Alladi Ramakrishnan Hall

Moment map and the vortex equation

Sushmita Venugopalan

Chennai Mathematical Institute

A moment map is a function associated to the Hamiltonian action of a Lie
group on a symplectic manifold. It is a function from the symplectic
manifold to the dual of the Lie algebra. The symplectic quotient is the
zero level set of this map quotiented out by the group action. By the Kempf-Ness theorem, when a complex reductive group acts on a non-singular variety, the GIT quotient coincides with the symplectic quotient.

In gauge theory, there are infinite dimensional analogues of this phenomenon. I present one such case involving the space of holomorphic fibre bundles on curves with some additional data. In this case the moment map corresponds to the vortex equation.