Towards a Mathematical Theory of 3D Mirror Symmetry - Justin Hilburn

Slides are available here: www.math.stonybrook.edu/PDFs/...
Stony Brook University
Mathematics Colloquium
December 14, 2023
Speaker: Justin Hilburn, Perimeter Institute
Abstract:
Dualities between supersymmetric quantum field theories are like secret passages connecting different areas of mathematics. Two celebrated examples are (2d) mirror symmetry, a duality of 2d QFTs which connects algebraic geometry and symplectic topology, and S-duality, a duality of 4d QFTs which underlies the geometric Langlands program. 3d mirror symmetry is a closely related duality of 3d N=4 QFTs discovered by Intirilligator and Seiberg.
One striking mathematical consequence of 3d mirror symmetry is a conjectural equivalence between two 2-categories associated to a pair of 3d mirror hyperkahler manifolds. The first 2-category is of an algebro-geometric flavor and was studied by Kapustin-Rozansky-Saulina. The second 2-category is more mysterious and is expected to be constructed by counting solutions to the 3d generalized Seiberg-Witten equations.
In this talk I will explain what is known about this conjecture and the role it plays in mathematics. In particular we will see that it categorifies a mysterious phenomenon in geometric representation theory known as symplectic duality.