Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its algebra. We develop theory of quasi Poisson structures relative to not necessarily non-degenerate Ad-invariant symmetric 2-tensor in the tensor square one general Hamiltonian bilinear form on g, structure being given by skew bracket two variables such that suitable data defined terms as symmetry involving measure how fails satisfy Jacobi identity. The present approach involves novel concept momentum mapping yields, case, bijective correspondence between structures. new applies various non-singular moduli spaces yields thereupon, via reduction with respect an appropriately mapping, ordinary Among these are representation spaces, possibly twisted, fundamental Riemann surface, punctured, semistable holomorphic vector bundles well Higgs bundle spaces. In comes down stratified symplectic kind explored literature recovers, e.g., part K\"ahler introduced Narasimhan Seshadri for stable curve. algebraic setting, arise affine varieties. A side result is explicit equivalence extended independently gauge theory.