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Loop operators and S-duality from curves on Riemann surfaces

Nadav Drukker, David R. Morrison, Takuya Okuda

TL;DR

The authors classify 1/2 BPS Wilson-'t Hooft loop operators in a class of 4D N=2 superconformal theories engineered from M5-branes on a Riemann surface, showing a precise correspondence between gauge-theory loop charges (magnetic $p_j$ and electric $q_j$, with Dirac and Weyl constraints) and Dehn–Thurston parameters $(p_j;q_j)$ describing non-self-intersecting curves on the associated punctured surface. They prove that Dehn’s theorem provides a one-to-one labeling of loop operators, and that duality acts via the mapping class group on these curves, yielding explicit transformation rules in several examples (notably SU(2) with $N_F=4$, and simple quivers). The work thus connects 4D N=2 loop operators to 2D surface geometry (via Fenchel–Nielsen coordinates and Penner’s PIL transformations) and furnishes concrete S-duality predictions while situating the results within M-theory and AdS/CFT contexts. Potential extensions to higher-rank groups (SU(N)) and to $T_N$ theories, as well as implications for Liouville theory and localization, are discussed as future directions.

Abstract

We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface--the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

Loop operators and S-duality from curves on Riemann surfaces

TL;DR

The authors classify 1/2 BPS Wilson-'t Hooft loop operators in a class of 4D N=2 superconformal theories engineered from M5-branes on a Riemann surface, showing a precise correspondence between gauge-theory loop charges (magnetic and electric , with Dirac and Weyl constraints) and Dehn–Thurston parameters describing non-self-intersecting curves on the associated punctured surface. They prove that Dehn’s theorem provides a one-to-one labeling of loop operators, and that duality acts via the mapping class group on these curves, yielding explicit transformation rules in several examples (notably SU(2) with , and simple quivers). The work thus connects 4D N=2 loop operators to 2D surface geometry (via Fenchel–Nielsen coordinates and Penner’s PIL transformations) and furnishes concrete S-duality predictions while situating the results within M-theory and AdS/CFT contexts. Potential extensions to higher-rank groups (SU(N)) and to theories, as well as implications for Liouville theory and localization, are discussed as future directions.

Abstract

We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface--the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

Paper Structure

This paper contains 8 sections, 65 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prelude: SU(2) gauge theory with NF=4
  3. Classification of loop operators in gauge theory
  4. Examples
  5. Curves on punctured Riemann surfaces
  6. Examples: checks and predictions
  7. Conclusions and discussion
  8. M2-branes as holographic loop operators

Figures (15)

  • Figure 1: (a) A "quiver" diagram that represents the $SU(2)$ gauge theory with $N_F=4$ flavors. (b) A sphere with four punctures obtained by fattening the quiver diagram. The geodesics we discuss are for the hyperbolic metric where each puncture has a $2\pi$ deficit angle and is at the end of an infinite tube. The red curve represents $\gamma$ represents the Wilson loop in the fundamental representation, while the green curve $\delta$ corresponds to the minimal 't Hooft loop.
  • Figure 2: Examples of $SU(2)$ quiver diagrams. (a) The ${\mathcal{N}}=2^*$ gauge theory (a mass deformation of ${\mathcal{N}}=4$ SYM) corresponds to a once-punctured torus. (b) Two quiver gauge theories that are dual to each other. They correspond to a genus two surface with no punctures.
  • Figure 3: (a) A once-punctured torus. (b) A genus two Riemann surface. Two S-dual realizations of the theory based on this surface are given in Figure \ref{['fig:SU(2)quivers']}(b).
  • Figure 4: A pair of pants with seams
  • Figure 5: A pair of pants, shown as a disk with two holes
  • ...and 10 more figures