If G is a Lie group whose adjoint representation preserves a nondegenerate symmetric bilinear form on its Lie algebra (e.g. a semisimple group) and F is the fundamental group of a closed oriented surface S, then the spaces of equivalence classes of representations F-greater than G (equivalently gauge-equivalence classes of flat G-connections over S) enjoys a rich symplectic geometry. These symplectic manifolds generalize the Kähler structures on the Jacobi variety, moduli of holomorphic vector bundles, and Teichmüller space. This talk will describe this geometry and several open questions about these symplectic manifolds.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
