Boundary Chiral Algebras and Holomorphic Twists

TL;DR

The paper investigates the holomorphic twist of 3d ${ m N}=2$ theories with boundaries, revealing that bulk local operators form a commutative chiral (vertex) algebra with a degree-1 shifted Poisson structure, while boundary operators organize into noncommutative chiral algebras that are modules over the bulk algebra. It develops a BV-BRST framework to compute bulk and boundary algebras in explicit Lagrangian theories, including free chirals, Landau-Ginzburg models, and gauge theories with matter and CS terms, and identifies a bulk-boundary map into the center of the boundary algebra. A key feature is the derived structure: bulk-boundary centers, shifted symplectic geometry, and potential higher $A_ abla$-like operations on boundary algebras, suggesting rich homological and deformation-theoretic content. The work demonstrates RG-flow invariance of these algebras, tests dualities via explicit examples (e.g., SQED–XYZ and level-rank dualities), and connects to half-indices and 3d indices through character formulas. Overall, the framework provides a robust algebraic handle on boundary phenomena in holomorphic twists, with wide-ranging implications for dualities, line operators, and boundary degrees of freedom in 3d gauge theories.

Abstract

We study the holomorphic twist of 3d ${\cal N}=2$ gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral algebras (\emph{a.k.a.} vertex operator algebras). The bulk algebra is commutative, endowed with a shifted Poisson bracket and a "higher" stress tensor; while the boundary algebra is a module for the bulk, may not be commutative, and may or may not have a stress tensor. We explicitly construct bulk and boundary algebras for free theories and Landau-Ginzburg models. We construct boundary algebras for gauge theories with matter and/or Chern-Simons couplings, leaving a full description of bulk algebras to future work. We briefly discuss the presence of higher A-infinity like structures.

Figures (3)

• Figure 1: Separating operators in the $t$-direction to show that their OPE must be nonsingular.
• Figure 2: Bringing a bulk operator ${\cal O}$ to the boundary defines the bulk-boundary map $\beta({\cal O})$. The OPE of $\beta({\cal O})$ any other boundary operator ${\cal O}'$ must be nonsingular, since the two operators can be separated in $t$.
• Figure 3: Showing that $\beta({\cal O}_1)=\beta({\cal O}_2)=0$ implies $\beta(\{\!\!\{{\cal O}_1,{\cal O}_2\}\!\!\})=0$, up to $Q$-exact terms.

Theorems & Definitions (1)

• Conjecture