3d N=4 TQFT's

The research in this Fellowship lies at the interface of pure mathematics (algebra geometry and topology) and theoretical physics (quantum field theory). I will construct a new class of three-dimensional topological quantum field theories, the eponymous 3d N=4 TQFT's, and use a combination of techniques from physics, algebra, and geometry to understand and define their structure.

A hallmark of a 3d TQFT is that its physical properties only depend on the shape --- but not the size --- of three-dimensional spacetime. A classic example of such a TQFT, called Chern-Simons theory, was constructed in the 90's. Its quantum expectation values were used to distinguish shapes of knots and three-dimensional spaces.

The 3d N=4 TQFT's I construct come from taking sectors of supersymmetric gauge theories that behave topologically. They are similar to Chern-Simons in some ways, but infinitely richer and more complicated in others. On one hand, their spaces of quantum states are infinite rather than finite-dimensional, and their expectation values will take a great deal of care to properly define. On the the hand they come in pairs, related by a duality (an equivalence) called 3d Mirror Symmetry, which roughly implies that any one computation can be done in at least two completely different ways, from two different perspectives. In physical terms, 3d Mirror Symmetry says that particles and vortices moving around in three dimensions basically look the same, and will probe the shape of a three-dimensional space in equivalent ways.

My research takes such intuitive statements and turns them into rigorous mathematics. It turns out that the mathematical structure of 3d N=4 TQFT's is related to an astounding number of other areas of mathematics --- the fields of vertex operator algebras, geometric representation theory, mirror symmetry (an older type, inspired by string theory), and topology all get related in surprising new ways to 3d N=4 TQFT's, to 3d physics, and ultimately to each other.