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Opers, surface defects, and Yang-Yang functional

Mathematical Physics Seminar

Speaker: Saebyeok Jeong, SUNY Stony Brook/Simons Center
Location: 2112 MSB
Start time: Fri, Oct 19 2018, 12:00PM

In this talk, I will introduce a gauge theoretical verification of a correspondence which relates two seemingly distinct quantization schemes for the Hitchin integrable systems [1].

First, I will briefly review the hyper-Kahler geometry of the Hitchin moduli space, especially as the moduli space of flat connections when viewed as a holomorphic symplectic reduction. There is a distinct Lagrangian submanifold, spanned by certain differential operators called opers, in the moduli space of flat connections. The quantization procedure provided by Beilinson and Drinfeld [2] identifies the holomorphic functions on the space of opers with the (off-shell) spectra of the quantum Hitchin Hamiltonians. The problem is how to understand this mysterious 'quantum/classical duality' in gauge theoretical language.

In the context of gauge theory, the Hitchin integrable systems are associated to the low-energy descriptions of the d=4, N=2 gauge theories of class S. The Bethe/gauge correspondence developed by Nekrasov and Shatashvili [3] accomplishes the quantization of the Hitchin system by putting the class S theory on the two-dimensional Omega-background. I will show how the aforementioned quantization procedures are actually related to each other (the correspondence is firstly conjectured by Nekrasov, Rosly, and Shatashvili [4] for the lowest rank case).

More precisely, I will show the effective twisted superpotential of the class S theory on the two-dimensional Omega-background is identical to the generating function for the space of opers in a specific Darboux coordinate system (of higher-rank Fenchel-Nielsen type, roughly). The verification involves the following key ingredients: 1) The half-BPS codimension two (surface) defects of the class S theories can be used to construct the opers and their solutions. 2) The solutions constructed in this way have convergence domains, but they are connected to each other by analytic continuations. 3) A Darboux coordinate system, which generalizes the Fenchel-Nielsen coordinate system to arbitrary ranks, is constructed.

The comparison between the holonomies of flat connections expressed in the suggested Darboux coordinates and the monodromies of opers expressed in exact gauge theoretical terms establishes the desired equality.



Seminar starts at 12:00 (not 12:10)