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Darboux coordinates, Yang-Yang functional, and gauge theory

Nikita Nekrasov, Alexey Rosly, Samson Shatashvili

TL;DR

The paper establishes a geometric framework in which the Yang–Yang functional of quantum Hitchin systems is realized as the generating function of SL(2) oper varieties in holomorphic Darboux coordinates on the moduli space of flat connections. By embedding this construction into the 4d N=2 gauge theory with a 2d Omega-background, the twisted superpotential is shown to encode the Yang–Yang function up to a τ-independent piece, yielding a Bethe/Gauge correspondence for the rank one case. The work ties the quantum spectrum to intersections of Lagrangian submanifolds in the local moduli space M^{loc}, with explicit Darboux charts provided via pant decompositions and detailed oper descriptions in genus zero and one, including degenerate limits and boundary branes. It further connects to instanton counting, conformal blocks, and Liouville theory, offering concrete checks in SU(2) theories and proposing a construction of the second brane L_γ through dilogarithm-generating functions. The framework invites generalization to other groups, richer Omega-backgrounds, and irregular singularities, potentially illuminating facets of the geometric Langlands program and quantum Liouville/Toda theories.

Abstract

The moduli space of SL(2) flat connections on a punctured Riemann surface 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 SL(2)-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 Omega-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.

Darboux coordinates, Yang-Yang functional, and gauge theory

TL;DR

The paper establishes a geometric framework in which the Yang–Yang functional of quantum Hitchin systems is realized as the generating function of SL(2) oper varieties in holomorphic Darboux coordinates on the moduli space of flat connections. By embedding this construction into the 4d N=2 gauge theory with a 2d Omega-background, the twisted superpotential is shown to encode the Yang–Yang function up to a τ-independent piece, yielding a Bethe/Gauge correspondence for the rank one case. The work ties the quantum spectrum to intersections of Lagrangian submanifolds in the local moduli space M^{loc}, with explicit Darboux charts provided via pant decompositions and detailed oper descriptions in genus zero and one, including degenerate limits and boundary branes. It further connects to instanton counting, conformal blocks, and Liouville theory, offering concrete checks in SU(2) theories and proposing a construction of the second brane L_γ through dilogarithm-generating functions. The framework invites generalization to other groups, richer Omega-backgrounds, and irregular singularities, potentially illuminating facets of the geometric Langlands program and quantum Liouville/Toda theories.

Abstract

The moduli space of SL(2) flat connections on a punctured Riemann surface 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 SL(2)-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 Omega-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.

Paper Structure

This paper contains 32 sections, 120 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Bethe ansatz
  3. Gauge theories and quantum integrability
  4. Gauge theories from six dimensions
  5. The $\Omega$-background
  6. Two dimensional flat connections and higher dimensional gauge theory
  7. The hyperkähler structure
  8. From ${\mathcal{N}}=2$ 4d gauge theory to 2d ${\mathcal{N}}=(4,4)$ sigma model
  9. $\Omega$-deformation as the boundary condition
  10. Two two dimensional theories
  11. Twisted superpotential as the generating function
  12. Geometry of the moduli space of flat connections
  13. Moduli of flat connections: explicit description
  14. The symplectic form
  15. The case of a sphere with four punctures
  16. ...and 17 more sections

Figures (9)

  • Figure 1: Generators of ${\pi}_{1} \left( {\bf S}^{2} \backslash \{ z_{1}, \ldots , z_{n} \} \right)$
  • Figure 2: The intersecting loops ${\gamma}_{1}$, ${\gamma}_{2}$ and the simple loops ${\gamma}_{x,1,2}^{\pm}$
  • Figure 3: The $A$ and $B$ loops
  • Figure 4: The $C^{+}$ and $C^{-}$ loops
  • Figure 5: The genus $0$ edges
  • ...and 4 more figures