Abstract
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).
Highlights
Along this paper S = Sg,b denotes a compact, oriented, connected surface with nonempty boundary, of genus g and with b 1 boundary components
We assume that G is connected; notice that since π1(S) is free, its representations lift to the universal covering of the Lie group
Using the computation of the symplectic form in [Law09], we prove in Proposition 5.7 that, on R∗(F2, SL3(C)) \ {τ21 ̄2 ̄1 = τ12 ̄1 ̄2} the volume form is ΩSFL2√3(C) = T ∗Ω, where
Summary
By Fricke–Klein–Vogt Theorem [Fri[96], FK97, Vog89] (see [Gol[09], Mag80] for a modern treatment), X(F2, SL2(C)) ∼= C3 and the coordinates are precisely the traces of the peripheral elements γ1, γ2, and γ1γ2, denoted by t1, t2, and t12 respectively In this case the relative character variety is just a point, and√the symplectic form is trivial.
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