• Date:  2023-02-21 (Tue) 10:30 ~ 12:00 

           2023-02-23 (Thu) 10:30 ~ 12:00 

                       2023-02-28 (Tue) 10:30 ~ 12:00 

• Place:  129-104 (SNU)
• Title:  An introduction to geometric representation theory and 3d mirror symmetry
• Speaker:  Justin Hilburn (Perimeter Institute)
• Abstract:

The Beilinson-Bernstein theorem, which identifies representations of semi-simple Lie algebra \mathfrak{g} with D-modules on the flag variety G/B, makes it possible to use powerful techniques from algebraic geometry, especially Hodge theory, to attack problems in representation theory. Some successes of this program are the proofs of the Kazhdan-Lusztig and Jantzen conjectures as well as discovery that the Bernstein-Gelfand-Gelfand categories O for Langlands dual Lie algebras are Koszul dual.

 

The modern perspective on these results places them in the context of deformation quantizations of holomorphic symplectic manifolds: The universal enveloping algebra U(\mathfrak{g}) is isomorphic to the ring of differential operators on G/B which is a non-commutative deformation of the ring of functions on the cotangent bundle T^*G/B. Thanks to work of Braden-Licata-Proudfoot-Webster it is known that an analogue of BGG category O can be defined for any associative algebra which quantizes a conical symplectic resolution. Examples include finite W-algebras, rational Cherednik algebras, and hypertoric enveloping algebras.

 

Moreover BLPW collected a list of pairs of conical symplectic resolutions whose categories O are Koszul dual. Incredibly, these “symplectic dual” pairs had already appeared in physics as Higgs and Coulomb branches of the moduli spaces of vacua in 3d N=4 gauge theories.  Moreover, there is a duality of these field theories known as 3d mirror symmetry which exchanges the Higgs and Coulomb branch. Based on this observation Bullimore-Dimofte-Gaiotto-Hilburn showed that the Koszul duality of categories O is a shadow of 3d mirror symmetry.

 

In this series of lectures I will give an introduction to these ideas assuming only representation theory of semi-simple Lie algebras and a small amount of algebraic geometry.