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Mirror Symmetry And Loop Operators

Benjamin Assel, Jaume Gomis

TL;DR

The paper resolves how Wilson loops in 3d N=4 theories transform under three-dimensional mirror symmetry by identifying them with Vortex loop operators in the mirror theory. It develops a UV framework where both Wilson and Vortex loops are realized via 3d/1d coupled systems and uses Type IIB brane constructions, S-duality, and Hanany-Witten moves to derive explicit mirror maps for a wide class of linear and circular quivers, including T[SU(N)] and SQCD. The authors validate these maps with exact S^3 computations, combining 3d localization with 1d SQM indices and revealing intricate structures such as mass-FI exchange, background Wilson factors, and hopping duality. The results provide a robust, brane-driven algorithm for translating loop operators across mirror pairs, with potential implications for other dualities such as Seiberg duality in 4d.

Abstract

Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely different gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic definition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically different. Our predictions for the mirror map, which we derive guided by branes in string theory, are confirmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.

Mirror Symmetry And Loop Operators

TL;DR

The paper resolves how Wilson loops in 3d N=4 theories transform under three-dimensional mirror symmetry by identifying them with Vortex loop operators in the mirror theory. It develops a UV framework where both Wilson and Vortex loops are realized via 3d/1d coupled systems and uses Type IIB brane constructions, S-duality, and Hanany-Witten moves to derive explicit mirror maps for a wide class of linear and circular quivers, including T[SU(N)] and SQCD. The authors validate these maps with exact S^3 computations, combining 3d localization with 1d SQM indices and revealing intricate structures such as mass-FI exchange, background Wilson factors, and hopping duality. The results provide a robust, brane-driven algorithm for translating loop operators across mirror pairs, with potential implications for other dualities such as Seiberg duality in 4d.

Abstract

Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely different gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic definition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically different. Our predictions for the mirror map, which we derive guided by branes in string theory, are confirmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.

Paper Structure

This paper contains 32 sections, 155 equations, 40 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Loop Operators in 3d $\cN=4$ Theories
  3. Wilson Loop Operators
  4. Vortex Loop Operators
  5. Brane Realization Of Loop Operators and Mirror Map
  6. Brane Realization Of Wilson Loop Operators
  7. Mirror Of Wilson Loops From S-duality
  8. Mirror Symmetry and Loop Operators: Examples
  9. $T[SU(2)]$
  10. Wilson Loops In The $U(N-1)$ Node Of $T[SU(N)]$
  11. The Other $T[SU(N)]$ Loops
  12. Loops And Mirror Map In Circular Quivers
  13. Loops In SQCD with $2N$ Quarks And Its Mirror
  14. Loop Operators In 3d $\cN=4$ Theories On $S^3$
  15. Wilson Loops On $S^3$
  16. ...and 17 more sections

Figures (40)

  • Figure 1: General linear and circular quiver diagrams.
  • Figure 2: Dotted arrows reflect the possibility of adding an adjoint chiral multiplet in a gauge node.
  • Figure 3: 3d/1d quiver realizing a Vortex loop in the $T[SU(N)]$ theory.
  • Figure 4: 3d/1d quiver theories realizing the same Vortex loop.
  • Figure 5: 1d quiver theory coupled to 3d quiver theory by gauging 1d flavor symmetries with 3d vector mutiplets (dynamical or weakly gauged).
  • ...and 35 more figures