On the variation of the Poisson structures
of certain moduli spaces
Johannes Huebschmann (Lille)
Abstract:
Given a Lie group G whose Lie algebra is endowed with a nondegenerate
invariant symmetric bilinear form, we construct a Poisson algebra of
continuous functions on a certain open subspace R of the space of
representations in G of the fundamental group of a compact connected
orientable topological surface with finitely many boundary circles; when
G is compact and connected, R may be taken dense in the space of all
representations. The space R contains spaces of representations where
the values of those generators of the fundamental group which correspond
to the boundary circles are constrained to lie in fixed conjugacy classes
and, on these representation spaces, the Poisson algebra restricts to
stratified symplectic Poisson algebras constructed elsewhere earlier.
When G is the unitary group, these smaller spaces arise as moduli spaces
of semistable holomorphic parabolic vector bundles with flags and weights
determined by the chosen conjugacy classes. In the general case,
the Poisson algebra on R gives a description of the variation of the
stratified symplectic Poisson structures on the smaller representation spaces
as the chosen conjugacy classes move. The space R is obtained by symplectic
reduction applied to a suitable extended moduli space whose construction,
in turn, involves, an appropriate fundamental groupoid.