Quantum difference equations, monodromies and mirror symmetry

Abstract: An important enumerative invariant of a symplectic variety $X$ is its vertex function. The vertex function is the analog of J-function in Gromov-Witten theory: it is the generating function for the numbers of rational curves in $X$.

In representation theory these functions feature as solutions of various $q$-difference and differential equations associated with $X$, with examples including qKZ and quantum dynamical equations for quantum loop groups, Casimir connections for Yangians and other objects.

In this talk I explain how these equations can be extracted from algebraic topology of symplectic dual variety $X^!$, also known as $3D$-mirror of $X$. This can be summarized as "identity" $$ \text{Enumerative geometry of }X = \text{algebraic topology of }X^! $$ The talk is based on work in progress with Y.Kononov arXiv:2004.07862; arXiv:2008.06309.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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