Representations, Moduli and Duality

The notion of duality in physics has a rich history, going back at least as far as the observation that Maxwell’s theory of elctromagnetism is symmetric after swapping electric and magnetic fields. More generally, dualities in physics can provide a way of relating two seemingly very different physical theories via a nontrivial duality transformation. For instance, S-duality in physics provides a way to swap strongly coupled physical theories for weakly coupled ones [MO77], for which we may use perturbative methods to exactly solve the equations governing the system. Such dualities in physics are often translated into deep conjectures and programs of research in mathematics [KW07]. This proposal centers on one in particular: 3d mirror symmetry [IS96]. Via this symmetry, the Coulomb branch of the physical theory is exchanged 1 with the Higgs branch. This idea has deep implications, cutting across multiple areas of mathematics, since various spaces that have become central objects in mathematics over the last forty years turn out to be moduli spaces of states in these two branches. Moduli spaces arise in mathematics from the observation that a set of objects that we may be studying (be they representations, sheaves, or spaces themselves) can often be naturally arranged into something richer than a set: a space, with perhaps an algebraic, or symplectic, or even “higher” structure. The subject of geometric representation theory is the study of these spaces, along with geometrically defined homomorphisms between the invariants of these spaces [CG97]. Tracking through all this mathematics, the study of dualities in mathematics which formalise 3d mirror symmetry in physics provides deep new insights into moduli spaces and representation theory.