# Seminars & Colloquia Calendar

## 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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