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Poisson structures on certain moduli spaces for bundles on a surface. (English) Zbl 0819.58010

Summary: Let \(\Sigma\) be a closed surface, \(G\) a compact Lie group, with Lie algebra \(g\), and \(\xi \colon P \to \Sigma\) a principal \(G\)-bundle. In earlier work we have shown that the moduli space \(N(\xi)\) of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from \(N(\xi)\) onto a certain representation space \(\text{Rep}_{\xi}(\Gamma,G)\), in fact a diffeomorphism, with reference to suitable smooth structures \(C^{\infty}(N(\xi))\) and \(C^{\infty}\left(\text{Rep}_{\xi}(\Gamma,G)\right)\), where \(\Gamma\) denotes the universal central extension of the fundamental group of \(\Sigma\). Given a coadjoint action invariant symmetric bilinear form on \(g^*\), we construct here Poisson structures on \(C^{\infty}(N(\xi))\) and \(C^{\infty}\left(\text{Rep}_{\xi}(\Gamma,G)\right)\) in such a way that the mentioned diffeomorphism identifies them. When the form on \(g^*\) is non-degenerate the Poisson structures are compatible with the stratifications where \(\text{Rep}_{\xi}(\Gamma,G)\) is endowed with the corresponding stratification and, furthermore, yield structures of a stratified symplectic space, preserved by the induced action of the mapping class group of \(\Sigma\).

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
81T13 Yang-Mills and other gauge theories in quantum field theory
58D27 Moduli problems for differential geometric structures
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

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