Symplectic Duality and 3d Mirror Symmetry

Symplectic Duality and 3d Mirror Symmetry

###### String Theory Seminar

#### Justin Hilburn (University of Pennsylvania)

**Thursday, April 19, 2018 -3:30pm to 4:30pm**

Three of the most important results in geometric representation theory are

- The Borel-Weil-Bott theorem (1954) which says that the finite dimensional representations of any semi-simple Lie group G can be realized as global sections of G-equivariant line bundles on the flag variety G/B.
- The Beilinson-Bernstein theorem (1981) which says that the category of (not necessarily finite dimensional) representations of the Lie algebra g is equivalent to the category of modules over the ring of differential operators (D-modules) on the flag variety. Beilinson and Bernstein also identified a certain subcategory of g-mod known as category O with the category D-modules whose restrictions to Schubert cells are flat connections.
- The Koszul duality theorem of Beilinson-Ginzburg-Soergel (1996) which proved a highly non-trivial equivalence between category O for g and category O for the Langlands dual Lie algebra g^L.

Building on work of Kashiwara-Rouquier, Gan-Ginzburg, and others Braden-Licata-Proufoot-Webster argued that the natural context for these results was the deformation quantization of conical symplectic resolutions: The ring of differential operators on G/B is a deformation quantization of the symplectic manifold T^*G/B and the Beilinson-Bernstein theorem is equivalent to the fact that T^*G/B resolves the singularities of the cone of nilpotent elements in g^*. Moreover, they showed that one can define a version of category O for any conical symplectic resolution X and conjectured that each symplectic resolution X had a symplectic dual X^! such that O(X) is Koszul dual to O(X^!).

Soon, it became clear that BLPW's list of conjectural symplectic dual pairs coincided with the list of known Higgs and Coulomb branches of 3d N=4 theories but the physical interpretation of category O was unclear until Bullimore, Dimofte, and Gaiotto, and I identified it with the category of (2,2)-boundary conditions. I will show how to interpret this work terms of the construction of the Coulomb branch by Braverman-Finkelberg-Nakajima. Part of this work is joint with Kamnitzer-Weekes.