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Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory

G. Vartanov, J. Teschner

TL;DR

The work develops a cohesive framework connecting four-dimensional N=2 class S gauge theories to Liouville conformal field theory by viewing the instanton partition function ${\\mathcal{Z}}^{\text{inst}}$ as a wavefunction in the zero-mode quantum mechanics of the quantized moduli space ${\\mathcal{M}}_{\\text{flat}}(C)$. It constructs two natural Darboux coordinate systems for ${\\mathcal{M}}_{\\text{flat}}(C)$, derives explicit kernels for duality transformations (S-, B-, F-, Z-moves) forming a projective unitary representation of the Moore–Seiberg groupoid, and identifies these kernels with Liouville conformal blocks. The paper then shows how the quantum Teichmüller theory provides a common language for both the gauge-theoretic and Liouville pictures, yielding a detailed Riemann–Hilbert problem whose solution is provided by Liouville theory. By embedding these results into the Omega-deformed gauge theory framework of Nekrasov and Witten, it offers a conceptual derivation of the AGT correspondence and clarifies the scheme-dependence arising from UV regularization. Overall, the work forges deep connections between gauge theory, quantum geometry of flat connections, Teichmüller theory, and Liouville CFT, with significant implications for non-perturbative dynamics and duality in supersymmetric systems.

Abstract

We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks.

Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory

TL;DR

The work develops a cohesive framework connecting four-dimensional N=2 class S gauge theories to Liouville conformal field theory by viewing the instanton partition function as a wavefunction in the zero-mode quantum mechanics of the quantized moduli space . It constructs two natural Darboux coordinate systems for , derives explicit kernels for duality transformations (S-, B-, F-, Z-moves) forming a projective unitary representation of the Moore–Seiberg groupoid, and identifies these kernels with Liouville conformal blocks. The paper then shows how the quantum Teichmüller theory provides a common language for both the gauge-theoretic and Liouville pictures, yielding a detailed Riemann–Hilbert problem whose solution is provided by Liouville theory. By embedding these results into the Omega-deformed gauge theory framework of Nekrasov and Witten, it offers a conceptual derivation of the AGT correspondence and clarifies the scheme-dependence arising from UV regularization. Overall, the work forges deep connections between gauge theory, quantum geometry of flat connections, Teichmüller theory, and Liouville CFT, with significant implications for non-perturbative dynamics and duality in supersymmetric systems.

Abstract

We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks.

Paper Structure

This paper contains 141 sections, 3 theorems, 344 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Riemann surfaces: Some basic definitions and results
  3. Complex analytic gluing construction
  4. The Moore-Seiberg groupoid
  5. Hyperbolic metrics vs. flat connections
  6. Hyperbolic pants decomposition and Fenchel-Nielsen coordinates
  7. Trace coordinates
  8. Topological classification of closed loops
  9. Trace functions
  10. Skein algebra
  11. Generators and relations
  12. Poisson structure
  13. Darboux coordinates for ${\mathcal{M}}_{\rm flat}(C)$
  14. Supersymmetric gauge theories
  15. Quantization of ${\mathcal{M}}_{\rm flat}(C)$ from gauge theory
  16. ...and 126 more sections

Key Result

Proposition 1

where the balancing condition is $\sum_{i=1}^3 \mu_i+\nu_i = Q.$

Figures (10)

  • Figure 1: A sphere with three punctures, and a choice of branch cuts for the definition of the logarithms of local coordinates around the punctures.
  • Figure 2: The B-move
  • Figure 3: The Z-move
  • Figure 4: The F-move
  • Figure 5: The S-move
  • ...and 5 more figures

Theorems & Definitions (4)

  • Claim 1
  • Proposition 1
  • Proposition 2
  • Proposition 3