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$SL(2,\mathbb{Z})$ dualities of boundary conditions in Abelian M2-brane SCFTs

Tadashi Okazaki, Douglas J. Smith

TL;DR

The paper investigates boundary conditions in Abelian M2-brane SCFTs and uncovers an action of SL(2,$\mathbb{Z}$) dualities on chiral boundary data. It develops and matches half-indices and ’t Hooft anomalies for a web of dual pairs, including ADHM theories, ADHM quivers, ABJM, and circular quiver CS theories, using brane realizations in Type IIB to motivate the dualities. A key result is that diagonal breaking of U(1) gauge factors in ABJM-like setups yields a residual $\mathbb{Z}_k$ gauge theory on the boundary, which pairs with dual ADHM boundary data under $S$- and $T$-transformations. The findings illuminate a rich boundary-duality structure in 3d $\mathcal{N}=4$ and $\mathcal{N}=2$ contexts, with precise, testable index and anomaly equivalences that could guide future holographic and defect-operator studies.

Abstract

We propose $SL(2,\mathbb{Z})$ dualities of supersymmetric boundary conditions in the three-dimensional supersymmetric field theories describing a semi-infinite M2-brane terminating on M5-branes. Specifically, we present dualities of boundary conditions for Abelian (quiver) ADHM theories and circular quiver Chern-Simons matter theories including the ABJM model. For the circular quiver Chern-Simons theories we take boundary conditions breaking a $U(1)_1 \times U(1)_{-1}$ gauge group to its diagonal subgroup which is decoupled. This can be generalized to break $U(1)_k \times U(1)_{-k}$, leaving a $\mathbb{Z}_k$ gauge theory. We find matching of the 't Hooft anomalies and supersymmetric half-indices for all the proposed dual boundary conditions.

$SL(2,\mathbb{Z})$ dualities of boundary conditions in Abelian M2-brane SCFTs

TL;DR

The paper investigates boundary conditions in Abelian M2-brane SCFTs and uncovers an action of SL(2,) dualities on chiral boundary data. It develops and matches half-indices and ’t Hooft anomalies for a web of dual pairs, including ADHM theories, ADHM quivers, ABJM, and circular quiver CS theories, using brane realizations in Type IIB to motivate the dualities. A key result is that diagonal breaking of U(1) gauge factors in ABJM-like setups yields a residual gauge theory on the boundary, which pairs with dual ADHM boundary data under - and -transformations. The findings illuminate a rich boundary-duality structure in 3d and contexts, with precise, testable index and anomaly equivalences that could guide future holographic and defect-operator studies.

Abstract

We propose dualities of supersymmetric boundary conditions in the three-dimensional supersymmetric field theories describing a semi-infinite M2-brane terminating on M5-branes. Specifically, we present dualities of boundary conditions for Abelian (quiver) ADHM theories and circular quiver Chern-Simons matter theories including the ABJM model. For the circular quiver Chern-Simons theories we take boundary conditions breaking a gauge group to its diagonal subgroup which is decoupled. This can be generalized to break , leaving a gauge theory. We find matching of the 't Hooft anomalies and supersymmetric half-indices for all the proposed dual boundary conditions.

Paper Structure

This paper contains 30 sections, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure
  3. Future works
  4. Boundary conditions for M2-brane SCFTs
  5. M-theory brane setup
  6. $U(N)$ ADHM boundary conditions
  7. ADHM theory with $(\mathcal{N},N,N)+\widetilde{\Gamma} + \widetilde{\eta}$
  8. ADHM theory with $(\mathcal{D},D,D)$
  9. ADHM quiver boundary conditions
  10. ADHM quiver theory with $(\mathcal{N},N,N)+\Gamma+\eta$
  11. ADHM quiver theory with $(\mathcal{D},D,D)$
  12. ABJM and circular quiver Chern-Simons boundary conditions
  13. Diagonal gauge group breaking -- $\mathbb{Z}_k$ gauge theory
  14. Half-indices
  15. $U(N)$ ADHM theory
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Abelian dualities with mapping of boundary $(p, q)$ 5-branes for the ADHM and ABJM theories.
  • Figure 2: Abelian dualities with mapping of boundary $(p, q)$ 5-branes for the ADHM theories and circular quiver Chern-Simons theories.