cid:1) [[1]] = 0 , • (cid:16) C ~θ τ ⋆ C ~θ ′ τ (cid:17) [[1]] Z = D Q~θ τ , P ~θ ′ τ E . (4.219)In the computation of the first and last of the above relations we used ( j A Ra ∧ j λ Rb ) [[1 R ]] Z = 0which derives from the following regularization. First, we chose the support of the annuli A Ra in such a way that ( j A Ra ∧ j λ Rb ) [[1 R ]] = 0 when a = b . Then, when a = b we considera framing ˘ λ Ra of λ Ra and set ( j A Ra ∧ j λ Ra ) [[1 R ]] := ( j A Ra ∧ j ˘ λ Ra ) [[1 R ]]. This last term is aKronecker index and as such an integer, that is to say zero modulo integers. This is anexample of what we will call the zero regularization procedure in the next property. Thethird generalized product in the above list might be ill-defined. However, if the singularsupport of δ ⋆ − S does not intersect the support of any of the j λ Ra , then the correspondinggeneralized DB product is well-defined. In fact, it is even sufficient to require that thesingular support of ( δ ⋆ − S ) ◦ R = d †◦◦ S ◦ R does not intersect the support of any of the j λ Ra .Thus, this third term is not hard to manage, for instance by putting some simple supportrestriction on the set of S fields. The last kind of generalized product which can be ill-defined are those of the form ( S ⋆ S ′ ) [[1]]. Identifying the “good fields” to consider seems ahard task. Fortunately, we will see in the last section that finding a precise set of “goodfields” is a rather irrelevant question in the context of Chern-Simons and BF abeliantheories. Nonetheless, because they will play a role in these theories, the more specificcase of DB 1-currents associated with 1-cycles requires some attention.Let J ∂c = (cid:16) j c ◦ L , j c ◦ R , j c Σ R (cid:17) and J ∂c ′ = (cid:16) j c ′ L ◦ , j c ′ R ◦ , j c ′ R Σ (cid:17) be DB 1-currents associated withtwo homologically trivial 1-cycles ∂c and ∂c ′ , respectively. Then, we have:( J ∂c ⋆ J ∂c ′ ) [[1]] =( j c ◦ L ∧ d †◦◦ j c ′ L ◦ ) [[1 L ]] + ( j c ◦ R ∧ d †◦◦ j c ′ R ◦ ) [[1 R ]] ++ (∆ Σ R C ∧ ( δ ⋆ − C ′ ) Σ R ) [[1 Σ R ]] − ( d † Σ j c Σ R ∧ f ( d † Σ ◦ j c ′ L ◦ )) [[1 Σ R ]] . (4.220)If the supports of ∂c and ∂c ′ do not intersect then ( δ ⋆ − C ) Σ R ∧ ( δ ⋆ − C ′ ) Σ R = 0 so that theright-hand side of the above expression coincides with expression (3.118) of the linking44umber of ∂c with ∂c ′ and as such is an integer. Thus, we have:( J ∂c ⋆ J ∂c ′ ) [[1]] = lk ( ∂c, ∂c ′ ) Z = 0 . (4.221)Of course, it might be necessary to chose c and c ′ carefully for the above generalizedDB product to be well-defined, a precaution which must also be taken when the linkingnumber is written as a Kronecker index.Let z and z ′ be two homologically non-trivial 1-cycles. There exist p, p ′ ∈ Z andtwo 2-chains c and c ′ such that pz = ∂c and p ′ z ′ = ∂c ′ . We already seen that the DB1-currents associated with z (resp. z ′ ) is J z = C ~θ τ − J ~θ τ (resp. J z ′ = C ~θ τ − J ~θ τ ) where ~θ τ (resp. ~θ ′ τ ) is such that P ~θ τ (resp. P ~θ ′ τ ) represents the homology class of z (resp. z ′ ).Now, from (4.219) we deduce that:( J z ⋆ J z ′ ) [[1]] = (cid:16) C ~θ τ ⋆ C ~θ ′ τ (cid:17) [[1]] + (cid:16) J ~θ τ ⋆ J ~θ ′ τ (cid:17) [[1]] Z = 0 . (4.222)In addition, we know that there is a bilinear pairing Γ : T ( X L ∪ f X R ) × T ( X L ∪ f X R ) → Q / Z , referred to as the linking form of X L ∪ f X R and which is defined as follows. Let z τ and z ′ τ be the two non-trivial torsion 1-cycles we have just introduced. Following [15],we set: Γ( z τ , z ′ τ ) = 1 p ( c ⊙ z ′ τ ) . (4.223)It is not hard to check that this definition is independent of the choice of c . Since p ′ z ′ = ∂c ′ , we can also write:Γ( z τ , z ′ τ ) = 1 pp ′ ( c ⊙ ∂c ′ ) = 1 pp ′ lk ( ∂c, ∂c ′ ) . (4.224)This bilinear pairing becomes an Q / Z -valued bilinear pairing at the level of classes. Now,since we have j z = d † C ~θ τ and j z ′ = d † C ~θ ′ τ , we deduce that:Γ( z τ , z ′ τ ) = (cid:16) C ~θ τ ⋆ C ~θ ′ τ (cid:17) [[1]] Z = D Q~θ τ , P ~θ ′ τ E . (4.225)See [11] for the particular case of a rational homology sphere. Let us point out that, thelinking number of z τ and z ′ τ , now defined as Γ( z τ , z ′ τ ), is not an integer and hence notzero in Q / Z . Furthermore, this rational linking number is not (cid:0) J z τ ⋆ J z ′ τ (cid:1) [[1]] since thisgeneralized DB product is zero. Of course, the linking form Γ is nothing but the linkingform introduced in Property 2.Finally, let z and z ′ be two homologically non-trivial free 1-cycles with associated DB1-currents J z = J ~n f + C and J z ′ = J ~n ′ f + C ′ , respectively. By combining the fundamentalevaluations (4.219) together with property (4.221) we deduce that:( J z ⋆ J z ′ ) [[1]] Z = 0 . (4.226)We end with the following property. 45 roperty 18.
1) Let z and z ′ be two -cycles of X L ∪ f X R and let J z and J z ′ be two DB -currentassociated with z and z ′ , respectively. Then we have: ( J z ⋆ J z ′ ) [[1]] Z = 0 . (4.227) Moreover, we define the zero regularization procedure by setting: ( J z ⋆ J z ) [[1]] = ( J z ⋆ J ˘ z ) [[1]] Z = 0 , (4.228) where ˘ z is any framing of z .2) The linking form of X L ∪ f X R is defined with the help of generalized DB productsaccording to: Γ( z τ , z ′ τ ) = (cid:16) C ~θ τ ⋆ C ~θ ′ τ (cid:17) [[1]] Z = D Q~θ τ , P ~θ ′ τ E , (4.229) where z τ and z ′ τ are torsion -cycles whose homology classes are given by P ~θ τ and P ~θ ′ τ ,respectively. For homologically trivial 1-cycles, the zero regularization procedure can be identifiedwith the framing procedure which allows to define the linking of an homologically trivial1-cycle z with itself, aka the self-linking of z . The other cases are an extension of thisframing procedure to generalized DB products. Let us start with the BF theory as the Chern-Simons action can be derived from the BFone. We already know [4, 5] that on a closed 3-manifold M the the U (1) BF action canbe defined as the functional: S BF,k (cid:0) ¯ A, ¯ B (cid:1) = k Z R ¯ A ⋆ ¯ B , (5.230)where ¯ A and ¯ B are elements of H D ( M ) and k is a coupling constant. This action cannotbe considered as classical since it is R / Z -valued. Nevertheless, this choice of definitionfor the BF action is twofold. Firstly it goes straightforwardly to the quantum worldand secondly it automatically implies that the coupling constant k is an integer, i.e. isquantized. Following the same path, the Chern-Simons action is defined by: S CS,k (cid:0) ¯ A (cid:1) = S BF,k (cid:0) ¯ A, ¯ A (cid:1) = k Z R ¯ A ⋆ ¯ A , (5.231)46ith ¯ A ∈ H D ( M ) and k ∈ Z .These actions where studied in various contexts [1, 3, 4]. Our purpose here is to findexpressions of the CS and BF actions in the context of a Heegaard spliting X L ∪ f X R when H D ( M ) is replaced by H D ( M ) ⋆ , that is to say when we deal with generalized DBclasses instead of smooth ones. Let us recall that we want to do that in order to includelinks into the set of fields. Indeed, a link can be see as a 1-cycle and hence as a generalizedDB class.We shall concentrate on the BF action and try to find its expression in term of thevariables which appear in the decomposition formula (4.164). In other words, we wantto compute: S BF,k ( ~n f , ~θ, S, ~n ′ f , ~θ ′ , S ′ ) = k (cid:16) J ( ~n f ,~θ,S ) ⋆ J ( ~n ′ f ,~θ ′ ,S ′ ) (cid:17) [[1]] , (5.232)where the generalized DB product was given in Definition 3. By using the sets of funda-mental generalized DB products (4.219) we straightforwardly get: S BF,k ( ~n f , ~θ, S, ~n ′ f , ~θ ′ , S ′ ) = k ( S ⋆ S ′ ) [[1]] ++ k g X a =1 (cid:16) n af j λ Σ Ra ∧ S ′ R ◦ + n ′ fa j λ Σ Ra ∧ S ◦ R (cid:17) [[1 R ]] ++ k D Q~θ, ~n ′ f E + k D Q~θ ′ , ~n f E − k D P ~θ τ , Q~θ ′ τ E . (5.233)The Chern-Simons action is then: S CS,k ( ~n f , ~θ, S ) = k ( S ⋆ S ) [[1]] + 2 k g X a =1 (cid:16) n af j λ Σ Ra ∧ S ◦ R (cid:17) [[1 R ]]+ 2 k D Q~θ, ~n f E − k D P ~θ τ , Q~θ ′ τ E . (5.234)Of course, these expressions are formal because the various product appearing in themmight be ill-defined. These expressions, although formal, are very close to the expressionsobtained in the smooth case in [1]. In particular the inner products are the same. Let’s have a look at the functional measures which are involved in the determination ofthe CS and BF partition function as well as of the expectation values of U (1) Wilsonloops. These observables will be discussed in the last subsection. Decomposition (4.164)provides us with the most natural expression of the functional measures: dµ CS,k ( ~n f , ~θ, S ) = DS X ~n f X ~θ τ d b ~θ f e iπS CS,k ( ~n f ,~θ,S ) . (5.235)47ll integrals and sums are taken over classes, or rather over the representatives of Prop-erty 13. Note that the only infinite dimensional, and so formal, measure is the one dealingwith the variables S . All the others are finite dimensional. Of course the Bf measure is: dµ BF,k ( ~n f , ~θ, S, ~n ′ f , ~θ ′ , S ′ ) = DSDS ′ X ~n f ,~n ′ f X ~θ τ ,~θ ′ τ d b ~θ f d b ~θ ′ f e iπS BF,k ( ~n f ,~θ,S,~n ′ f ,~θ ′ ,S ′ ) . (5.236)Let us consider the shift: ~n f → ~n f + ~t f = ~u f , (5.237)where ~t f is fixed. Since T ( X L ∪ f X R ) is a free abelian group, this shift does not changethe sums over ~n f in the functional measure dµ CS,k . However, under this shift the CSaction (5.234) changes according to: S CS,k ( ~u f − ~t f , ~θ, S ) = S CS,k ( ~u f , ~θ, S ) − k (cid:16) J ( ~u f ,~θ,S ) ⋆ J ~t f (cid:17) [[1]] , (5.238)with: J ~t f = , g X a =1 t af j A Ra , ζ ~t f ! . (5.239)This is a simple consequence of the relation: (cid:16) J ( ~u f ,~θ,S ) ⋆ J ~t f (cid:17) [[1]] = g X a =1 (cid:0) t af j λ Ra ∧ S ◦ R (cid:1) [[1 R ]] + D Q~θ, ~t f E = (cid:16) J ( ~n f ,~θ,S ) ⋆ J ~t f (cid:17) [[1]] , (5.240)which itself can be deduced from (5.233). The second line follows from the fact thatthe right-hand side of the first equality is independent of ~u f . The zero regularizationassumption implying that (cid:16) J ~t f ⋆ J ~t f (cid:17) [[1]] = 0, we finally find: S CS,k ( ~u f − ~t f , ~θ, S ) = k (cid:16) ( J ( ~u f ,~θ,S ) − J ~t f ) ⋆ ( J ( ~u f ,~θ,S ) − J ~t f ) (cid:17) [[1]] . (5.241)Hence, since the sum over ~n f is invariant under the shift (5.237), the CS measure isinvariant by a shift of the form: J ( ~u f ,~θ,S ) → J ( ~u f ,~θ,S ) − J ~t f , (5.242)Conversely, if we add to the CS action a term like the second term appearing in the righthand side of (5.238) then this contribution can be totally reabsorbed in the CS measure48hanks to the shift invariance. This property is referred to as the color 2k-periodicity of the CS theory with coupling constant k . Let us point out that the color periodicity ofthe Chern-Simons theory is related to the Cameron-Martin property of the action. Anequivalent result is obtained for the BF action when considering shifts of the form: ( ~n f → ~n f + ~t f = ~u f ~n ′ f → ~n ′ f + ~t ′ f = ~u ′ f , (5.243)which infers the color k-periodicity of the BF theory with coupling constant k .A dual symmetry comes from the term 2 k D Q~θ f , ~n f E in the CS action. For any ~m ∈ Z g and any ~n f we have: 2 k (cid:28) Q (cid:18) ~m k (cid:19) , ~n f (cid:29) = h Q ~m, ~n f i Z = 0 , (5.244)which means that the CS action is invariant with respect to the shift: ~θ f → ~θ f + ~m k . (5.245)Then, since the finite measure d b ~θ f is obviously invariant under this shift, we concludethat the CS functional measure is invariant under (5.245). The directions ~m/ k gener-ating this symmetry are called zero modes of the CS theory with coupling constant k .Similarly, the BF functional measure is invariant under ~θ f → ~θ f + ~mk~θ ′ f → ~θ ′ f + ~m ′ k . (5.246)The directions ~m/k generating this symmetry are called the zero modes of the BFtheory with coupling constant k . In the next subsections we shall see how the symme-try on color and the dual one provided by zero modes dual symmetries couple in thefunctional integration procedure which yield partition functions and expectation valuesof observables. Let us start again with the BF theory. The BF partition function is defined by: Z BF,k = 1 N BF Z dµ BF,k ( ~n f , ~θ, S, ~n ′ f , ~θ ′ , S ′ )= 1 N BF Z DSDS ′ X ~n f ,~n ′ f X ~θ τ ,~θ ′ τ d b ~θ f d b ~θ ′ f e iπS BF,k ( ~n f ,~θ,S,~n ′ f ,~θ ′ ,S ′ ) , (5.247)49here N BF is a normalization factor supposed to take care of the divergences that mayoccur due to the infinite dimensional part in the functional integration. It expression willbe chosen almost at the end of the computation.The integration is formally performed over H D ( X L ∪ f X R ) ⋆ × H D ( X L ∪ f X R ) ⋆ . How-ever, the BF action is not well-defined on this configuration space. Hence, we shouldrestrict the integration to a (maximal) subspace E on which the action is well-defined.Nevertheless, it turns out that only the finite integrals will provide us with quantityrelated to the U (1) Turaev-Viro invariant.Let us first perform the integration over the circular variables ~θ f and ~θ ′ f . By lookingat the expression (5.233) of the BF action it is quite obvious that this integration yieldsthe factor: δ Q † ~n f ,~ δ Q † ~n ′ f ,~ , (5.248)in the remaining sums and integrals. Thanks to Property 1 and its corollary, this factorcan be replaced by the factor: δ ~n f ,~ δ ~n ′ f ,~ , (5.249)Let us recall that we have assumed that the trivial class is represented by ~n = ~
0. Now,taking into account this factor when performing the sum of ~n f and ~n ′ f then gives: Z BF,k = 1 N BF Z DSDS ′ X ~θ τ ,~θ ′ τ e iπS BF,k ( ~ ,~θ τ ,S,~ ,~θ ′ τ ,S ′ ) , (5.250)with: S BF,k ( ~ , ~θ τ , S, ~ , ~θ ′ τ , S ′ ) = k ( S ⋆ S ′ ) [[1]] − k D P ~θ τ , Q~θ ′ τ E . (5.251)The two terms in this expression of the action are totally decoupled so that we have: Z BF,k = 1 N BF (cid:18)Z DSDS ′ e iπk ( S⋆S ′ )[[1]] (cid:19) X ~θ τ ,~θ ′ τ e − iπk h P ~θ τ ,Q~θ ′ τ i . (5.252)We now decide to set: N BF = Z DSDS ′ e iπk ( S⋆S ′ )[[1]] , (5.253)and we conclude that: Z BF,k = X ~θ τ ,~θ ′ τ e − iπk h P ~θ τ ,Q~θ ′ τ i . (5.254)50ecalling that D P ~θ τ , Q~θ ′ τ E = Γ( ~θ τ , ~θ ′ τ ) where Γ : T × T → R / Z induces the linking formof X L ∪ f X R according to Property 2, we recover the standard partition function of the U (1) BF theory which is related to a Turaev-Viro invariant [4].With exactly the same steps, we obtain the following expression for the CS partitionfunction: Z CS,k = 1 N BF (cid:18)Z DSe iπk ( S⋆S )[[1]] (cid:19) X ~θ τ e − iπk h P ~θ τ ,Q~θ τ i , (5.255)and by setting: N CS = Z DSe iπk ( S⋆S )[[1]] , (5.256)we finally get: Z CS,k = X ~θ τ e − iπk h P ~θ τ ,Q~θ τ i . (5.257)This is the standard expression for the CS partition function which is related to aReshetikhin-Turaev invariant [2–4].Let us stress out that thanks to the form we have chosen for the normalization factors N BF and N CS the partition functions are obtained from finite integrals. In particular,this means that beside the zero regularization assumption, we didn’t need to check if thevarious generalized DB product are well defined. In the CS theory, the reduced expectation value of an observable O CS is: hhO CS ii = 1 N CS Z O CS dµ CS,k ( ~n f , ~θ, S ) , (5.258)and in the BF theory, for an observable O BF , by: hhO BF ii = 1 N BF Z O BF dµ BF,k ( ~n f , ~θ, S, ~n ′ f , ~θ ′ , S ′ ) . (5.259)As already noticed, the observables in the CS and BF theories are Wilson loop, that isto say U (1) holonomies. Let us consider a framed link L = L ∪ L ∪ · · · ∪ L n in X L ∪ f X R .We represent this link as a 1-cycle z L and denote by ~N = ~N f + ~N τ its homology class,with ~N τ = ˆ P ~ Θ τ and ~ Θ τ = ~M /p . The observable defined by L is then: W CS ( L , J ) = e iπ R L J . (5.260)51s a 1-cycle of X L ∪ f X R , z L defines a generalized DB class J L so that we can rewritethe above Wilson loop observable as: W CS ( L , J ) = e iπ ( J ⋆ J L )[[1]] . (5.261)As usual, the quantity ( J ⋆ J L ) [[1]] is not always well-defined so we have to assume that thespace of configuration we are working with only contains fields for which this generalizedDB product is well-defined. In fact, we must chose our link L properly because the spaceof configuration have already been chosen for the action to be well-defined. In reality, allthese subtleties will prove quite irrelevant since, as with the partition function, it is thefinite part of the functional integral that will yield a link invariant.In order to compute hh W CS ( L , J ) ii we must compute the generalized DB product J ⋆ J L . We also want to do that in the variables ( ~n f , ~θ, S ) used to write the CS func-tional measure. This means that we must decompose the DB 1-current representing J L according to (4.164). Thanks to Property 15 we can write: J L = J ~N f − J Θ τ + C Θ τ + C = J ( ~N f , − Θ τ ,C T ) , (5.262)with C T = C Θ τ + C . Then, by noticing that ( J ⋆ J L ) [[1]] is very close from the BF action,we can rely on expression (5.233) to obtain:( J ⋆ J L ) [[1]] =( J ( ~n f ,~θ,S ) ⋆ J ( ~N f , − Θ τ ,C T ) ) [[1]]=( S ⋆ C T ) [[1]] + g X a =1 ( n af j λ Ra ∧ C ◦ T,R + N af j λ Ra ∧ S ◦ R ) [[1 R ]] ++ D Q~θ, ~N f E − D Q~ Θ τ , ~n f E + D P ~θ τ , Q~ Θ τ E =( S ⋆ C T ) [[1]] + g X a =1 ( N af j λ Ra ∧ S ◦ R ) [[1 R ]] + D Q~θ, ~N f E + D P ~θ τ , Q~ Θ τ E , (5.263)where C = ( C ◦ L , C ◦ R , C Σ R ) and C T = C Θ τ + C . We used the fact that: g X a =1 ( n af j λ Ra ∧ C ◦ T,R ) [[1 R ]] = g X a =1 ( n af j λ Ra ∧ ( C ~ Θ τ + C ) ◦ R ) [[1 R ]] Z = D Q~ Θ τ , ~n f E , (5.264)in order to compensate the term − D Q~ Θ τ , ~n f E in the expression of ( J ⋆ J L ) [[1]]. Notethat P ga =1 (cid:0) n af j λ Ra ∧ C ◦ R (cid:1) [[1 R ]] Z = 0 because C ◦ R is a 2-chain.52ow, we add the above result to the expression (5.234) of the CS action thus obtaining: k ( J ( ~n f ,~θ,S ) ⋆ J ( ~n f ,~θ,S ) ) [[1]] + ( J ( ~n f ,~θ,S ) ⋆ J ( ~N f , − ~ Θ τ ,C T ) ) [[1]] = k ( S ⋆ S ) [[1]] + 2 k D Q~θ τ , ~n f E − k D P ~θ τ , Q~θ τ E ++2 k g X a =1 (cid:0) n af j λ Ra ∧ S ◦ R (cid:1) [[1 R ]] + 2 k D Q~θ f , ~n f E ++( S ⋆ C T ) [[1]] + D Q~θ τ , ~N f E + D P ~θ τ , Q~ Θ τ E ++ g X a =1 ( N af j λ Ra ∧ S ◦ R ) [[1 R ]] + D Q~θ f , ~N f E . (5.265)The terms containing Q~θ f can be gathered into: D Q~θ f , k~n f + ~N f E . (5.266)Then, as in the computation of the partition function, we first perform the integrationover the circular variables ~θ f which yields, thanks to Property 1 and its corollary, theconstraint 2 k~n f + ~N f = ~
0. Hence, instead of (5.265) we must consider: k ( J ( ~n f ,~θ,S ) ⋆ J ( ~n f ,~θ,S ) ) [[1]] + ( J ( ~n f ,~θ, s ) ⋆ J ( ~N f , − ~Mp ,C T ) ) [[1]] | k~n f + ~N f = ~ = k ( S ⋆ S ) [[1]] + ( S ⋆ C T ) [[1]] − k D P ~θ τ , Q~θ τ E + D P ~θ τ , Q~ Θ τ E , (5.267)and limit the sum over the free variables ~n f to one contribution. Hence, we obtain thefollowing expression of the involved expectation value: hh W CS ( L , J ) ii = δ k~n f + ~N f N CS Z D S X ~θ τ e iπ ( k S ⋆ S + S ⋆ C T )[[1]] e − iπ ( k h P ~θ τ ,Q~θ τ i − h P ~θ τ ,Q~ Θ τ i ) . (5.268)The integrals once more decouple thus yielding: hh W CS ( L , J ) ii = δ k~n f + ~N f N CS (cid:18)Z D S e iπ ( k S ⋆ S + S ⋆ C T )[[1]] (cid:19) ×× X ~θ τ e − iπ ( k h P ~θ τ ,Q~θ τ i − h P ~θ τ ,Q~ Θ τ i ) . (5.269)The last step is to deal with the argument of the first exponential accoridng to:( k S ⋆ S + S ⋆ C T ) [[1]] = k (cid:18)(cid:18) S + C T k (cid:19) ⋆ (cid:18) S + C T k (cid:19)(cid:19) [[1]] − k (cid:18) C T k ⋆ C T k (cid:19) [[1]] . (5.270)53ince by construction C T is a DB 1-current of type S the division by 2 k is allowed andwe can make the following change of variables: U = S + C T k . (5.271)By referring to the discussion made right after Property 13 this redefinition of the ”infinitedimensional” variables involve a redefinition of the free angular variables ~θ f . However, wealready integrated over these variables so a change in the angular variable is irrelevant atthis stage. If we assume that the formal measure D ∫ is invariant under the above shift,then we obtain: hh W CS ( L , J ) ii = δ [2 k ] ~N f ,~ N CS (cid:18)Z D U e iπk ( U ⋆ U )[[1]] (cid:19) ×× e − iπk ( C T k ⋆ C T k )[[1]] X ~θ τ e − iπ ( k h P ~θ τ ,Q~θ τ i − h P ~θ τ ,Q~ Θ τ i ) . (5.272)Taking into account the expression of normalization factor N CS we get: hh W CS ( L , J ) ii = δ [2 k ] ~N f ,~ e − iπk ( C T k ⋆ C T k )[[1]] X ~θ τ e − iπ ( k h P ~θ τ ,Q~θ τ i − h P ~θ τ ,Q~ Θ τ i ) . (5.273)From what we saw in the previous section, we have: k ( C T k ⋆ C T k ) [[1]] Z = 14 kp lk ( Z p , ˘ Z p ) , (5.274)with Z p being the p -fold of Z and ˘ Z p a framing of this p -fold, the cycle Z being thegeneralized curvature of C T = C ~ Θ τ + C , where ~ Θ τ = ~M /p . If we inject this last relationin (5.273), the resulting expression is: hh W CS ( L , J ) ii = δ [2 k ] ~N f ,~ e − iπ kp lk ( Z p , ˘ Z p ) X ~θ τ e − iπ ( k h P ~θ τ ,Q~θ τ i − h P ~θ τ ,Q~ Θ τ i ) , (5.275)in full agreement with formula (90) of [3]. Up to the factor − iπ , we can write theargument of the first exponential in the above expression as:14 k ( C ⋆ C ) [[1]] + 12 kp g X a =1 ( P ~M ) a (cid:0) j λ Ra ∧ j c ◦ R (cid:1) [[1 R ]] + 14 kp D P ~M , Q ~M E . (5.276)54ue to the presence of the factor 1 /k , which is the inverse of the coupling constant k ,this contributions is referred to as the perturbative component of the reduced expectationvalue [3].In the case of BF observables, we consider: W BF ( L , L , J , J ′ ) = e iπ H L J · e iπ H L J ′ , (5.277)for two links L and L . We then consider the DB 1-currents: ( J L = J ~N f − J Θ τ + C Θ τ + C = J ( ~N f , − Θ τ ,C T ) J L = J ~N f − J Θ τ + C Θ τ + C = J ( ~N f , − Θ τ ,C T ) , (5.278)associated with L and L , respectively. By following the same procedure as in theChern-Simons case, and with obvious notations, we end with the reduced expectationvalue: hh W BF ( L , L , J , J ′ ) ii = δ [ k ] ~N f ,~ δ [ k ] ~N f ,~ e − iπkp p lk ( Z p , Z p ) X ~θ τ ,~θ ′ τ e − iπ ( k h P ~θ τ ,Q~θ ′ τ i − h P ~θ τ ,Q~ Θ τ i − h P ~θ ′ τ ,Q~ Θ τ i ) . (5.279)This expression coincides with formula (2.53) of [5]. Of course, if the two links are equalthen the linking number in the first exponential is actually a self-linking and thus isdefined with the help of a framing as in the Chern-Simons case. We leave it to the readerto find the equivalent of the perturbative component (5.276) in the BF case. In this second article dedicated to the use of a Heegaard splitting in the determination ofthe partition functions and expectation values in U (1) CS and BF theories we saw howthe use of DB 1-currents yield to results which were obtained by different approaches.However, although the results are already known, the use of a Heegaard splitting allowsto explicitly see the role played by the Riemann surface on which the spitting is based.All the degree of freedom which enter in the final results actually come from Property 2and are referring to the gluing diffeomorphism f : Σ L → Σ R which defines the Heegaardsplitting X L ∪ f X R thought the endomorphism P : Z g → Z g .Another virtue of the Heegaard splitting approach is that it shows how the exactsequence based on (generalized) curvatures can be used in the context of CS and BFtheories. Usually [2, 4], it is the exact sequence based on the second cohomology of M z canonically defines a singularDB class η z thanks to which integration of a smooth DB class A along z takes the form( J A ⋆ η z ) [[1]].The so-called standard approach can be extended to U (1) CS and BF theories ona smooth closed (4 n + 3)-dimensional manifold which also yield invariants [4]. It istherefore natural to wonder whether there exists some equivalent of the Heegaard splittingapproach for such higher dimensional manifolds. Unfortunately, a smooth closed (4 n +3)-dimensional manifold ( n >
0) cannot always be decomposed according to V ∪ f W with V and W diffeomorphic manifolds and f : ∂V → ∂W a diffeomorphism. However, as the U (1) CS and BF invariants discussed in this article all rely on T n +1 ( M ) and the linkingform Γ M we could consider the following trick: first, we look for V , W and f as aboveand in such a way that T n +1 ( M ′ ) = T n +1 ( M ) and Γ ′ M = Γ M , where M ′ = V ∪ f W ;then, we apply the Heegaard splitting approach to M ′ . Another interesting point wouldbe to see if it is possible to extend the Heegaard construction to a decomposition V ∪ f W with the only constraint that f : ∂V → ∂W is a diffeomorphism. More generally, asany smooth manifold admits a handle decomposition, we investigate how to adapt theHeegaard splitting approach to such a decomposition. A fourth and last point of interestconcern the Kirk and Klassen construction in which a BF type action interpolates betweennon-abelian flat CS actions [16]. We can wonder what remains of this in the U (1) contextwhere the space of equivalence classes of U (1)-connections is not connected.The use of a Heegaard splitting can also be seen as a first step in order to understandhow CS and BF theories could be considered as QFT on a closed 3-manifold, whereasQFT in its standard approach, which is perturbative by essence, only deal with R .These topological models are of particular interest in such a search since we know theresults concerning their partition functions and the expectation values of some of theirobservables. Of course, the identification of the relevant degrees of freedom entering thefinal expression of the partition functions and expectation values drives us to the questionsof these degrees of freedom in the non-abelian case. In that case too all should come fromthe gluing diffeomorphism f which defines the Heegaard splitting. Unfortunately, thesedegrees of freedom still seem mysterious. Nevertheless, the abelian degrees of freedomshould be somewhere among them as they are for instance in the case of lens spaces [17].Let us conclude this article with a questioning. It is pretty amazing that U (1) CSand BF theories are almost absent from the usual lectures on QFT in R (or even R n +3 .)Indeed, the propagators of these theories have nice geometrical interpretations for a largeclass of gauge fixing procedure [18], while partition functions and expectation values ofholonomies provide manifold invariants. As to the computation of these quantities theyalso rely on geometry [19, 20]. It is not usual at all for propagators, partition functionsand expectation values to have such nice interpretations in QFT.56 cknowledgments: The author would like to thank Eric Pilon for fruitful discus-sions concerning currents and distributions.57 eferences [1] F. Thuillier,
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