Darboux coordinates, Yang-Yang functional, and gauge theory

Abstract

The moduli space of SL2 flat connections on a punctured Riemann surface Σ with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL2-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N = 2 supersymmetric theory subject to the two-dimensional Ω-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case.TCD MATH 11-05 HMI-11-04 IHES/P/11/16 ITEP-TH-16/11

Knot invariants from four-dimensional gauge theory

“…It has been argued [63] that the S-duality between these two branes is important in understanding the AGT correspondence [64] as well as recent developments relating supersymmetric gauge theory and integrable systems [65]. The role of the S-duality between B cc and B oper is more explicit in [66].…”

“…Recently [66], a "noncompact" version of the Gaudin model has been described in which the finite-dimensional representations R a are replaced by infinite-dimensional ones. The eigenvectors of the commuting Hamiltonians are again expressed as opers, now with certain conditions on their monodromies.…”

Supersymmetric field theories and geometric Langlands: The other side of the coin

“…. , a 3g−3+n ) representing a Poisson-commuting half of the system of complex Fenchel-Nielsen coordinates g) used in a related context in [5]. The orbits are mapped via holonomy map and the NAH correspondence to the Hitchin moduli space M H (G).…”

“…• Y σ (a; τ ; ǫ 1 ) is the generating function for the Lagrangian submanifold Op sl 2 (C τ ) of opers, considered in a related context in [5], with a being the complex Fenchel-Nielsen coordinates for Op sl 2 (C τ ) defined from the traces of holonomies along a set of curves on C defining the pants decomposition σ.…”

“…It overlaps with another family of results for which Nekrasov and collaborators have introduced the name BPS/CFT-correspondence, see [2] for the first in a series of papers on this subject and references to previous work in this direction. An older development, the relations between the N = 4 super-Yang-Mills theory (SYM) and the geometric Langlands correspondence exhibited by Kapustin and Witten [3], naturally fits into the emerging picture [4,5]. One may note, however, that the approach of Beilinson and Drinfeld to the geometric Langlands correspondence has close connections to CFT, see [6] for a review and further references, which are not obvious in the approach of Kapustin and Witten. Some connections between these subjects have been discussed in [4,7,5,8], and the recent work [9,10] indicates that at least a part of these relations admit a further deformation, motivated by supersymmetric gauge and string theory.…”