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FIELD Math:Geometry
DATE February 18 (Thu), 2021
TIME 16:00-17:00
PLACE Online
SPEAKER Wen, Yaoxiong
HOST Bhamidi, S.S Sreedhar
INSTITUTE Peking University
TITLE 3d mirror symmetry
ABSTRACT One of the recent remarkable discoveries is the connection between quantum K-theory and 3d TQFT. For a long time, quantum K-theory has been viewed as a variant to quantum cohomology, which comes from a 2d TQFT. 3d physics has its own mirror symmetry phenomenon, which falls into two versions, for $\mathcal{N}=4$ theories versus $\mathcal{N}=2$ theories. There are many mathematical results on the enumerative-geometric aspect of 3d $\mathcal{N}=4$ mirror symmetry by Okounkov's group. Our main interest is in 3d $\mathcal{N}=2$ theories, which apply to a general K?hler manifold. Its mirror symmetry was poorly understood even in physics.

In this talk, I will introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N}=2$ abelian mirror symmetry construction in physics introduced by Dorey-Tong. More precisely, for a short exact sequence
0 -> Z^k -> Z^n -> Z^{n-k} -> 0,
we consider the toric Artin stack $[C^n/(C^*)^k]$, and its mirror is given by the Gale dual of the above exact sequence, i.e., $[C^n/(C^*)^{n-k}]$. We introduce the modified equivariant K-theoretic I-functions for the mirror pair; they are defined by the contribution of fixed points. Under the mirror map, which switches the K?lher parameters and equivariant parameters and maps $q$ to $q^{-1}$, we see that modified I-functions with the effective level structure of mirror pair coincide. This talk is based on the joint work with Yongbin Ruan and Zijun Zhou.
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