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Dual Boundary Conditions in 3d SCFT's

Tudor Dimofte, Davide Gaiotto, Natalie M. Paquette

TL;DR

The paper develops a comprehensive framework for half-BPS boundary conditions in 3d ${\mathcal N}=2$ gauge theories, preserving ${\mathcal N}=(0,2)$ at the boundary and matching UV boundary data through IR dualities via duality interfaces. Central to the approach is the 3d half-index, extended to Dirichlet boundary conditions and boundary monopole sectors, which serves as a nonperturbative check of dualities across abelian and nonabelian theories, including Aharony duality, level-rank dualities, and particle-vortex triality. The authors systematically construct dual boundary conditions and interfaces for a large class of dual pairs (SQED/XYZ, SQCD/detYZ, level-rank pairs, Aharony dual pairs) and verify anomaly matching and half-index identities, often via explicit monopole-sum computations and boundary Fermi/chiral multiplet couplings. The results illuminate deep links between 3d boundary dynamics and 2d boundary CFTs (e.g., WZW and coset models), and outline a broad program to extend boundary dualities to more general SUSY settings and to non-supersymmetric dualities.

Abstract

We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d $\mathcal{N}=2$ gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation of supersymmetric indices. Examples include basic abelian dualities, level-rank dualities, and Aharony dualities.

Dual Boundary Conditions in 3d SCFT's

TL;DR

The paper develops a comprehensive framework for half-BPS boundary conditions in 3d gauge theories, preserving at the boundary and matching UV boundary data through IR dualities via duality interfaces. Central to the approach is the 3d half-index, extended to Dirichlet boundary conditions and boundary monopole sectors, which serves as a nonperturbative check of dualities across abelian and nonabelian theories, including Aharony duality, level-rank dualities, and particle-vortex triality. The authors systematically construct dual boundary conditions and interfaces for a large class of dual pairs (SQED/XYZ, SQCD/detYZ, level-rank pairs, Aharony dual pairs) and verify anomaly matching and half-index identities, often via explicit monopole-sum computations and boundary Fermi/chiral multiplet couplings. The results illuminate deep links between 3d boundary dynamics and 2d boundary CFTs (e.g., WZW and coset models), and outline a broad program to extend boundary dualities to more general SUSY settings and to non-supersymmetric dualities.

Abstract

We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation of supersymmetric indices. Examples include basic abelian dualities, level-rank dualities, and Aharony dualities.

Paper Structure

This paper contains 64 sections, 342 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Future directions and additional motivations
  3. 2d ${\mathcal{N}}=(0,2)$ boundary conditions
  4. Free chiral
  5. Boundary conditions encoded in the action
  6. Boundary fields and boundary superpotentials
  7. Flips
  8. Multiple chirals
  9. Bulk superpotentials
  10. Example: XYZ
  11. Boundary 't Hooft anomalies
  12. Nonabelian symmetries
  13. Gauge fields
  14. Reduction of 3d gauge multiplets
  15. Basic ${\mathcal{N}}$, ${\mathcal{D}}$ boundary conditions
  16. ...and 49 more sections

Figures (3)

  • Figure 1: Using collision with a duality interface to generate dual boundary conditions. Note that both collisions and the bulk flows to the IR are RG flows that hold different parameters fixed. If the diagram commutes, then the collision of ${\mathcal{I}}$ and ${\mathcal{B}}$ defines a dual to ${\mathcal{B}}$.
  • Figure 2: Boundary local operators at the end of a line operator ${\mathcal{L}}$ form a module $M_{\mathcal{L}}$ for the boundary chiral algebra.
  • Figure 3: Defining a duality interface ${\mathcal{I}}$ between dual 3d theories ${\mathcal{T}}^\vee$ and ${\mathcal{T}}$ by starting with a factorization of the identity interface in ${\mathcal{T}}$ (on the LHS) and dualizing half the space. The interface has the necessary property (that it flows to the identity in the IR) provided that the "diagram commutes"; in particular the coupling on the RHS must involve the appropriate duals of the operators on the LHS.