The complex-symplectic geometry of SL(2,C)-characters over surfaces

By William M. Goldman. 2003

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The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the Fenchel-Nielsen twist flows on Teichmueller space and the complex quakebend flows on quasi-Fuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space of complex-projective structures between different projective structures with the same holonomy (grafting). A pants decomposition determines a holomorphic completely integrable system on X. This integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fbae8954bbd0c9fc5fc9bff4d0847ffbe0aa46cc

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Chern-Simons theory and S-duality (2011, methodology): “A more intrinsic, geometric definition of complex Fenchel-Nielsen coordinates for Mflat(GC;C) is given in [90, 91], and further analyzed by Goldman [92].”