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Chiral life on a slab

Sergey Alekseev, Mykola Dedushenko, Mikhail Litvinov

Abstract

We study chiral algebra in the reduction of 3D $\mathcal{N} = 2 $ supersymmetric gauge theories on an interval with the $\mathcal{N}=(0,2)$ Dirichlet boundary conditions on both ends. By invoking the 3D ``twisted formalism'' and the 2D $βγ$-description we explicitly find the perturbative $\overline{Q}_+$ cohomology of the reduced theory. It is shown that the vertex algebras of boundary operators are enhanced by the line operators. A full non-perturbative result is found in the abelian case, where the chiral algebra is given by the rank two Narain lattice VOA, and two more equivalent descriptions are provided. Conjectures and speculations on the nonperturbative answer in the non-abelian case are also given.

Chiral life on a slab

Abstract

We study chiral algebra in the reduction of 3D supersymmetric gauge theories on an interval with the Dirichlet boundary conditions on both ends. By invoking the 3D ``twisted formalism'' and the 2D -description we explicitly find the perturbative cohomology of the reduced theory. It is shown that the vertex algebras of boundary operators are enhanced by the line operators. A full non-perturbative result is found in the abelian case, where the chiral algebra is given by the rank two Narain lattice VOA, and two more equivalent descriptions are provided. Conjectures and speculations on the nonperturbative answer in the non-abelian case are also given.
Paper Structure (19 sections, 2 theorems, 113 equations, 5 figures, 2 tables)

This paper contains 19 sections, 2 theorems, 113 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation via VOA$[M_4]$.
  3. The rest of this paper is structured as follows.
  4. Basics
  5. Conventions:
  6. Basic Supersymmetry
  7. Holomorphic-topological twist
  8. beta-gamma System
  9. (0,2) Cohomology And Beta-Gamma System
  10. 3D Perspective
  11. Vector Multiplet
  12. The non-perturbative corrections
  13. U(1)
  14. 2D Perspective or Beta-Gamma System
  15. U(1)
  16. ...and 4 more sections

Key Result

Proposition 1

For any $[\alpha]\in \mathcal{Z}_d\left( \Omega^{3,0}\oplus \Omega^{2,1} \right) /d \Omega^{2,0}$ there exists $\beta\in {\mathcal{Z}}_d\left(\Omega^{3,0}\right)$ such that so there is an isomorphism: where we have made use of a slight abuse of notation, and $d \Omega^{2,0}$ in the last quotient should be understood as $d\Omega^{2,0}\cap \Omega^{3,0}$.

Figures (5)

  • Figure 1: Two boundaries with a line operator connecting them.
  • Figure 2: The diagram connecting single $\mathcal{A}$ in the Wilson line to the operator $B$ at the boundary.
  • Figure 3: Possible codimension one surfaces over which $T_{z\nu}$ is integrated to generate holomorphic translations along the boundary.
  • Figure 4: The bulk monopole $M_p$ is connected by a Wilson line of charge $kp$ to the Dirichlet boundary. In the $Q$-cohomology, due to the topological invariance in the $t$ direction, this is equivalent to the boundary monopole $M_p$.
  • Figure 5: Moving a monopole of magnetic charge $p$ from one boundary to another creates a Wilson line of electric charge $kp$.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof