Holomorphic volume forms on representation varieties of surfaces with boundary

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten's formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group SL(N,C).


Introduction
Along this paper S = S g,b denotes a compact, oriented, connected surface with nonempty boundary, of genus g and with b ≥ 1 boundary components. We assume that χ(S) = 2 − 2g − b < 0. The fundamental group π 1 (S) is a free group F k of rank k = 1 − χ(S) ≥ 2.
Let G be a connected, semisimple, complex, linear group with compact real form G R , e.g. G = SL N (C) and G R = SU(N). We also assume that G is simply connected; notice that since π 1 (S) is free, their representations lift to the universal covering of the Lie group.
Fix a nondegenerate symmetric bilinear G-invariant form on the Lie algebra B∶ g × g → C , such that the restriction of B to g R , the Lie algebra of G R , is positive definite. This means that B is a negative multiple of the Killing form. Let R(S, G) denote the set of conjugacy classes of representations of π 1 (S) ≅ F k into G. We are only interested in irreducible representations for which the centralizer coincides with the center of G. Following Johnson and Millson [12] we call such representations good (see Definition 10), and we use the notation R * (S, G) to denote the corresponding open subset of R(S, G).
For surfaces with boundary S, we need to consider also R(S, ∂S, G) ρ 0 , the relative set of conjugacy classes of representations (for each peripheral curve we require its image to be in a fixed conjugacy class), see Subsection 2.2. Let R * (S, ∂S, G) ρ 0 denote the corresponding open subset of good representations. The holomorphic volume form Ω S is defined on R * (S, G) but the holomorphic symplectic form ω is defined on R * (S, ∂S, G) ρ 0 . To relate both spaces and both forms, we need to deal with each component of ∂S, which are circles.
We identify the variety of representations of the circle S 1 with G, by mapping each representation to the image of a fixed generator of π 1 (S 1 ). We restrict to regular representations, namely that map the generator of π 1 (S 1 ) to regular elements. Then R reg (S 1 , G) ≅ G reg G. Using that G is simply connected (see Remark 20 when G is not simply connected), one of the consequences of Steinberg's theorem [25] is that where r = rank G, and that there is a natural isomorphism (Corollary 19): In Section 4.3 we show the existence of a form ν∶ ⋀ r H 1 (S 1 , Ad ρ) → C defined by the formula (1) ν(∧v) = ± TOR(S 1 , Ad ρ, o, u, v) ⟨∧v, ∧u⟩.
Proposition 2. When G is simply connected, then ν = ±C d σ 1 ∧ ⋯ ∧ d σ r , for some constant C ∈ C * depending on G and B. In addition, for G = SL N (C) and B(X, Y ) = − tr(XY ) for X, Y ∈ sl N (C), Let ρ 0 ∈ R * (S, G) be ∂-regular, i.e. the image of each peripheral curve is a regular element of G (Definitions 15 and 16). We have an exact sequence (Corollary 19): where ∂S = ∂ 1 ⊔ ⋯ ⊔ ∂ b denote the boundary components of S. For a ∂-regular representation ρ∶ π 1 (S) → G we let ν i denote the form corresponding to the restriction ρ π 1 (∂ i ) ∶ π 1 (∂ i ) → G as in (1) on ∂ i ≅ S 1 . Set d = dim G, r = rank G, and b > 0 be the number of components of ∂S. The following generalizes Theorem 1 to surfaces with boundary, [29] see also [1,Theorem 5.40].
The formula of Theorem 3 is homogeneous in the bilinear form B∶ g × g → C: if B is replaced by λ 2 B for some λ ∈ C * , then ω is replaced by λ 2 ω, ν i by λ r ν i and Ω π 1 (S) by λ 2 n+b r Ω π 1 (S) , as 2 n + b r = −χ(S)d = dim R(S, G).
We focus now on G = SL N (C), which is simply connected and has rank r = N − 1. We fix a bilinear form on the Lie algebra: We compute explicit volume forms for spaces of representations of free groups in SL 2 (C) and SL 3 (C). We start with a pair of pants S 0,3 . The fundamental group π 1 (S 0,3 ) ≅ F 2 is free on two generators γ 1 and γ 2 . By Fricke-Klein theorem, X(F 2 , SL 2 (C)) ≅ C 3 and the coordinates are precisely the traces of the peripheral elements γ 1 , γ 2 , and γ 1 γ 2 , denoted by t 1 , t 2 , and t 12 respectively. In this case the relative character variety is just a point, and the symplectic form is trivial. Thus, by applying Theorem 3 and equality ν = ± √ 2 d tr γ (Proposition 2), we have on R * (F 2 , SL 2 (C)). By [8], for k ≥ 3, the 3k − 3 trace functions t 1 , t 2 , t 12 , t 3 , t 13 , t 23 , . . . , t k , t 1k , t 2k define a local parameterization Next we deal with SL 3 (C). To avoid confusion with SL 2 (C), the trace functions in SL 3 (C) are denoted by τ i 1 ⋯i k ; notice that τ¯i ≠ τ i . Lawton obtains in [14] an explicit description of the variety of characters X(F 2 , SL 3 (C)). It follows from his result that defines a local parameterization. Using the computation of the symplectic form in [15], we prove in Proposition 39 that, on R * (F 2 , SL 3 (C)) ∖ {τ 2121 = τ 1212 } the volume form is Ω This is generalized to a free group of arbitrary rank. First we start with the generic local parameters: Proposition 6. For k ≥ 3, the 8k − 8 trace functions T = (τ 1 , τ1, τ 2 , τ2, . . . , τ k , τk, τ 12 , τ12, τ 13 , τ13, . . . , τ 1k , τ1k, τ 23 , τ23, . . . , τ 2k , τ2k, ). Next we provide the holomorphic volume form: and ∆ 1 2i is as in Proposition 6. The paper is organized as follows. In Section 2 we review the results on spaces of representations that we need, in particular we describe the relative variety of representations. In Section 3 we recall the tools of Reidemeister torsion, including the duality formula, on which Theorem 3 is based. In Section 4 we describe all forms and we prove Theorem 3. Section 5 is devoted to formulas for SL N (C), the form ν and as well as the volume form for the free groups of rank 2 in SL 2 (C) and SL 3 (C). In Section 6 we use Goldman's formula for the Poisson bracket to give the symplectic form in terms of trace functions for the relative varieties of representations of S 0,4 and S 1,1 in SL 2 (C). Finally, in Section 7 we compute volume forms on spaces of representations of free groups of higher rank in SL 2 (C) and SL 3 (C). Acknowledgements. We are indebted to Simon Riche for helpful discussions and for pointing out Steinberg results to us.

Varieties of representations
Throughout this article G denotes a simply-connected semisimple complex linear Lie group. We let d denote the dimension of G, and r its rank. Also recall that along this paper S = S g,b denotes a compact, oriented, connected surface with nonempty boundary, of genus g and with b ≥ 1 boundary components, ∂S = ∂ 1 ⊔ ⋯ ⊔ ∂ b . We assume that 2.1. The variety of good representations. The set of representations of π 1 (S) ≅ F k in G is R(S, G) = hom(π 1 (S), G) ≅ G k . It follows from [20, Chap.4, §1.2] that G is algebraic, and hence R(S, G) is an affine algebraic set (it has a natural algebraic structure independent of the choice of the isomorphism π 1 (S) ≅ F k ).
The group G acts on R(S, G) by conjugation and we are interested in the quotient This is not a Hausdorff space, so we need to restrict to representations with some regularity properties. Following [12], we define: For ρ ∈ R(S, G), its centralizer is For the proof, see for instance [12, §1], or [19, Proposition 3.8] for injectivity, as irreducibility is equivalent to stability in GIT [12, §1]. For smoothnesses see [5].
Given a representation ρ ∈ R(S, G), the Lie algebra g turns into an π 1 (S)-module via Ad ○ρ. If there is no ambiguity this module is denoted just by g, and the coefficients in cohomology are denoted by Ad ρ.
Proposition 13. Let ρ ∈ R * (S, G) be a good representation. Then there is a natural isomorphism This proposition can be found for instance in [24, Corollary 50], but we sketch the proof as it may be useful for the relative case.
Proof. Let Z 1 = Z 1 (S; Ad ρ) denote the space of crossed morphisms from π 1 (S) to g, i.e. maps d∶ π 1 (S) → g satisfying d(γµ) = d(γ) + Ad ρ(γ) d(µ), ∀γ, µ ∈ π 1 (S). Let B 1 = B 1 (S; Ad ρ) denote the subspace of inner crossed morphisms: for a ∈ g the corresponding inner morphism maps γ ∈ π 1 (S) to Ad ρ(γ) (a) − a. Weil's construction identifies Z 1 with T ρ R(S, G) (usually Z 1 is the Zariski tangent space to a scheme, possibly non-reduced, but as π 1 (S) is free, R(S, G) is a smooth algebraic variety). The subspace B 1 corresponds to the tangent space to the orbit Ad G (ρ). Then, in order to identify the tangent space to R * (S, G) with the cohomology group H 1 (S; Ad ρ) = Z 1 B 1 , we use a slice, for instance anétale slice provided by Luna's theorem [23, Theorem 6.1], or an analytic slice (cf. [12]). In the setting of a good representation ρ, a slice is a subvariety S ⊂ R(S, G) containing ρ, invariant by Z(ρ) = Z(G), such that the conjugation map is locally bi-analytic at (e, ρ) and the projection S → X(S, G) is also bi-analytic at ρ. (If ρ was not good, we should take care of the action of Z(ρ) Z(G). In addition, for Γ not a free group the description is more involved). Then the assertion follows easily from the properties of the slice.

The relative variety of representations. Let
denote the decomposition in connected components. By abuse of notation, we also let ∂ i denote an element of the fundamental group represented by the corresponding oriented peripheral curve. This is well defined only up to conjugacy in π 1 (S), but our constructions do not depend on the representative in the conjugacy class.
Here O(ρ 0 (∂ i )) denotes the conjugacy class of ρ 0 (∂ i ). We also denote Besides considering good representations, we restrict our attention to representation which map peripheral elements to regular elements of G.
We use a slice at ρ 0 , S ⊂ R(M) as in the proof of Proposition 13. The fact that H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) is a surjection means that the restriction map is transverse to the products of orbits by conjugation Now the proposition follows from the properties of the slice.

Steinberg map.
In order to understand the space of conjugacy classes of regular representations of Z we identify each representation with the image of its generator, so that where σ 1 , ⋯, σ r denote the characters corresponding to a system of fundamental representations (for SL N (C) those are the coefficients of the characteristic polynomial).
Theorem 18. (Steinberg, [25]) If G is simply connected, then the map (2) is a surjection and has a section s∶ C r → G reg so that s(C r ) is a subvariety that intersects each orbit by conjugation in G reg precisely once.
For instance, when G = SL N (C) the section in Theorem 18 can be chosen to be the companion matrix (see [26, p. 120] and [11,Sec. 4.15]).

Corollary 19.
If G is simply connected, then: (i) The map (2) induces natural isomorphisms between the space of regular orbits by conjugation, the variety of characters, and C r : (ii) The Steinberg map induces a natural isomorphism Moreover, for each good, ∂-regular representation ρ 0 ∈ R * (S, G) and Proof. For (i), notice that what we aim to prove is the isomorphism G reg G ≅ G G ≅ C r ; which is straightforward from the existence of the section in Theorem 18. For (ii), by the existence of the section we also know that the differential of Steinberg's map Z 1 (Z, Ad ρ) ≅ g → C r is surjective whenever ρ is regular [11, §4.19]. In addition it maps B 1 (Z, Ad ρ) to 0, because Steinberg map is constant on orbits by conjugation.
Thus we have a well defined surjection H 1 (S 1 , Ad ρ) → C r , which is an isomorphism because of the dimension. The exact sequence follows from the long exact sequence in cohomology of the pair (S, ∂S) and the identification of cohomology groups with tangent spaces, cf. Proposition 17.
Remark 20. When G is not simply connected, then the universal coveringG → G is finite and π 1 (G) can be identified with a (finite) central subgroup Z ofG. The center of G acts on the quotientG G and we obtain a commutative diagram Notice that ϕ is a finite branched covering. Then part (ii) of Corollary 19 can be easily adapted for those [g] ∈ G G which are outside the branch set of ϕ.

Reidemeister torsion
Let ρ ∈ R(S, G) be a representation; recall that we consider the action of π 1 (S) on g via the adjoint of ρ. Most of the results in this section apply not only to g but to its real form g R , provided that the image of the representation is contained in G R . Recall also that we assume that B restricted to the compact real form g R is positive definite.
Consider a cell decomposition K of S. If C * (K; Z) denotes the simplicial chain complex on the universal covering, one defines We consider the so called geometric basis. Start with a B-orthonormal C-basis {m 1 , . . . , m d } of g. For each i-cell e i j of K we choose a liftẽ i j to the universal coveringK, then is also a basis for C i (K; Ad ρ). Notice that we are interested in the case where the zero and second cohomology groups vanish, so we assume thath 0 =h 2 = ∅.
Reidemeister torsion is defined as Here, for two bases a and b of a vector space, [a ∶ b] denotes the determinant the matrix whose colons are the coefficients of the vectors of a as linear combination of b.
Remark 21. The choice of the bilinear form B is relevant, as we use a B-orthonormal basis for g and χ(S) ≠ 0. Namely, if we replace B by λ 2 B, then the orthonormal basis will be 1 λ {m 1 , . . . , m d } and the torsion will be multiplied by a factor λ −χ(S)d = λ dim R(S,G) . For an ordered basis a = {a 1 , . . . , a m } of a vector space, denote is often used in the literature (cf. [18]).
3.1. The holomorphic volume form. The tangent space to R * (S, G) at [ρ] is identified to t H 1 (S; Ad ρ), by Proposition 13. There is a natural holomorphic volume form on H 1 (S; Ad ρ): The surface S has the simple homotopy type of a graph. Moreover, graphs that are homotopy equivalenet are also simple-homotopy equivalent, thus this volume form depends only on the fundamental group is a Lie group covering and we get an isomorphismf ρ ∶ G Z(G) → O(ρ), and hence The next lemma justifies why Reidemeister torsion is the natural choice of volume form on the variety of representations up to conjugation.
Proof. We use a graph G with one vertex and k edges to compute the torsion of S. The Reidemeister torsion of this graph is tor If we make the choiceb 1 = c 0 , which is a basis for g, then We identify the 1-cells with the generators of F k , so that every element in c 1 is viewed as a tangent vector to the variety of representations, and c 1 has volume one, (θ G ) k (∧c 1 ) = 1 because we started with an B-orthonormal basis for g. Thus As δc 0 is a basis of the tangent space to the orbit π * (δc 0 ) = 0. Moreover, using π * (h) = h: By (6) and (7), to conclude the proof of the lemma we claim that θ O(ρ) (∧δc 0 ) = 1. For that purpose, we use the canonical identification T ρ O(ρ) ≅ B 1 (π 1 (S); Ad ρ). Using this identification, the tangent map of the orbit map for γ ∈ π 1 (S).
Therefore for the basis δc 0 of B 1 (π 1 (S); Ad ρ) we obtain by (5): This concludes the proof of the claim and the lemma.
3.2. The nondegenerate pairing. Consider K ′ the cell decomposition dual to K: . The complex C * (K ′ , ∂K ′ ; Z) yields the relative cohomology of the pair (S, ∂S). This can be generalized to cohomology with coefficients. If C * (K; Z) denotes the simplicial chain complex on the universal covering, recall from (3) that and we similarly define where π 1 (S) acts on g by the adjoint representation.
Following Milnor [18], there is a paring Here "⋅" denotes the intersection number in the universal covering. The main properties of this paring are that for η ∈ Zπ 1 (S) we have: Here the bar . ∶ Zπ 1 (S) → Zπ 1 (S) denotes the anti-involution that extends Z-linearly the anti-morphism of π 1 (S) that maps γ ∈ π 1 (S) to γ −1 . Notice that the sign ± in equation (8) depends only on the dimension of the chains. For each i-dimensional cell e i j we fix a liftẽ i j toK. Also, we chose a (2 − i)-dimensional cellf 2−i j which projects to f 2−i j . By replacingf 2−i j by a translate, we can assume that We obtain, for each i-chain c ∈ C i (K; Z) and each Given α ∈ C i (K; Ad ρ) and defines a nondegenerate pairing By using equation (8), it is easy to see that this pairing satisfies (10) ⟨δα, α ′ ⟩ = ±⟨α, δα ′ ⟩ , and therefore it induces a non-singular pairing in cohomology Given a basis h = {h i } i of H 1 (S; Ad ρ) and h ′ = {h ′ i } i a basis of H 1 (S, ∂S; Ad ρ), we introduce the notation (12) ⟨∧h, ∧h ′ ⟩ ∶= det ⟨h i , h ′ j ⟩ ij which is the natural extension of the pairing (11) to 3.3. The duality formula. Let ρ ∈ R(π 1 (S), G) be a representation. This is E. Witten's generalization of the duality formula of W. Franz and J. Milnor. We reproduce the proof for completeness. In Witten's article [29] the proof of this particular formula is only given in the closed case, and Milnor [18] and Franz [2] consider only the acyclic case.
Proof. We chose the geometric basis of C i (K; Ad ρ) and C 2−i (K ′ , ∂K ′ ; Ad ρ) to be dual to each other, by choosing dual lifts of the cells and a B-orthonormal basis of the Lie algebra g. In this way, the matrix of the intersection form (9) with respect the geometric basis is the identity, in particular its determinant is 1: ⟨∧c i , ∧(c 2−i ) ′ ⟩ = 1. Thus we view the product of torsions in the statement of the proposition as three changes of basis, one for each intersection form: Next, following Witten, we may chose the lift of the coboundaries to be orthogonal to the lift of the cohomology of the other complex: In addition, by direct application of (10): Thus the numerator in (13) is the determinant of a matrix with some vanishing blocks, and (13) becomes: (10). Hence (14) equals ±⟨∧h, ∧h ′ ⟩, concluding the proof.
Remark 24. Notice that the proof generalizes in any dimension, after changing the product by a quotient in the odd dimensional case, and taking care of the intersection product in all cohomology groups. The fact that ω is bilinear and alternating is clear from construction, non-degeneracy follows from Poincaré duality, and the deep result is to prove dω = 0. When S is closed this was proved by Goldman in [5]. When ∂S ≠ ∅, the result with real coefficients is due to Guruprasad, Huebschmann, Jeffrey, and Weinstein [10], and in [15] Lawton explains why it works in the complex case.

4.2.
Sign refined Reidemeister torsion for the circle. Let V be a finite dimensional real or complex vector space, and ϕ∶ π 1 (S 1 ) → SL(V ) be a representation. In what follows we use the refined torsion with sign due to Turaev, that we denote TOR(S 1 , ϕ, o, u, v) [28, §3]. This torsion depends on the choice of an orientation o in cohomology with constant coefficients of S 1 and the choice of respective basis u for H 0 (S 1 ; ϕ) and v for H 1 (S 1 ; ϕ). For a circle S 1 , the choice of an orientation determines a fundamental class, hence an orientation in homology.
We start with a cell decomposition K of S 1 , with i-cells e i 1 , . . . , e i a , i = 0, 1, and a (real or complex) basis {m 1 , . . . , m k } for the vector space V . The geometric basis for C i (K; ϕ) is then c i = {(ẽ i 1 ) * ⊗ m 1 , (ẽ i 1 ) * ⊗ m 2 , . . . , (ẽ i a ) * ⊗ m k }. As before, let B 1 = Im(δ∶ C 0 (K; ϕ) → C 1 (K; ϕ)) denote the coboundary space and chose b 1 as basis for B 1 and lift it tob 1 in C 0 (K; ϕ). Consider alsoṽ ⊂ C 1 (K; ϕ) a representative of v and similarlyũ ⊂ C 0 (K; ϕ) for u. Then we define the torsion: Notice that there is no sign indeterminacy, because we include c i in the notation. In fact sign indeterminacy comes from changing the order or the orientation of the cells of K.
The sign is not affected by the choice of a basis for V , because χ(S 1 ) = 0. Following [28, §3] we consider This quantity is invariant under subdivision of the cells of K, but it still depends on their ordering and orientation. To make this quantity invariant, Turaev introduces the notion of cohomology orientation, i.e. an orientation of the R-vector space H 0 (S 1 ; R) ⊕ H 1 (S 1 ; R). We consider a geometric basis the complex with trivial coefficients C i (K; R), c i = {(e i 1 ) * , . . . , (e i a ) * }, with the same ordering and orientation of cells. We chose any basis h i of H i (S 1 ; R) that yield the orientation o.

Definition 26. The sign determined torsion is
Let −o denote the homology orientation opposite to o. It is straightforward from construction that (16) TOR In particular, we do not need the homology orientation when dim ϕ is even. For a circle S 1 , the choice of an orientation determines a fundamental class, hence an orientation in cohomology. Let ϕ i ∶ π 1 (S 1 ) → SL(V i ) be representations into finite dimensional vector spaces, for i = 1, 2. Then H * (S 1 ; ϕ 1 ⊕ ϕ 2 ) ≅ H * (S 1 ; ϕ 1 ) ⊕ H * (S 1 ; ϕ 2 ). Let u i and v i denote basis for H 0 (S 1 ; ϕ i ) and H 1 (S 1 ; ϕ i ) respectively. The following lemma reduces to an elementary calculation: Lemma 27. Let ϕ i ∶ π 1 (S 1 ) → SL(V i ) be representations into finite dimensional vector spaces, for i = 1, 2. Then

4.3.
An holomorphic volume form on R reg (S 1 , G). As in the introduction we let G denote a simply-connected, semisimple, complex, linear Lie group, d = dim G, and r = rk G.
Definition 28. We call a representation ρ∶ π 1 (S 1 ) → G regular if the image of the generator of π 1 (S 1 ) is a regular element g ∈ G. The set of conjugacy classes of regular representations is denoted by R reg (S 1 , G).
In the next lemma we use the refined torsion with sign due to Turaev (see Section 4.2). By (16) changing the orientation of S 1 changes the torsion TOR(S 1 , Ad ρ, o, u, v) by a factor (−1) d = (−1) r , as well as ⟨∧v, ∧u⟩ by the same factor.
Let G R denote the compact real form of the semisimple complex linear group G. We will assume that the restriction of the nondegenerate symmetric bilinear G-invariant form B on the Lie algebra to g R is positive definite. This means that B is a negative multiple of the Killing form. In what follows we will denote by Ad R ∶ G R → Aut(g R ) the restriction of Ad to the real form G R .
Lemma 29. If ρ∶ π 1 (S 1 ) → G is a regular representation, and if u and v are bases of H 0 (S 1 ; Ad ρ) and H 1 (S 1 ; Ad ρ) respectively, then the product Lemma 30. If ρ∶ π 1 (S 1 ) → G R is a regular representation and if u and v are bases of H 0 (S 1 ; Ad R ρ), and H 1 (S 1 ; Ad R ρ) respectively, then Proof of Lemma 29. Let u and u ′ be bases for H 0 (S 1 ; Ad ρ), and v and v ′ , for H 1 (S 1 ; Ad ρ). We change bases by means of the following formulas: Hence This proves independence of u.
Proof of Lemma 30. We are assuming that the image of ρ is contained in the compact real form, ρ(π 1 (S)) ⊂ G R . By (17) in the proof of Lemma 29, the sign is independent of v. By regularity, H 0 (S 1 ; Ad R ρ) ⊂ g R is a Cartan subalgebra h, and B restricted to h is positive definite. Hence we may chose an R-basis of g R compatible with the orthogonal decomposition g R = h ⊥ h ⊥ . This is also a decomposition of π 1 (S 1 )-modules, and by Lemma 27 the torsion decomposes accordingly as a product of torsions. We compute the torsion of h first. Since the adjoint action of H on h is trivial, we have natural isomorphisms We chose a cell decomposition of S 1 with a single (positively oriented) cell in each dimension. In particular, as the adjoint action of H on h is trivial, the boundary operator δ∶ C 0 (K; h) → C 1 (K; h) vanishes. Chose a B-orthonormal basis for h, this provides geometric basis c 1 and c 0 , and since δ = 0, those are also representatives of basis in cohomology. By choosing those bases (u = c 0 and v = c 1 ), Following the construction in Section 4.2, we compute α 0 = β 0 = r and α 1 = β 1 = 2r ≡ 0 mod 2. Thus N ≡ r 2 ≡ r mod 2 and As the torsion for the trivial representation corresponds to the case r = 1, Tor for the trivial representation is −1 and Also, by construction, ⟨∧c 1 , ∧c 0 ⟩ = 1.
Next we compute the torsion of h ⊥ . We have H * (S 1 ; h ⊥ ) = 0 and, since dim h ⊥ is even, where g ∈ G is the image of a generator of π 1 (S 1 ). Notice that, as dim h ⊥ is even, the sign is independent of the cohomology orientation. Let ∆ G be the Weyl function [9]. Then [9, (7.47)] for details). This finishes the proof of the lemma.
Definition 31. Let ρ∶ π 1 (S 1 ) → G be a regular representation. The form for any basis u of H 1 (S 1 ; Ad ρ). (By Lemma 29, it is independent of u.) We are interested in understanding ν as a differential form on R reg (S 1 , G) for G simply connected. Recall from §2.3 that when G is simply connected, the Steinberg map has coordinates the fundamental characters (σ 1 , . . . , σ r )∶ G → C r .
Proposition 32. For G simply connected, there exists a constant C ∈ C * and a choice of sign for ν such that ν = C dσ 1 ∧ ⋅ ⋅ ⋅ ∧ dσ r .
Proof. Using Steinberg's section s∶ C r → G reg (Theorem 18), consider for each p ∈ C r the subagebra g Ad s(p) of elements fixed by Ad s(p). By the constant rank theorem this defines an algebraic vector bundle g Ad ○s → E(s) → C r .
Since algebraic vector bundles over C r are trivial [22,27], there is a trivialization u = (u 1 , . . . , u r )∶ C r → E(s), so that {u 1 (p), . . . , u r (p)} is a basis for g Ad s(p) , for each p ∈ We claim that these forms are both algebraic. Assuming the claim, they are a polynomial multiple of dz 1 ∧ ⋯ ∧ dz r , for (z 1 , . . . , z r ) the standard coordinate system for C r . Since they vanish nowhere in C r , both forms in (21) are a constant multiple of dz 1 ∧ ⋯ ∧ dz r . Viewed as as forms on R reg (S 1 , G), they are both a constant multiple of dσ 1 ∧ ⋯ ∧ dσ r and the proposition follows, once we have shown the claim.
To prove that the forms in (21) are algebraic, use a CW-decomposition K of S 1 with a 1 and a 0-cell, so that the groups of cochains C i (K, Ad s(p)), for i = 0, 1, are naturally identified with g. We also have a natural isomorphism R s(p) −1 * ∶ T s(p) G → g, which is precisely the tangent map to righ multiplicatiuon by s(p) −1 . This identification maps which is a map algebraic on p ∈ C r . Hence the intersection product is which is polynomial on p ∈ C r . To show that the torsion is algebraic, using again triviality of algebraic bundles on C r , complete u to a section of the trivial bundle (u 1 , . . . , u r , . . . , u d )∶ C r → g. Setting b 1 = {u r+1 , . . . , u d }, then u(p)⊔b 1 (p) is a basis for g, for each p ∈ C. We view u(p)⊔b 1 (p) as a basis for C 0 (K, Ad s(p)), so that u(p) projects to a basis for H 0 (S 1 , Ad s(p)), for every p ∈ C r . Fix c 0 = c 1 a basis for g. By construction: where the sign depends on the orientation in homology, but it is constant on p. Thus this is a quotient of algebraic polynomial functions on C r , but since it is defined everywhere, it is polynomial.

4.4.
Witten's formula. Let ρ∶ π 1 (S) → G be a good ∂-regular representation. Let ν i denote the peripheral form of the i-th component of ∂S (Definition 31), and let ω denote the symplectic form of the relative character variety (15). We aim to prove Theorem 3, namely, that Ω π 1 (S) = ± 1 n! ω n ∧ ν 1 ∧ ⋯ ∧ ν b . Proof of Theorem 3. We apply the duality formula (Proposition 23) and the formula of the torsion for the long exact sequence of the pair, Equation (23) below. For this purpose we discuss the bases in cohomology. Start with u a basis for H 0 (∂S; Ad ρ). If β denotes the connecting map of the long exact sequence, then complete β(u) to a basis for H 1 (S, ∂S; Ad ρ): β(u) ⊔h. Next we chose v a basis for H 1 (∂S; Ad ρ) that we lift tõ v by i, and if we set j(h) = h, then h ⊔ṽ is a basis for H 1 (S; Ad ρ) (and h is a basis for ker(i) = Im(j)). The bases are organized as follows: (22) 0 → H 0 (∂S; Ad ρ) As the bases have been chosen compatible with the maps of the long exact sequence, the product formula for the torsion [17] gives: (23) tor(S, Ad ρ, h ⊔ṽ) = ± tor(S, ∂S, Ad ρ, β(u) ⊔h) tor(∂S, Ad ρ, u, v).
Notice that on the right hand side we use Turaev's sign refined torsion. Next we claim that the sign of this formula is + and not −. It suffices to determine the sign in the compact case. Then the formula will follow in the complex case by a connectedness argument (the variety of characters of a free group is connected and irreducible, and ∂-regularity and being good are Zariski open properties, hence they fail in a set of real codimension ≥ 2). We show that the sign is + in the compact case by showing that all terms are positive. Since TOR(∂S, Ad ρ, u, v)⟨u, v⟩ is positive by Lemma 30, the sign will follow from the equality (26) ω(∧h, ∧h) = 1 n! ω n (∧h) 2 , that will also complete the proof of the theorem. We give self-contained proof of (26) by completeness. By Darboux's theorem there are local coordinates so that Let A be a matrix of size 2n × 2n whose colons are the components of the vectors of h in this coordinate system. Then, if J denotes the matrix of the standard symplectic form, On the other hand ω n = n! dx 1 ∧ dx 2 ∧ ⋯ ∧ dx 2n , hence 1 n! ω n (∧h) = det A and we are done.

Formulas for the group SL N (C)
If G = SL N (C) we can give explicit formulas for several volume forms.
5.1. The form ν for SL N (C). We know that ν is a constant multiple of d σ 1 ∧ ⋯ ∧ d σ r and we shall determine the constant, completing the proof of Proposition 2. Recall that we chose the C-bilinear form on sl N (C) to be In SL N (C) the invariant functions are the symmetric functions on the spectrum: if the eigenvalues of A ∈ SL N (C) are λ 1 , . . . , λ N , then Those symmetric functions are characterized by Cayley-Hamilton theorem: We identify R(S 1 , SL N (C)) with the group SL N (C) by mapping a representation to the image of a generator of π 1 (S), so that σ i is a function on R(S 1 , SL N (C)) invariant under conjugation. On the other hand, σ 1 , . . . , σ N −1 are the coordinates of the isomorphism: Proposition 33. Let ν∶ ⋀ N −1 H 1 (S 1 , Ad ρ) → C denote the volume form in Definition 31.
By direct application of the proposition, we get: where γ is a generator of π 1 (S 1 ).
Proof of Proposition 33. We identify the variety of representations of the cyclic group π 1 (S 1 ) with SL N (C) by considering the image of a generator, that we call g. To simplify, we may assume that g is semisimple, by Proposition 32. After diagonalizing: with u 1 + ⋯ + u N = 0 and all u i are pairwise different mod 2π √ −1Z. The Cartan algebra h is the subalgebra of diagonal matrices. Since the decomposition sl N (C) = h ⊥ h ⊥ is preserved by the adjoint action of g, the torsion is the corresponding product of torsions, by Lemma 27. By looking at the action on non-diagonal entries of sl N (C), the torsion of the adjoint representation on h ⊥ is: which is the product ∆ G (g)∆ G (g −1 ) of Weyl functions [9, §7]. Thus We use coordinates for the Cartan algebra via the entries of the diagonal matrices: or, equivalently, of Proof. In order to compute TOR(S 1 , h, o, ∧v, ∧u) we proceed as in the proof of Lemma 30. In particular we chose a cell decomposition of S 1 with a single (positively oriented) cell in each dimension, and bases in homology represented by the geometric bases. With this choice of u and v, by (19), TOR(S 1 , h, o, v, u) = 1.
Next we compute ⟨∧v, ∧u⟩. The basis u and v are constructed from dual basis in H * (S 1 ; Z) tensorized by a basis of h. We choose a basis for the Cartan subalgebra, e = {e 1 , . . . , e N −1 }: Since the cells of S 1 are positively oriented, ⟨∧v, ∧u⟩ = det(B(e i , e j )) i,j ).
In addition, as B(e i , e i ) = −2 and B(e i , e j ) = −1 for i ≠ j, det(B(e i , e j )) i,j ) = (−1) N −1 N. Thus On the other hand, direct computation yields: By the natural identification of H 1 (S 1 ; h) with the Cartan algebra h we get the lemma.

The form ν for SU(N). An element in SU(N) is conjugate to a diagonal element
A matrix is regular if and only if e iθ j ≠ e iθ k for j ≠ k. By identifying R reg (S 1 , SU(N)) with the image of the generator (or its conjugacy class), functions on θ 1 , . . . , θ N invariant under permutations are well defined on R reg (S 1 , SU(N)). Also the form d θ 1 ∧ ⋯ ∧ d θ N −1 is well defined up to sign by the relation ∑ θ i ∈ 2πZ.
Proof. From the proof of Proposition 33, if g ∈ SU(N) is the image of the generator of π 1 (S 1 ), ν = ∆ G (g) θ H . By [9,Exercise 7.3.5], On the other hand, by Lemma 35, which proves the formula.
Remark 37. We may consider also the restriction to SL N (R). Then the expression of the volume form is just the restriction of Proposition 33. It may be either real valued or √ −1 times real, because B is not positive definite on sl N (R). The restriction of B to so N is positive definite, but its restriction to its orthogonal so ⊥ N ⊂ sl N (R) is negative definite. Notice that dim so ⊥ N = (N − 1)(N + 2) 2 ≡ ǫ(N) mod 2, that determines whether it is real or √ −1 times real.

5.3.
Volume form for representation spaces of F 2 . In this subsection we compute the volume form on the space of representations of a free group of rank 2, F 2 = ⟨γ 1 , γ 2 ⟩, in SL 2 (C) and SL 3 (C). We use the notation t i 1 ⋯i k for the trace functions of γ i 1 ⋯γ i k in SL 2 (C), with the convention γ¯i = γ −1 i . For instance, the trace function of γ 1 γ −1 2 will be denoted by t 12 .
Thus, as γ 1 , γ 2 and γ 1 γ 2 represent the peripheral elements of a pair of pants S 0,3 , a generic subset of the relative variety of representations is locally parameterized by (τ 12 , τ1 2 ); in the subset of points where there is no branching, i.e. τ 1212 ≠ τ 2121 . Lawton has computed in [15,Thm. 25] the Poisson bracket: As (τ 12 , τ1 2 ) are local coordinates, an elementary computation yields Therefore On the other hand, by Proposition 33, the form ν 1 corresponding to γ 1 is and similarly for γ 2 and γ 12 . Using Theorem 3 and these computations we get: the restriction of the holomorphic volume form on R * (F 2 , SL 3 (C)) ∖ {τ 2121 = τ 1212 } is Ω

Symplectic forms
Let ρ 0 ∈ R * (S, SL 2 (C)) be a good, ∂-regular representation. In this section we discuss the symplectic from on the relative character variety R * (S, ∂S, SL 2 (C)) ρ 0 for the two surfaces S 1,1 and S 0,4 , which are the surfaces with 2-dimensional relative character variety R * (S, ∂S, SL 2 (C)) ρ 0 . We use Goldman's product formula for the Poisson bracket for surfaces [6], as well as Lawton's generalization [15,Sec. 4] to the relative character variety.
For this purpose, let f ∶ G → C be an invariant function (i.e. a function on G invariant under conjugation). Following Goldman [7], its variation function (relative to B) is defined as the unique map F ∶ G → g such that for all X ∈ g, A ∈ G, When G = SL 2 (C) and f = tr, the corresponding variation formula T∶ SL 2 (C) → sl 2 (C) must satisfy, by (35), tr(A X) = − tr T(A) X , ∀X ∈ sl 2 (C) and ∀A ∈ SL 2 (C). Thus Notice that T(A) ∈ sl 2 (C) is invariant by the adjoint action of A, and T(A) ≠ 0 for A ≠ ± Id.
For later computations, it is useful to recall (cf. [8]) that for all A, B ∈ SL 2 (C)

6.1.
A torus minus a disc. Let S 1,1 denote a surface of genus 1 with a boundary component. Its fundamental group is freely generated by two elements γ 1 and γ 2 that are represented by curves that intersect at one point. The peripheral element is the commutator [γ 1 , γ 2 ] = γ 1 γ 2 γ −1 1 γ −1 2 . The variety of characters X(S 1,1 , SL 2 (C)) is the variety of characters of the free group on two generators, and it is isomorphic to C 3 with coordinates t 1 , t 2 , t 12 , by Fricke-Klein (32). Equality (36) implies that t 1 t 2 = t 12 + t 12 .
We compute next the symplectic form.
Remark 42. From Proposition 41 we can compute again the volume form on R * (F 2 , SL 2 (C)), already found in Corollary 38. Namely, by Theorem 3, Proposition 41, and Corollary 34, since the commutator γ 1 γ 2 γ −1 1 γ −1 2 is the peripheral element, Differentiating (38), we get (41) d t 1212 = (2t 1 − t 2 t 12 )d t 1 + (2t 2 − t 1 t 12 )d t 2 + (2t 12 − t 1 t 2 )d t 12 , thus, as t 1 t 2 = t 12 + t 12 , by replacing (41) in (40): 6.2. A planar surface with four boundary components. Let S 0,4 denote the planar surface with four boundary components and let λ and µ be two simple closed curves so that each one divides S 0,4 in two pairs of pants and they intersect in precisely two points. Chose also one of the intersections points as a base point for the fundamental group. Orient the curves λ and µ and obtain two new oriented curves α and β, by changing both crossings in a way compatible with the orientation, according to Figures 1 and 2. Since the curves are oriented, we may talk about the elements they represent in π 1 (S 0,4 ), in particular the products λµ and αβ and their trace functions, t λµ and t αβ , that depend on the orientations.
Lemma 43. Up to sign, the difference t λµ −t αβ is independent of the choice of orientations of λ and µ. The sign depends on whether we change one (-) or both (+) orientations.