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Exact results for vortex loop operators in 3d supersymmetric theories

Nadav Drukker, Takuya Okuda, Filippo Passerini

TL;DR

The paper develops an exact framework for vortex loop operators in 3d ${\cal N}=2$ theories by applying SUSY localization on $S^3$, $S^3_b$, and $S^1\times S^2$. It classifies 1/2-BPS loops, derives their BPS backgrounds, and computes exact partition functions and loop correlators via spectral analysis and transversally elliptic index theory, revealing systematic imaginary shifts in localization parameters induced by vorticity. It further demonstrates how abelian mirror symmetry exchanges flavor vortex loops and gauge Wilson loops, validating duality predictions through precise matching of shifted indices and correlators across dual pairs. Collectively, the results establish a robust, exact toolkit for vortex loop observables in 3d gauge theories and pave the way for exploring more general line and surface operators across dimensions.

Abstract

Three dimensional field theories admit disorder line operators, dubbed vortex loop operators. They are defined by the path integral in the presence of prescribed singularities along the defect line. We study half-BPS vortex loop operators for N=2 supersymmetric theories on S^3, its deformation S^3_b and S^1 x S^2. We construct BPS vortex loops defined by the path integral with a fixed gauge or flavor holonomy for infinitesimal curves linking the loop. It is also possible to include a singular profile for matter fields. For vortex loops defined by holonomy, we perform supersymmetric localization by calculating the fluctuation modes, or alternatively by applying the index theory for transversally elliptic operators. We clarify how the latter method works in situations without fixed points of relevant isometries. Abelian mirror symmetry transforms Wilson and vortex loops in a specific way. In particular an ordinary Wilson loop transforms into a vortex loop for a flavor symmetry. Our localization results confirm the predictions of abelian mirror symmetry.

Exact results for vortex loop operators in 3d supersymmetric theories

TL;DR

The paper develops an exact framework for vortex loop operators in 3d theories by applying SUSY localization on , , and . It classifies 1/2-BPS loops, derives their BPS backgrounds, and computes exact partition functions and loop correlators via spectral analysis and transversally elliptic index theory, revealing systematic imaginary shifts in localization parameters induced by vorticity. It further demonstrates how abelian mirror symmetry exchanges flavor vortex loops and gauge Wilson loops, validating duality predictions through precise matching of shifted indices and correlators across dual pairs. Collectively, the results establish a robust, exact toolkit for vortex loop observables in 3d gauge theories and pave the way for exploring more general line and surface operators across dimensions.

Abstract

Three dimensional field theories admit disorder line operators, dubbed vortex loop operators. They are defined by the path integral in the presence of prescribed singularities along the defect line. We study half-BPS vortex loop operators for N=2 supersymmetric theories on S^3, its deformation S^3_b and S^1 x S^2. We construct BPS vortex loops defined by the path integral with a fixed gauge or flavor holonomy for infinitesimal curves linking the loop. It is also possible to include a singular profile for matter fields. For vortex loops defined by holonomy, we perform supersymmetric localization by calculating the fluctuation modes, or alternatively by applying the index theory for transversally elliptic operators. We clarify how the latter method works in situations without fixed points of relevant isometries. Abelian mirror symmetry transforms Wilson and vortex loops in a specific way. In particular an ordinary Wilson loop transforms into a vortex loop for a flavor symmetry. Our localization results confirm the predictions of abelian mirror symmetry.

Paper Structure

This paper contains 50 sections, 309 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Half-BPS loop operators
  3. Half-BPS Wilson loop
  4. Half-BPS vortex loop operator: Vector multiplet
  5. Half-BPS vortex loop operator: Matter multiplet
  6. Abelian theory
  7. Non-abelian theory
  8. Vortex and Wilson loop operators for flavor symmetries
  9. Localization on $\mathbb{S}^3$ and harmonic analysis
  10. Classical factor
  11. Fluctuation determinant
  12. Spectral analysis
  13. Localization on $\mathbb S^3_b$ by index theory
  14. Partition function
  15. Vortex loop expectation values
  16. ...and 35 more sections

Figures (3)

  • Figure 1: Fluctuation modes of a chiral multiplet. The lattices represent states with principle quantum numbers $n=0,1,2$ and the allowed values of $m$ and $m'$. In the presence of the vortex loop these multiplets are broken to smaller ones encapsulated by the ovals. Only the short representations (with two modes) contribute to the determinant.
  • Figure 2: After introducing $\eta=1/2$ the entire spectrum in Figure \ref{['fig:lattice']} is shifted by $m\to m+\eta/2$ and $m'\to m'+\eta/2$. For the multiplets under the dashed line the principle quantum number is shifted $n\to n-\eta$ and above the dashed line $n\to n+\eta$, which effects the determinant.
  • Figure 3: For $\eta=1$ the spectrum is shifted by a full integer. Here are the new states with principle quantum numbers 0, 1 and 2. The states above the dashed line come from the original multiplet with $n-\eta$ and those below from $n+\eta$. Compared to the spectrum in Figure \ref{['fig:lattice']}, with the same value of $n$, there are the same number of $\phi$ modes, but an extra $\{\psi^+,F\}$ short multiplet, and one $\{\phi,\psi^+\}$ short multiplet gets enlarged by an extra $\psi^-$ and $F$ mode.