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  of a
correspondence which relates quantization of integrable system to classical
symplectic geometry .
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 . 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  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 , 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
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
The direct comparison of the results establishes the desired identity.