This research project concerns the study of examples of topological gauge theories in low dimensions. Gauge theory, in special cases, is believed to underline the mathematical description of the universe; for example, the standard model for particle physics is a gauge theory in 4 dimensions. Much mathematical research into quantum physics over the last half-century has focussed on the analysis required to make accurate experimental predictions, but what was missed until the last decade or two was that topological (distance-independent) features of the theory are governed by an underlying algebra, which has some novel and unexplored features. The PI will study fundamental questions pertaining to this algebra in low dimension.
The main project aims at a rigid description of topological gauge theory (the so-called A-model) by the tested technique of mirror symmetry, which converts "soft" topological questions, defined up to coherent systems of homotopies, into rigid questions of analytic or algebraic geometry. The PI has a proposal for the mirror of pure gauge theory in 3 dimensions, which controls the gauging and 2-dimensional gauge theories such as arise from symplectic manifolds with Hamiltonian group actions. Another component of the research involves gauge theory in 3 dimensions, specifically Chern-Simons theory, which seems to lie half-way between the soft A- and the rigid B-model. Finally, a small project involving representation theory (geometric Langlands duality for algebraic curves) will be undertaken: the construction of a generic Langlands kernel for simply laced groups, which relies on cohomological and algebro-geometric techniques, but the subject is informed and inspired by electric-magnetic duality for gauge theory in 4 dimensions.
