]>NUPHB14199S0550-3213(17)30351-610.1016/j.nuclphysb.2017.10.021The Author(s)Quantum Field Theory and Statistical SystemsFig. 1Attaching homemorphism f:∂HL→∂HR.Fig. 1Fig. 2Generators {γ1,γ2} and meridinal discs {D1,D2} in a handlebody of genus 2.Fig. 2Fig. 3Example of a genus 2 Heegaard diagram.Fig. 3Fig. 4Disc Dj and the neighbourhood Nj of Dj.Fig. 4Fig. 5Heegaard diagram for the lens space L(5,2), with base point xb displayed.Fig. 5Fig. 6Values of the connections inside one intersection region.Fig. 6Fig. 7Heegaard diagram for Σ3, with base point xb displayed.Fig. 7Fig. 8Values of the map U0 in the region where f⁎AL0 and AR0 are vanishing.Fig. 8Fig. 9Images of the regions {F2,F4,G2,G4} parametrised in equation (6.7).Fig. 9Fig. 10Closed surface specified by ΦR−1Φf⁎L:∂HR→SU(2).Fig. 10Fig. 11U˜0 images in B1 of the boundaries of the regions {F2,F4,G2,G4}.Fig. 11Fig. 12Heegaard diagram for the Poincaré sphere.Fig. 12Fig. 13Values of U0 in the region where f⁎AL0 and AR0 vanish.Fig. 13Flat connections in three-manifolds and classical Chern–Simons invariantEnoreGuadagniniab⁎enore.guadagnini@unipi.itPhilippeMathieucFrankThuilliercaDipartimento di Fisica “E. Fermi” dell'Università di Pisa, ItalyDipartimento di Fisica “E. Fermi” dell'Università di PisaItalybINFN Sezione di Pisa, Largo B. Pontecorvo 2, 56127 Pisa, ItalyINFN Sezione di PisaLargo B. Pontecorvo 2Pisa56127ItalycLAPTh, Université de Savoie, CNRS, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, FranceLAPThUniversité de SavoieCNRSChemin de BellevueBP 110Annecy-le-Vieux CedexF-74941France⁎Corresponding author.Editor: Hubert SaleurAbstractA general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern–Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of π1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern–Simons invariants are illustrated.1IntroductionEach SU(N)-connection, with N≥2, in a closed and oriented 3-manifold M can be represented by a 1-form A=Aμdxμ which takes values in the Lie algebra of SU(N). The Chern–Simons function S[A],(1.1)S[A]=∫MLCS(A)=18π2∫MTr(A∧dA+i23A∧A∧A)=18π2∫Md3xϵμνλTr(Aμ(x)∂νAλ(x)+i23Aμ(x)Aν(x)Aλ(x)), can be understood as the Morse function of an infinite dimensional Morse theory, on which the instanton Floer homology [1] and the gauge theory interpretation [2] of the Casson invariant [3] are based. Under a local gauge transformation(1.2)Aμ(x)⟶AμΩ(x)=Ω−1(x)Aμ(x)Ω(x)−iΩ−1(x)∂μΩ(x), where Ω is a map from M into SU(N), the function S[A] transforms as(1.3)S[AΩ]=S[A]+IΩ, where the integer IΩ∈Z,(1.4)IΩ=124π2∫MTr(Ω−1dΩ∧Ω−1dΩ∧Ω−1dΩ), can be used to label the homotopy class of Ω. The stationary points of the function (1.1) correspond to flat connections, i.e. connections with vanishing curvature F(A)=2dA+i[A,A]=0. We shall now concentrate on flat connections exclusively. Let A be a flat connection in M, and let γ⊂M be an oriented path connecting the starting point x0 to the final point x1. The associated holonomy γ→hγ[A]∈SU(N) is given by the path-ordered integral(1.5)hγ[A]=Pei∫γA, which is computed along γ. Under a gauge transformation A→AΩ, one finds(1.6)hγ[AΩ]=Ω−1(x0)hγ[A]Ω(x1). Let us consider the set of holonomies which are associated with the closed oriented paths such that x0=x1=xb, for a given base point xb. Since the element hγ[A]∈SU(N) is invariant under homotopy transformations acting on γ, this set of holonomies specifies a matrix representation of the fundamental group π1(M) in the group SU(N). Because of equation (1.3), the classical Chern–Simons invariant cs[A],(1.7)cs[A]=S[A]mod Z, is well defined for the gauge orbits of flat SU(N)-connections on M, and it is well defined [4] for the SU(N) representations of π1(M) modulo the action of group conjugation. If the orientation of M is modified, one gets cs[A]→−cs[A].In the case of the structure group SU(2), methods for the computation of cs[A] have been presented in References [5–10], where a few non-unitary gauge groups have also been considered. In all the examples that have been examined, cs[A] turns out to be a rational number. In the case of three dimensional hyperbolic geometry, the associated PSL(2,C) classical invariant [7,11–13] combines the real volume and imaginary Chern–Simons parts in a complex geometric invariant. The Baseilhac–Benedetti invariant [14] with group PSL(2,C) represents some kind of corresponding quantum invariant.Precisely because flat connections represent stationary points of the function (1.1), flat connections and the corresponding value of cs[A] play an important role in the quantum Chern–Simons gauge field theory [15]. For instance, the path-integral solution of the abelian Chern–Simons theory has recently been produced [16,17]. In this case, flat connections dominate the functional integration and the value of the partition function is given by the sum over the gauge orbits of flat connections of the exponential of the classical Chern–Simons invariant. The classical abelian Chern–Simons invariant is strictly related [16,17] with the intersection quadratic form on the torsion group of M, which also enters the abelian Reshetikhin–Turaev [18,19] surgery invariant.In general, the precise expression of the flat connections is an essential ingredient for the computation of the observables of the quantum Chern–Simons theory by means of the path-integral method. In this article we shall mainly be interested in nonabelian flat connections. We will show that, given a representation ρ of π1(M) and a Heegaard splitting presentation [20] of M (with the related Heegaard diagram), by means of a general construction one can define a corresponding smooth flat connection A on M. The method that we describe is related with the deduction [21] of a presentation of the fundamental group of a manifold M by means of a Heegaard splitting of M. Then the associated invariant cs[A] assumes a canonical form, which can be written as the sum of two contributions. The first term is determined by the intersections of the curves in the Heegaard diagram and can be interpreted as a sort of “coloured intersection form”. Whereas the second term is the Wess–Zumino volume of a region in the structure group SU(N) which is determined by the representation of π1(M) and by the Heegaard gluing homeomorphism.The procedure that we present for the determination of the flat connections can find possible applications also in the description of the topological states of matter [22,23]. A discussion on the importance of topological configurations and of the holonomy operators in gauge theories can be found for instance in Ref. [24].Our article is organised as follows. Section 2 contains a brief description of the main results of the present article. The general construction of flat connections in a generic 3-manifold M by means of a Heegaard splitting presentation of M is discussed in Section 3. The canonical form of the corresponding classical Chern–Simons invariant is derived in Section 4, where a two dimensional formula of the Wess–Zumino group volume is also produced. In the remaining sections, our method is illustrated by a few examples. Flat connections in lens spaces are discussed in Section 5 and a non-abelian representation of the fundamental group of a particular 3-manifold is considered in Section 6; computations of the corresponding classical Chern–Simons invariants are presented. The case of the Poincaré sphere is discussed in Section 7. One example of a general formula of the classic Chern–Simons invariant for a particular class of Seifert manifolds is given in Section 8. Finally, Section 9 contains the conclusions.2OutlookThe main steps of our construction can be summarised as follows. For any given SU(N) representation ρ of π1(M),(2.1)ρ:π1(M)→SU(N), one can find a corresponding flat connection A on M whose structure is determined by a Heegaard splitting presentation M=HL∪fHR of M. In this presentation, the manifold M is interpreted as the union of two handlebodies HL and HR which are glued by means of the homeomorphism f:∂HL→∂HR of their boundaries, as sketched in Fig. 1.Let the fundamental group of M be defined with respect to a base point xb which belongs to the boundaries of the two handlebodies. Then the representation ρ of π1(M) canonically defines a representation of the fundamental group of each of the two handlebodies HL and HR. As shown in Fig. 2, in each handlebody the generators of its fundamental group can be related with a set of corresponding disjoint meridinal discs. To each meridinal disc is associated a matrix which is specified by the representation ρ; this matrix can be interpreted as a “colour” which is attached to each meridinal disc. With the help of these coloured meridinal discs, one can construct a smooth flat connection AL0 in HL—and similarly a smooth flat connection AR0 in HR—whose holonomies correspond to the elements of the representation ρ in the handlebody HL (or HR). The precise definition of AL0 and AR0 is given in Section 3.In general, AL0 and AR0 do not coincide with the restrictions in HL and HR of a single connection A in M, because the images—under f—of the boundaries of the meridinal discs of HL are not the boundaries of meridinal discs of HR. So, in order to define a connection A which is globally defined in M, one needs to combine AL0 with AR0 in a suitable way. In facts, the exact matching of the gauge fields AL0 and AR0 in M is specified by the homeomorphism f through the Heegaard diagram, which shows precisely how the boundaries of the meridinal discs of HL are pasted onto the surface ∂HR, in which the boundaries of the meridinal discs of HR are also placed. Let us denote by f⁎AL0 the image of AL0 under f. The crucial point now is that, on the surface ∂HR, the connections AR0 and f⁎AL0 are gauge related(2.2)f⁎AL0=U0−1AR0U0−iU0−1dU0, on ∂HR, because their holonomies define the same representation of π1(∂HR). The value of the map U0 from the surface ∂HR on the group SU(N) is uniquely determined by equation (2.2) and by the condition U0(xb)=1. In facts, we will demonstrate that(2.3)U0(x)=ΦR−1(x)Φf⁎L(x), for x∈∂HR, where ΦR and Φf⁎L denote the developing maps associated respectively with AR0 and f⁎AL0 from the universal covering of ∂HR into the group SU(N). The definition of the developing map will be briefly recalled in Section 3.3. Then the map U0 can smoothly be extended to the whole handlebody HR; this extension will be denoted by U. The values of U:HR→SU(N) inside HR are not constrained and can be chosen without restrictions apart from smoothness. As far as the computation of the classical Chern–Simons invariant is concerned, the particular choice of the extension U of U0 turns out to be irrelevant. To sum up, the connection A—which is well defined in M and whose holonomies determine the representation ρ—takes the form(2.4)A={AL0 in HL; U−1AR0U−iU−1dU in HR; the correct matching of these two components is ensured by equation (2.2). The expression (2.4) of the connection impliesProposition 1The classical Chern–Simons invariant (1.7), evaluated for the SU(N) flat connection (2.4), takes the form(2.5)cs[A]=X[A]+Γ[U]modZ, where(2.6)X[A]=18π2∫∂HRTr[U0−1AR0U0∧f⁎AL0], and(2.7)Γ[U]=124π2∫HRTr[U−1dU∧U−1dU∧U−1dU].The function X[A] is defined on the surface ∂HR, and similarly the value of the Wess–Zumino volume Γ[U] mod Z only depends [25–27] on the values of U in ∂HR (i.e., it only depends on U0). A canonical dependence of Γ on U0 will be produced in Section 4.4. Therefore both terms in expression (2.5) are determined by the data on the two-dimensional surface ∂HR of the Heegaard splitting presentation M=HL∪fHR exclusively. This is why the particular choice of the extension of U0 inside HR is irrelevant. The remaining part of this article contains the proof of Proposition 1 and a detailed description of the construction of the flat connection A. Examples will also be given, which elucidate the general procedure and illustrate the computation of cs[A].3Flat connectionsGiven a matrix representation ρ of π1(M), we would like to determine a corresponding flat connection A on M whose holonomies agree with ρ; then we shall compute S[A].In order to present a canonical construction which is not necessarily related with the properties of the representation space, we shall use a Heegaard splitting presentation M=HL∪fHR of M. The construction of A is made of two steps. First, in each of the two handlebodies HL and HR we define a flat connection, AL0 and AR0 respectively, whose holonomies coincide with the elements of the matrix representation of the fundamental group of the handlebody which is induced by ρ. Second, the components AL0 and AR0 are combined according to the Heegaard diagram to define A on M.3.1Heegaard splittingLet us recall [4,20] that the fundamental group of a three-dimensional oriented handlebody H of genus g is a free group with g generators {γ1,γ2,...,γg}. A disc D in H is called a meridinal disc if the boundary of D belongs to the boundary of H, ∂D⊂∂H, and ∂D is homotopically trivial in H. Let {D1,D2,...,Dg} be a set of disjoint meridinal discs in H such that H−{D1,D2,...,Dg} is homeomorphic with a 3-ball with 2g removed disjoint discs in its boundary. These meridinal discs {D1,D2,...,Dg} can be put in a one-to-one correspondence with the g handles of the handlebody H or, equivalently, with the generators of π1(H), and can be oriented in such a way that the intersection of γj with Dk is δjk. For instance, in the case of a handlebody of genus 2, a possible choice of the generators {γ1,γ2} and of the discs {D1,D2} is illustrated in Fig. 2, where the base point xb is also shown.By means of a Heegaard presentation M=HL∪fHR of the 3-manifold M, which is specified by the homeomorphism(3.1)f:∂HL→∂HR, one can find a presentation of the fundamental group π1(M). Suppose that the two handlebodies HL and HR have genus g. Let {D1,D2,...,Dg} be a set of disjoint meridinal discs in HL which are associated with the g handles of HL. The homeomorphism f:∂HL→∂HR is specified—up to ambient isotopy—by the images Cj′=f(Cj) in ∂HR of the boundaries Cj=∂Dj, for j=1,2,..,g. Thus each Heegaard splitting can be described by a diagram which shows the set of the characteristic curves {Cj′} on the surface ∂HR. One example of Heegaard diagram is shown in Fig. 3.Let {γ1,γ2,...,γg} be a complete set of generators for π1(HR) which are associated to a complete set of meridinal discs of HR. The fundamental group of M is specified by adding to the generators {γ1,γ2,...,γg} the constraints which implement the homotopy triviality condition of the curves {Cj′}. Indeed, since each curve Cj is homotopically trivial in M, the fundamental group of M admits [20,21] the presentation(3.2)π1(M)=〈γ1,γ2,...,γg|[C1′]=1,...,[Cg′]=1〉, where [Cj′] denotes the π1(HR) homotopy class of Cj′ expressed in terms of the generators {γ1,γ2,...,γg}. The classes [Cj′] are determined by the intersections of the boundaries of the meridinal discs of HL and HR, which can be inferred from the Heegaard diagram.3.2Flat connection in a handlebodyLet us consider the handlebody HL of the Heegard splitting M=HL∪fHR of genus g and a corresponding set {D1,D2,...,Dg} of disjoint meridinal discs in HL. For each j=1,2,...,g, consider a collared neighbourhood Nj of Dj in HL. As shown in Fig. 4, Nj is homeomorphic with a cylinder Dj×[0,ϵ] parametrised as (z∈C,|z|≤1)×(0≤t≤ϵ).The strip (|z|=1)×(0≤t≤ϵ) belongs to the surface ∂HL. The flat SU(N)-connection on HL we are interested in will be denoted by AL0; AL0 is vanishing in HL−{N1,N2,...,Ng} and, inside each region Nj, AL0 is determined by ρ(γj). More precisely, suppose that(3.3)ρ(γj)=eibj, where the hermitian traceless matrix bj belongs to the Lie algebra of SU(N). Let θ(t) be a C∞ real function, with θ′(t)=dθ(t)/dt>0, satisfying θ(0)=0 and θ(ϵ)=1. Then the value of AL0 in the region Nj is given by(3.4)AL0|Nj=bjθ′(t)dt. The orientation of the parametrisation (or the sign in equation (3.4)) is fixed so that the holonomy of the connection (3.4) coincides with expression (3.3). As a consequence of equation (3.4) one has dAL0=0 and also, since Nj∩Nk=∅ for j≠k, one finds AL0∧AL0=0.By construction, the smooth 1-form AL0 represents a flat connection on HL whose holonomies coincide with the matrices that represent the elements of the fundamental group of HL. The restriction of AL0 on the boundary ∂HL has support on g ribbons and its values are determined by equation (3.4); the j-th ribbon represents a collared neighbourhood of the curve Cj=∂Dj in ∂HL. The same construction can be applied to define a flat connection AR0 on HR.3.3Flat connection in a 3-manifoldLet us now construct a flat connection A in M=HL∪fHR which is associated with the representation ρ of π1(M). As far as the value of A on HL is concerned, one can put(3.5)A|HL=AL0. The image f⁎AL0 of AL0 under the homeomorphism f:∂HL→∂HR does not coincide in general with AR0 in ∂HR. But since f⁎AL0 and AR0 are associated with the same matrix representation of π1(∂HR), the values of f⁎AL0 and AR0 on ∂HR are related by a gauge transformation, f⁎AL0=U0−1AR0U0−iU0−1dU0, as shown in equation (2.2), in which U0 must assume the unit value at the base point xb. Then the map U0 can smoothly be extended in HR, let U denote this extension. The value of A on HR is taken to be(3.6)A|HR=U−1AR0U−iU−1dU. The value of U0 on the surface ∂HR represents a fundamental ingredient of our construction, so we now describe how it can be determined. To this end, we need to introduce the concept of developing map.Let us recall that any flat SU(N)-connection A defined in a space X can be locally trivialized because, inside a simply connected neighbourhood of any given point of X, A can be written as A=−iΦ−1dΦ. The value of Φ coincides with the holonomy of A. When the representation of π1(X) determined by A is not trivial, Φ cannot be extended to the whole space X. A global trivialisation of A can be found in the universal covering Xˆ of X; in this case, the map Φ:Xˆ→SU(N) represents the developing map. For any element γ of π1(X) acting on Xˆ by covering transformations, the developing map satisfies(3.7)Φ(γ⋅x)=hγ[A]⋅Φ(x), in agreement with equations (1.6). Now, on the surface ∂HR we have the two flat connections f⁎AL0 and AR0 which are related by a gauge transformation, equation (2.2). Thus, for each oriented path γ⊂∂HR connecting the starting point x0 with the final point x, the corresponding holonomies are related according to equation (1.6) which takes the form(3.8)U0−1(x0)hγ[AR0]U0(x)=hγ[f⁎AL0]. From this equation one obtains U0(x)=hγ−1[A0R]U0(x0)hγ[f⁎AL0]. When the starting point x0 coincides with the base point xb of the fundamental group, one has U(xb)=1, and then(3.9)U0(x)=hγ−1[AR0]hγ[f⁎AL0], for x∈∂HR. This equation is equivalent to the relation (2.3). Indeed, because of the transformation property (3.7), the combination ΦR−1Φf⁎L is invariant under covering translations acting on the universal covering of ∂HR (and then ΦR−1Φf⁎L is really a map from ∂HR into SU(N)), and locally coincides with the product hγ−1[AR0]hγ[f⁎AL0] appearing in equation (3.9).4The invariant4.1Proof of Proposition 1The Chern–Simons function S[A] of the connection (2.4)—whose components in HL and HR are shown in equations (3.5) and (3.6)—is given by(4.1)S[A]=∫MLCS(A)=∫HLLCS(A)+∫HRLCS(A). Since dAL0=0 and AL0∧AL0=0, one has(4.2)∫HLLCS(A)=∫HLLCS(A0L)=0. Moreover, a direct computation shows that(4.3)∫HRLCS(A)=∫HRLCS(A0R)−i8π2∫HRdTr[A0R∧dUU−1]+124π2∫HRTr[U−1dU∧U−1dU∧U−1dU]. As before, the first term on the r.h.s of equation (4.3) is vanishing(4.4)∫HRLCS(AR0)=0. By using equation (2.2), the second term can be written as the surface integral(4.5)X[A]=18π2∫∂HRTr[U0−1AR0U0∧f⁎AL0]. By combining equations (4.1)–(4.5) one finally gets(4.6)S[A]=18π2∫∂HRTr[U0−1AR0U0∧f⁎AL0]+124π2∫HRTr[U−1dU∧U−1dU∧U−1dU], which implies equation (2.5). This concludes the proof of Proposition 1.The term X[A] can be understood as a sort of coloured intersection form, because its value is determined by the trace of the representation matrices—belonging to the Lie algebra of the group—which are associated with the boundaries of the meridinal discs of the two handlebodies which intersect each other in the Heegaard diagram. Indeed, on the surface ∂HR, AR0 is different from zero inside collar neighbourhoods of the boundaries of the meridinal discs of HR, whereas f⁎AL0 is different from zero inside collar neighbourhoods of the images—under f—of the boundaries of the meridinal discs of HL. Thus, in the computation of X[A], only the intersection regions of the curves of the Heegaard diagram give nonvanishing contributions. But since the intersections of the boundaries of the meridinal discs of HL and HR determine the relations entering the presentation (3.2) of π1(M), an important part of the input, which is involved in the computation of X[A], is given by the fundamental group presentation (3.2). It turns out that the computation of X[A] can also be accomplished by means of intersection theory techniques by colouring the de Rham–Federer currents [28,29] of the disks {Dj}.When the representation ρ is abelian, Γ[U] vanishes and the classical Chern–Simons invariant is completely specified by X[A] which assumes the simplified form(4.7)cs[A]|abelian=X[A]|abelian=18π2∫∂HRTr[AR0∧f⁎AL0]modZ.4.2Group volumeThe term Γ[U] can be interpreted as the 3-volume of the region of the structure group which is bounded by the image of the surface ∂HR under the map ΦR−1Φf⁎L:∂HR→SU(N). In this case also, the combination ΦR−1Φf⁎L of the two developing maps, which are associated with f⁎AL0 and AR0, is characterized by the homeomorphism f:∂HL→∂HR which topologically identifies M.In general, the direct computation of Γ[U] is not trivial, and the following properties of Γ[U] turn out to be useful. When U(x) can be written as(4.8)U(x)=W(x)Z(x), where W(x)∈SU(N) and Z(x)∈SU(N), one obtains(4.9)Γ[U=WZ]=Γ[W]+Γ[Z]+18π2∫∂HRTr[dZZ−1∧W−1dW]. By means of equation (4.9) one can easily derive the relation(4.10)Γ[U=VHV−1]=Γ[H]−18π2∫∂HRTr[V−1dV∧(H−1dH+dHH−1)]+18π2∫∂HRTr[V−1dVH∧V−1dVH−1]. With a clever choice of the matrices V(x) and H(x), equation (4.10) assumes a simplified form. Indeed any generic map U(x)∈SU(N) can locally be written in the form U(x)=V(x)H(x)V−1(x) where(4.11)H(x)=exp(iC(x)), and C(x) belongs to the (N−1)-dimensional abelian Cartan subalgebra of the Lie algebra of SU(N). In this case, one has Γ[H]=0 and(4.12)H−1(x)dH(x)=dH(x)H−1(x)=idC(x). Therefore relation (4.10) becomes(4.13)Γ[VHV−1]=18π2∫∂HR{2iTr[dC∧V−1dV]+Tr[e−iCV−1dVeiC∧V−1dV]}, where it is understood that one possibly needs to decompose the integral into a sum of integrals computed in different regions of ∂HR where V(x) and H(x) are well defined [30]. Expression (4.13) explicitly shows that the value of Γ[U] (modulo integers) is completely specified by the value of U on the surface ∂HR.In the case of the structure group SU(2)∼S3, the computation of Γ[U] can be reduced to the computation of the volume of a given polyhedron in a space of constant curvature. Discussions on this last problem can be found, for instance, in the articles [31–38].4.3Canonical extensionThe reduction of the Wess–Zumino volume Γ[U] into a surface integral on ∂HR can be done in several inequivalent ways, which also depend on the choice of the extension of U0 from the surface ∂HR in HR. Let us now describe the result which is obtained by means of a canonical extension of U0. We shall concentrate on the structure group SU(2), the generalisation to a generic group SU(N) is quite simple.Suppose that the value of U0 on the surface ∂HR can be written as(4.14)U0(x,y)=ein(x,y)σ=ei∑a=13na(x,y)σa=cosn(x,y)+inˆ(x,y)σsinn(x,y), where (x,y) designate coordinates of ∂HR, n=[∑b=13nbnb]1/2, the components of the unit vector nˆ are given by nˆa=na/n, and {σa} (with a=1,2,3) denote the Pauli sigma matrices. The canonical extension of U0 is defined by(4.15)U(τ,x,y)=eiτn(x,y)σ, where the homotopy parameter τ takes values in the range 0≤τ≤1. When τ=1 one recovers the expression (4.14), whereas in the τ→0 limit one finds U=1. A direct computation gives(4.16)Tr(U−1∂τU[U−1∂xU,U−1∂yU])=2in2sin2(τn)Tr(Σ[∂yΣ,∂xΣ]), in which Σ(x,y)=∑a=13na(x,y)σa. Therefore, by using the identity(4.17)∫01dτsin2(τn)=12[1−sin(2n)2n], one gets(4.18)Γ[U]=−i8π2∫∂HR1n2[1−sin(2n)2n]Tr(ΣdΣ∧dΣ). This equation will be used in Section 6, Section 7 and Section 8.4.4RationalityAs it has already been mentioned, in all the considered examples the value of the SU(N) classical Chern–Simons invariant is given by a rational number. Let us now present a proof of this property for a particular class of 3-manifolds. Suppose that the universal covering M˜ of the three-manifold M is homeomorphic with S3, so that M can be identified with the orbit space [39] which is obtained by means of covering translations (acting on S3) which correspond to the elements of the fundamental group π1(M). Given a flat connection A on M, let us denote by A˜ the flat connection on M˜∼S3 which is the upstairs preimage of A. By construction, one has(4.19)S[A]|M=1|π1(M)|S[A˜]|S3, where |π1(M)| denotes the order of π1(M). On the other hand, since S3 is simply connected, one can find a map Ω:S3→SU(N) such that(4.20)A˜=−iΩ−1dΩ, and then(4.21)S[A˜]|S3=124π2∫S3Tr(Ω−1dΩ∧Ω−1dΩ∧Ω−1dΩ)=n, where n is an integer. Equations (4.19) and (4.21) imply(4.22)cs[A]|M=n|π1(M)|modZ, which shows that, for this type of manifolds, the value of cs[A] is indeed a rational number.Let us now present a few examples of computations of cs[A]; in the first instance, the representation of the fundamental group of the 3-manifold is abelian, whereas nonabelian representations are considered in the remaining examples.5First exampleIn order to illustrate how to compute X[A], let us consider the lens spaces L(p,q), where the coprime integers p and q verify p>1 and 1≤q<p. The manifolds L(p,q) admit [4,20] a genus 1 Heegaard splitting presentation, L(p,q)=HL∪fHR where HL and HR are solid tori. The fundamental group of L(p,q) is the abelian group π1(L(p,q))=Zp.5.1The representationLet us concentrate, for example, on L(5,2) whose Heegaard diagram is shown in Fig. 5, where the image C′ of a meridian C of the solid torus HL is displayed on the surface ∂HR. The torus ∂HR is represented by the surface of a 2-sphere with two removed discs +F and −F. The boundaries of +F and −F must be identified (the points with the same label coincide). A possible choice of the base point xb of the fundamental group is also depicted.In the solid torus HL, let the meridian C be the boundary of the meridinal disc DL⊂HL, which is oriented so that the intersection of DL with the generator γL⊂HL of π1(HL) is +1. Suppose that the representation ρ:π1(L(5,2))=Z5→SU(4) is specified by(5.1)ρ(γL)=exp[i2π5Y], where Y is given by(5.2)Y=(100001000010000−3). Let NL⊂HL be a collared neighbourhood of DL parametrised by (z∈C,|z|≤1)×(0≤t≤ϵ). The flat connection AL0 on HL is vanishing in HL−NL, whereas the value of AL0 in NL is given by(5.3)AL0|NL=2π5Yθ′(t)dt. The restriction of AL0 on the boundary ∂HL is nonvanishing inside a strip which is a collared neighbourhood of C. Therefore the image f⁎AL0 of AL0 on ∂HR is different from zero in a collared neighbourhood of C′.Let us now consider HR. The meridinal disc DR⊂HR can be chosen in such a way that the boundary of DR coincides with the boundaries of +F (and −F) of Fig. 5. The image on ∂HR of the corresponding generator γR of π1(HR) is associated to +F, and it can be represented by an arrow intersecting the boundary of the disc +F and oriented in the outward direction. As in the previous case, we introduce a collared neighbourhood NR⊂HR of DR parametrised by (z′∈C,|z′|≤1)×(0≤u≤ϵ). The flat connection AR0 is vanishing in HR−NR and, inside NR, one has(5.4)AR0|NR=Y˜θ′(u)du, where Y˜ represents an element of the Lie algebra of SU(N). The restriction of AR0 on the boundary ∂HR is nonvanishing inside a collared neighbourhood of ∂(+F). The value taken by AR0 must be consistent with the given representation ρ:π1(L(5,2))→SU(4) which is specified by equation (5.1). In order to determine AR0, one can consider a closed path γ⊂∂HR with base point xb. One needs to impose that the holonomy of AR0 along γ must coincide with the holonomy of f⁎AL0 along γ. One then finds Y˜=(4π/5)Y, and consequently(5.5)AR0|NR=4π5Yθ′(u)du.As shown in the Heegaard diagram of Fig. 5, the collar neighbourhood of C′ and the collar neighbourhood of ∂(+F)—where the connections f⁎AL0 and AR0 are nonvanishing—intersect in five (rectangular) regions of ∂HR. Only inside these rectangular regions is the 2-form AR0∧f⁎AL0 different from zero. As far as the computation of the Chern–Simons invariant is concerned, these five regions are equivalent and give the same contribution to X[A]. The values of the connections inside one of the five rectangular intersection regions are shown in Fig. 6.In the intersection region shown in Fig. 6, one then finds(5.6)∫one regionTr[AR0∧f⁎AL0]=−8π225∫0ϵdtθ′(t)∫0ϵduθ′(u)Tr[Y2]=−96π225. Therefore the value of the classical Chern–Simons invariant which, in this abelian case, takes the form(5.7)cs[A]=18π2∫∂HRTr[A0R∧f⁎AL0]modZ, is given by(5.8)cs[A]=5×(−96π2/25)8π2modZ=35modZ.5.2Lens spaces in generalFor a generic lens space L(p,q), the corresponding Heegaard diagram has the same structure of the diagram shown in Fig. 5. The curve C′ on ∂HR and the boundary of the disc (+F) give rise to p intersection regions. Since the group π1(L(p,q)) is abelian, the analogues of equations (5.3) and (5.5) take the form(5.9)AL0|NL=2πpMθ′(t)dt, and(5.10)AR0|NR=2πqpMθ′(u)du, where the matrix M belongs to the Lie algebra of SU(N) and satisfies(5.11)ei2πM=1. Therefore the expression of the classical Chern–Simons invariant (5.7) is given by(5.12)cs[A]=−18π2{(2π)2qp2Tr(M2)×p}=−qp[12Tr(M2)]modZ. Expression (5.12) is in agreement with the results [16,17] obtained in the case of the abelian Chern–Simons theory, where it has been shown that the value of the Chern–Simons action is specified by the quadratic intersection form on the torsion component of the homology group of the manifold.6Second exampleLet us consider the 3-manifold Σ3 which is homeomorphic with the cyclic 3-fold branched covering of S3 which is branched over the trefoil [20]. Σ3 admits a Heegaard splitting presentation of genus 2 and the corresponding Heegaard diagram is shown in Fig. 7. The surface ∂HR is represented by the surface of a 2-sphere with four removed discs: the boundaries of +F and −F (and similarly the boundaries of +G and −G) must be identified. In Fig. 7, the two characteristic curves C1′ and C2′ are represented by the continuous and the dashed curve respectively, and the base point xb is also shown.The two meridinal discs D1R and D2R of HR are chosen so that their boundaries coincide with the boundaries of the discs +F and +G respectively. The corresponding generators γ1 and γ2 of π1(HR) can be represented by two arrows which are based on the boundaries of +F and +G and oriented in the outward direction. By taking into account the constraints coming from the requirement of homotopy triviality of the curves C1′ and C2′, one finds a presentation of the fundamental group of Σ3,(6.1)π(Σ3)=〈γ1γ2|γ12=γ22=(γ1γ2)2〉. The group π(Σ3) is usually called [20] the quaternionic group; it has eight elements which can be denoted by {±1,±i,±j,±k}, in which ij=k, ki=j and jk=i.Let the representation ρ:π1(Σ3)→SU(2) be given by(6.2)γ1→g1=exp[i(π/2)σ1]=i(0110)=iσ1,γ2→g2=exp[i(π/2)σ2]=i(0−ii0)=iσ2. The corresponding flat connection AR0 on HR vanishes in HR−{N1R,N2R}, where N1R and N2R are collared neighbourhoods of the two meridinal discs {D1R,D2R} of HR, and(6.3)AR0={AR0|N1R=π2σ1θ′(u)du; AR0|N2R=π2σ2θ′(v)dv. With the choice of the base point xb shown in Fig. 7, the flat connection AL0 on HL turns out to be(6.4)AL0={AL0|N1L=π2σ1θ′(t)dt; AL0|N2L=π2σ2θ′(s)ds; where N1L and N2L are collared neighbourhoods of the two meridinal discs {D1L,D2L} of HL, and AL0 vanishes on HL−{N1L,N2L}. Note that, on the surface ∂HR, f⁎AL0 is nonvanishing inside the two ribbons which constitute collared neighbourhoods of the curve C1′ and C2′, whereas AR0 is nonvanishing inside the collared neighbourhoods of ∂D1R and ∂D2R. In the region of the surface ∂HR where both f⁎AL0 and AR0 are vanishing, the values taken by the map U0 entering equation (2.2) are shown in Fig. 8.We now need to specify the values of U0=ΦR−1Φf⁎L in the eight intersections regions of ∂HR where both f⁎AL0 and AR0 are not vanishing. The value of U0 is defined in equation (3.9). In each region, we shall introduce the variables X and Y according to a correspondence of the type(6.5)dX=θ′(t)dt,0≤X≤1dY=θ′(u)du,0≤Y≤1. The intersection regions are denoted as {F1,F2,F3,F4,G1,G2,G3,G4} with the convention that, for instance, the region F3 (or G3) is a rectangle in which one of the vertices is the point denoted by the number 3 of the boundary of the disk +F (or +G). The values of U0 in these eight regions are in order; in each of the corresponding pictures, the values of U0 at the vertices of the rectangle are also reported.Image 1Image 2Image 3Image 4Image 5Image 6Image 7Image 8By using the value of U0 in the eight intersections regions {F1,F2,F3,F4,G1,G2,G3,G4}, the contribution X[A], defined in equation (4.5), of the Chern–Simons invariant can easily be determined. One finds(6.6)X[A]=18π2Tr{−π4σ1σ1+π4σ1σ2+π4σ1σ1+π4σ1σ2−π4σ2σ2+π4σ2σ1+π4σ2σ2+π4σ2σ1}=0. Let us now consider the computation of the contribution Γ[A] of equation (2.7). Under the map U0=ΦR−1Φf⁎L:∂HR→SU(2), the images of the rectangles {F1,F3,G1,G3} are degenerate (they have codimension two). Whereas the images of the remaining four rectangles {F2,F4,G2,G4} constitute a closed surface of genus zero in SU(2)∼S3.As sketched in Fig. 9, the set of the images of {F2,F4,G2,G4} can be globally parametrised by new variables −1≤X≤1 and −1≤Y≤1 according to the relations(6.7)[G2]:U0=eiπ2(X+1)σ2eiπ2Yσ1=eiπ2Xσ2e−iπ2Yσ1iσ2=U˜0iσ2,[F4]:U0=e−iπ2Yσ1eiπ2(1+X)σ2=e−iπ2Yσ1eiπ2Xσ2iσ2=U˜0iσ2,[F2]:U0=e−iπ2Yσ1eiπ2(1+X)σ2=e−iπ2Yσ1eiπ2Xσ2iσ2=U˜0iσ2,[G4]:U0=eiπ2(X+1)σ2eiπ2Yσ1=eiπ2Xσ2e−iπ2Yσ1iσ2=U˜0iσ2.The images of {F2,F4,G2,G4} are glued as shown in Fig. 9; the edges which are labelled by the same symbol must be identified. Therefore, the closed surface which is specified by ΦR−1Φf⁎L:∂HR→SU(2) is topologically equivalent to the tetrahedron shown in Fig. 10. Relations (6.7) show that U0(X,Y) can globally be written as U0(X,Y)=U˜0(X,Y)iσ2, therefore if U˜ denotes the extension of U˜0 in HR, one has(6.8)Γ[U]=Γ[U˜]. In order to determine the value of Γ[U˜] one can use symmetry arguments.The manifold SU(2)∼S3 can be represented as the union of two equivalent (with the same volume) balls in R3 of radius π/2 with identified boundaries, SU(2)∼B1∪B2. Indeed each element of SU(2) can be written aseiθσ=cos(|θ|)+iθˆσsin(|θ|), where |θ|=[θθ]1/2 and θˆ=(θ/|θ|). The ball B1 contains the elements with 0≤|θ|≤π/2, and B2 contains the elements with (π/2)≤|θ|≤π.The application U˜0:∂HR→SU(2) maps the boundaries of the rectangles {F2,F4} and {G2,G4} into the eight edges in B1 shown in Fig. 11. Equation (6.7) and the picture of Fig. 11 demonstrate that the surface U˜0:∂HR→SU(2) is symmetric under rotations of π/2 around the σ3 axis and bounds a region R of SU(2) which is contained in half of the ball B1. According to the reasoning of Section 4.4, the volume of this region R must take the value n/8, where n is an integer. This integer n is less than 4 because R is contained inside B1 and satisfies n≤2 because R is contained inside half of B1. Finally, the value n=2 is excluded because a direct inspection shows that R does not cover the upper half-part of B1 completely. Therefore one finally obtains(6.9)Γ[U]=Γ[U˜]=18. In Section 8 it will be shown that equation (6.9) is also in agreement with a direct computation of Γ[U] that we have performed by means of the canonical expression (4.18). Finally, the validity of the result (6.9) has also been verified by means of a numerical evaluation of the integral (4.18). To sum up, in the case of the manifold Σ3 with the specified representation (6.2) of its fundamental group, the value of the classical Chern–Simons invariant is given by(6.10)cs[A]=18modZ.7Poincaré sphereThe Poincaré sphere P admits a genus 2 Heegaard splitting presentation. The corresponding Heegaard diagram [20] is shown in Fig. 12. One of the two characteristic curves, C1′=f(C1), is described by the continuous line, whereas the second curve C2′=f(C2) is represented by the dashed path; xb designates the base point for the fundamental group.Let the generators {γ1,γ2} of π1(HR) be associated with +F and +G respectively and oriented in the outward direction. According to the Heegaard diagram of Fig. 12, the homotopy class of C1′ is given by γ1−4γ2γ1γ2, whereas the class of C2′ is equal to γ2−2γ1γ2γ1. Therefore the fundamental group of P admits the presentation(7.1)π1(P)=〈γ1,γ2|γ15=γ23=(γ1γ2)2〉, which corresponds to the binary icosahedral (or dodecahedral) group of order 120. Since the abelianization of π1(P) is trivial, P is a homology sphere. A nontrivial representation ρ:π1(P)→SU(2) is given [40,41] by(7.2)ρ(γ1)=g1=eib1=exp[iπ5σ],ρ(γ2)=g2=eib2=exp[iπ3σ˜], where(7.3)σ=(100−1),σ˜=r(100−1)+1−r2(0110),r=cos(π/3)cos(π/5)sin(π/3)sin(π/5).Equation (7.2) specifies the values of AR0,(7.4)AR0={b1θ′(t1)dt1 inside a neighbourhood of +F; b2θ′(t2)dt2 inside a neighbourhood of +G; 0 otherwise . The values of AL0 are determined by equation (7.2) and by the choice of the base point. Indeed, let the generators {λ1,λ2} of π1(HL) be associated with C1 and C2 respectively. Then, from the Heegaard diagram and the position for the base point, one finds(7.5)ρ(λ1)=g1=eib1=exp[iπ5σ],ρ(λ2)=g2=eib2=exp[iπ3σ˜]. Consequently, the image of AL0 under the gluing homeomorphism f takes values(7.6)f⁎AL0={b1θ′(u1)du1 inside a neighbourhood of C1′; b2θ′(u2)du2 inside a neighbourhood of C2′; 0 otherwise .One can now determine the map U0=ΦR−1Φf⁎L:∂HR→SU(2). In the region of the surface ∂HR where both f⁎AL0 and AR0 are vanishing, the values of U0 are shown in Fig. 13. By using the method illustrated in the previous examples, one can compute the classical Chern–Simons invariant. The intersection component is given by(7.7)X[A]=18π2{−4Tr(b1b1)−2Tr(b2b2)+4Tr(b1b2)+Tr(b1g2b1g2−1)+Tr(b2g1b2g1−1)}=−215+12[15cos(π/3)sin(π/5)+13cos(π/5)sin(π/3)]2. The image of the map ΦR−1Φf⁎L:∂HR→SU(2) is a genus 0 surface in the group SU(2). We skip the details, which anyway can be obtained from the Heegaard diagram and equations (7.2)–(7.6). Numerical computations of the integral (4.18) give the following value of the Wess–Zumino volume (with 10−10 precision)(7.8)Γ[A]=0.0090687883⋯. Therefore, the value of the classical Chern–Simons invariant associated with the representation (7.2) of π1(P) turns out to be(7.9)cs[A]=−0.0083333333⋯=−1120modZ, where the last identity is a consequence of the fact that |π1(P)|=120. The result (7.9) has also been obtained by means of a complete computation of the integral (4.18); this issue is elaborated in Section 8.8Computations of the Wess–Zumino volumeThe computation of Γ[U] by means of the canonical expression (4.18) presents general features that are consequences of our construction of the flat connection A by means of a Heegaard splitting presentation of M. This allows the derivation of universal formulae of the classical Chern–Simons invariant for quite wide classes of manifolds. We present here one example; details will be produced in a forthcoming article.Let us consider the set of Seifert spaces Σ(m,n,−2) of genus zero with three singular fibres which are characterised by the integer surgery coefficients (m,1), (n,1) and (2,−1). The manifolds Σ(m,n,−2) admit [4,40] a genus two Heegaard splitting M=HL∪fHR and their fundamental group can be presented as(8.1)π1(M)=〈γ1,γ2|γ1m=γ2n=(γ1γ2)2〉, for nontrivial positive integers m and n. The manifold Σ3 discussed in Section 6 and the Poincaré manifold P considered in Section 7 are examples belonging to this class of manifolds. Let us introduce the representation of π1(M) in the group SU(2) given by(8.2)γ1→g1=exp[iθ1σ],γ2→g2=exp[iθ2σ˜], where σ and σ˜ are combinations of the sigma matrices satisfying σ2=1=σ˜2, and(8.3)g1m=g2n=(g1g2)2=−1. In this case, the value of the surface integral (4.5) is given by(8.4)X[A]=−14{m[θ1π]2+n[θ2π]2−2[(θ2π)cosθ1sinθ2+(θ1π)cosθ2sinθ1]2}. As it has been shown in the previous examples, the image of the map ΦR−1Φf⁎L:∂HR→SU(2) is a genus 0 surface in the group SU(2). The corresponding Wess–Zumino volume turns out to be(8.5)Γ[U]=14{12−2[(θ2π)cosθ1sinθ2+(θ1π)cosθ2sinθ1]2}. So that the value of the classical Chern–Simons invariant for the manifolds Σ(m,n,−2) reads(8.6)cs[A]=−14{m[θ1π]2+n[θ2π]2−12}mod Z. When m=n=2, expression (8.6) gives the value of the classical Chern–Simons invariant appearing in equation (6.9); and for m=5,n=3, expression (8.6) coincides with equation (7.9). Equation (8.6) is valid for generic values of m and n; for those particular values of m and n such that Σ(m,n,−2) is a Seifert homology sphere, our equation (8.6) is in agreement with the results of Fintushel and Stern [5] and Kirk and Klassen [6] for Seifert spheres.9ConclusionsGiven a SU(N) representation ρ of the fundamental group of a 3-manifold M, we have shown how to define a corresponding flat connection A on M such that the holonomy of A coincides with ρ. Our construction is based on a Heegaard splitting presentation of M, so that the relationship between A and the topology of M is displayed. The relative classical Chern–Simons invariant cs[A] is naturally decomposed into the sum of two contributions: a sort of coloured intersection form, which is specified by the Heegaard diagram, and a Wess–Zumino volume of a region of SU(N) which is determined by the non-commutative structure of the ρ representation of π1(M). A canonical expression for the Wess–Zumino volume, as function of the boundary data exclusively, has been produced. A few illustrative examples of flat connections and of classical Chern–Simons invariant computations have been presented.AcknowledgementsWe wish to thank R. Benedetti and C. Lescop for discussions.References[1]A.FloerCommun. Math. Phys.1181988215[2]C.TaubesJ. Differ. Geom.311990547[3]C.LescopGlobal Surgery Formula for the Casson–Walker Invariant1996Princeton Univ. Press[4]N.SavelievInvariants for Homology 3-Spheres2010Springer-VerlagBerlin, Heidelberg[5]R.FintushelR.SternProc. Lond. Math. Soc.611990109[6]P.A.KirkE.KlassenMath. 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