Abstract:

Abstract: Gromov-Witten theory for projective varieties has been the subject of intense research since its intruduction. Many important results in the area (eg. Givental's mirror theorem) were proven using equivariant localization techniques from topology, using natural group actions. For a symplectic manifold X, GW invariants are defined using moduli spaces pseudoholomophic maps u:Σ o X from a Riemann surface Σ to X. Suppose that a compact Lie group is acting on X in a Hamiltonian way. After an introduction to the general theory, I will introduce the "space of vortices" which are pairs (A,u) of a connection over a principal bundle P, and a section u:Σ satisfying certain "gauged" equations. Using these spaces one can define gauged Gromov-Witten invariants, which are an equivariant version of Gromov-Witten invariants. This theory depends on a choice of area form on Σ as well as Σ. I will describe joint work with C. Woodward regarding the dependency on the area form, as well as joint work with Ott, Ziltener and Woodward on punctured surfaces. I will also give relations with other equivariant theories and I will discuss a conjecture related to the Gromov-Witten theory of symplectic quotients.