Infinite dimensional geometric invariant theory and gauged Gromov-Witten theory

Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work with Andres Fernandez Herrero applying this general machinery to the stack of maps from a curve C to a quotient stack X/G, where G is a reductive group and X is an affine G-scheme. Our main immediate application is to compute generating functions for K-theoretic gauged Gromov-Witten invariants. The method we develop to analyze this moduli problem is an infinite dimensional analog of geometric invariant theory, which is potentially applicable to a much broader range of moduli problems.