**Title:** Symplectic Duality and Mirror Symmetry of 3d N=4 Theories

**Speaker:** Justin Hilburn

**Speaker Info:** University of Oregon

**Brief Description:**

**Special Note**:

**Abstract:**

By now it is well known that many features of the representation theory of semisimple Lie algebras can in fact be generalized to to any noncommutative algebra A that arises as the quantization of functions on a symplectic cone M_0. In particular, given a Hamiltonian C^*-action on M_0 with isolated fixed points, one can define a category of A-modules that is analogous to the classical BGG category O. Furthermore it has been observed that symplectic cones seem to appear in "symplectic dual" pairs such that the associated categories O are Koszul dual.It is the case that all known "symplectic dual" pairs have appeared the physics literatures as Higgs and Coulomb branches of the moduli space of vacua in 3d N=4 SUSY field theories. Furthermore mirror symmetry of such theories is known to exchange the Higgs and Coulomb branches so it seems clear that symplectic duality and 3d mirror symmetry should be the same phenomenon.

There are two major obstacles to making this statement precise. The first is that there was no mathematically precise definition of the Coulomb branch and the second is that the physical meaning of category O and its behavior under mirror symmetry were unknown.

In this talk I will describe the progress that has been made in overcoming these obstacles in the last year.

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