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The \(\operatorname{SU}(2)\)-character variety of the closed surface of genus 2. (English) Zbl 1396.53110

This paper studies the symplectic geometry of the moduli space \(\mathcal{M}= \mathrm{Hom}\left( \pi _{1}\left( \Sigma \right) ,\operatorname{SU}\left( 2\right) \right) /\operatorname{SU}\left( 2\right) \) of conjugacy classes of representations of the fundamental group \(\pi _{1}\left( \Sigma \right) \) into the compact Lie group \(K:=\operatorname{SU}\left( 2\right) \), where \(\Sigma \) is the compact oriented surface of genus \(2\).
A distinguished dense open subset of \(\mathcal{M}\) is given by the collection \(\mathcal{M}^{i}\) of conjugacy classes \(\left[ \rho \right] \) of irreducible representations \(\rho \in \mathrm{Hom}\left( \pi _{1}\left( \Sigma \right) ,\operatorname{SU}\left( 2\right) \right) \). The standard symplectic structure on \(\mathcal{M}^{i}\) is given at each \(\left[ \rho \right] \in \mathcal{M}^{i}\) by the alternating 2-form \(\omega \) on the cohomology group \(H^{1}\left( \pi _{1}\left( \Sigma \right) ,\mathfrak{k}\right) \) determined by the Poincaré duality and the Killing form on \(\mathfrak{k}\equiv \mathfrak{su}\left( 2\right) \), where \(\mathfrak{k}\) carries the \(\pi _{1}\left( \Sigma \right) \)-module structure defined by \(\mathrm{Ad}_{\rho \left( \cdot \right) }:\pi _{1}\left( \Sigma \right) \rightarrow \mathrm{Aut} \left( \mathfrak{k}\right) \) and \(H^{1}\left( \pi _{1}\left( \Sigma \right) , \mathfrak{k}\right) \) is canonically identified as the tangent space to \( \mathcal{M}^{i}\) at \(\left[ \rho \right] \).
Analyzing the moment map associated with the Hamiltonian \(\mathbb{T}^{3}\)-action on an open set \(\mathcal{M}^{\circ }\subsetneqq \mathcal{M}^{i}\) arising from the Goldman flow on \(\mathcal{M}\), the authors show that \( \mathcal{M}^{\circ }\) is dense in \(\mathcal{M}^{i}\) and by modding out the isotropy group, there is a free Hamiltonian \(\mathbb{T}^{3}\)-action on \( \mathcal{M}^{\circ }\), and more explicitly, \(\mathcal{M}^{\circ }\cong \Delta ^{\circ }\times \mathbb{T}^{3}\) as a trivial \(\mathbb{T}^{3}\)-bundle over an open \(3\)-simplex \(\Delta ^{\circ }\). By a similar analysis of the moment map associated with the standard Hamiltonian \(\mathbb{T}^{3}\)-action on the complex and hence symplectic manifold \(\mathbb{P}^{3}\left( \mathbb{C} \right) \), it is pointed out that this \(\mathbb{T}^{3}\)-action is also free on a dense open subset of \(\mathbb{P}^{3}\left( \mathbb{C}\right) \). As a consequence, the authors conclude that the dense open set \(\mathcal{M} ^{\circ }\) of \(\mathcal{M}^{i}\) can be identified symplectically with a dense open subset of \(\mathbb{P}^{3}\left( \mathbb{C}\right) \).
It is also shown that, up to a choice of a global section of the trivial bundle \(\mathcal{M}^{\circ }\rightarrow \Delta ^{\circ }\), there is a unique anti-symplectic involution on \(\mathcal{M}^{\circ }\) that is compatible with the Hamiltonian \(\mathbb{T}^{3}\)-action, i.e., preserving the Hamiltonian function \(\mu _{\xi }\) for all \(\xi \in \mathfrak{t}^{3}\). On the other hand, an anti-symplectic involution \(\sigma \) not compatible with the Hamiltonian \(\mathbb{T}^{3}\)-action is explicitly given and shown to have its fixed point set homeomorphic to \(\mathbb{P} ^{3}\left( \mathbb{R}\right) \).

MSC:

53D30 Symplectic structures of moduli spaces
53D20 Momentum maps; symplectic reduction
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[1] Choi, S.: Spherical triangles and the two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2. Int. J. Math. 22(9), 1261-1364 (2011) · Zbl 1261.57017 · doi:10.1142/S0129167X11007185
[2] Duistermaat, J.J.: Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution. Trans. Am. Math. Soc. 275(1), 417-429 (1983) · Zbl 0504.58020 · doi:10.1090/S0002-9947-1983-0678361-2
[3] Goldman, W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200-225 (1984) · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9
[4] Goldman, W.M.: Invariant functions in Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85, 263-302 (1986) · Zbl 0619.58021 · doi:10.1007/BF01389091
[5] Goldman, W.M.: Ergodic theory on moduli spaces. Ann. Math. (2) 146(3), 475-507 (1997) · Zbl 0907.57009 · doi:10.2307/2952454
[6] Goldman, W.M.: An ergodic action of the outer automorphism group of a free group. Geom. Funct. Anal. 17(3), 793-805 (2007) · Zbl 1139.57002 · doi:10.1007/s00039-007-0609-8
[7] Goldman, W.M.: Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In: Handbook of Teichmüller Theory, vol. II, pp. 611-684, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich (2009) · Zbl 1175.30043
[8] Jeffrey, L.C., Weitsman, J.: Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math. Phys. 150(3), 593-630 (1992) · Zbl 0787.53068 · doi:10.1007/BF02096964
[9] Jeffrey, L.C., Weitsman, J.: Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map. Adv. Math. 106, 151-168 (1994) · Zbl 0836.58004 · doi:10.1006/aima.1994.1054
[10] Johnson, D., Millson, J.J.: Deformation spaces associated to compact hyperbolic manifolds. In: Howe, R. (ed.) Discrete Groups in Geometry and Analysis (New Haven, CT, 1984), Progr. Math., 67, pp. 48-106. Birkhäuser Boston, Boston, MA (1987) · Zbl 0574.32032
[11] Narasimhan, M.S., Ramanan, S.: Moduli of vector bundles on a compact Riemann surface. Ann. Math. (2) 89, 14-51 (1969) · Zbl 0186.54902 · doi:10.2307/1970807
[12] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. (2) 82, 540-567 (1965) · Zbl 0171.04803 · doi:10.2307/1970710
[13] Sikora, A.S.: Character varieties. Trans. Am. Math. Soc. 364, 5173-5208 (2012) · Zbl 1291.14022 · doi:10.1090/S0002-9947-2012-05448-1
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