Mathematics Colloquia and Seminars

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Symplectic duality, as defined by Braden-Licata-Proudfoot-Webster, is a relationship between pairs of symplectic cones X and X^!. The most interesting part of this relationship is a Koszul duality between certain categories O and O^! of deformation quantization modules on X and X^! respectively. Many interesting algebras arise as deformation quantization of symplectic cones such as universal enveloping algebras of semisimple Lie algebras, W-algebras, spherical subalgebras of type A rational Cherednik algebras, and truncated shifted Yangians.

Physicists noticed that all known symplectic dual pairs arise as Higgs and Coulomb branches of the moduli space of vacua in 3d N=4 supersymmetric gauge theories. Moreover it is known that there is a 3d mirror symmetry such that the Higgs branch of a theory is the Coulomb branch of its mirror and vice versa. This leads to the question of whether symplectic duality is a manifestation of 3d mirror symmetry. I will survey some attempts to answer this question.