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Geometry, Symmetry and Physics Seminar

Opers, surface defects, and Yang-Yang functional

Saebyeok Jeong, Stony Brook University

Location:  Serin loung
Date & time: Thursday, 09 May 2019 at 2:30PM - 3:30PM

 In this talk, I will introduce a gauge theoretical derivation [1] of a
 correspondence which relates quantization of integrable system to classical
 symplectic geometry [2].
 First, I will begin with the construction of the class S theory T[C] by
 compactifying 6d N=(2,0) theory on a Riemann surface C, explaining the
 identification of the Coulomb moduli space of T[C] on R^3 X S^1 and the Hitchin
 moduli space on C. The Hitchin moduli space is hyper-Kahler, and its integrable
 structure becomes manifest when we view it through, say, the complex structure
 I. When this classical integrable system gets quantized, it becomes precisely
 the quantum integrable system which appears in the correspondence of Nekrasov
 and Shatashvili [3]. Meanwhile, we can also view the Hitchin moduli space
 through the complex structure J as the moduli space of (complex) flat
 connections on C. A natural question is how the quantization of the Hitchin
 integrable system is accounted for in this holomorphic symplectic geometry of
 the moduli space of flat connections.

 The conjecture of [2] states the following: there is a distinctive complex
 Lagrangian submanifold (called the submanifold of opers) of the moduli space of
flat connections, and the generating function of it is identical to the
 effective twisted superpotential of the corresponding class S theory T[C]. Since
 the effective twisted superpotential is also identified with the Yang-Yang
 functional of the Hitchin integrable system by the correspondence of [3], the
 conjecture establishes concrete connections between the quantum integrable
 system, supersymmetric gauge theory, and classical symplectic geometry.
 The gauge theoretical proof of the conjecture involves the following key
 ingredients:

     1) Use half-BPS codimension-two (surface) defects in the class S theory T[C]
 to construct the opers and their solutions.
     2) Analytically continue the surface defects partition functions to build
 connection formulas of the solutions.
     3) Construct a Darboux coordinate system relevant to the correspondence.
     4) Compute the monodromies of opers from 2) and compare with the expressions
 from 3).

 The direct comparison of the results establishes the desired identity.

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