[Paper Review] Chiral life on a slab

С. В. Алексеев, Mykola Dedushenko+1 more|arXiv (Cornell University)|Dec 30, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper studies chiral algebras in 3D $σ=2$ supersymmetric gauge theories compactified on an interval with $(0,2)$ boundary conditions. Using the holomorphic-topological twist and $β\gamma$ system formalism, it computes the perturbative $̅{Q}_+$ cohomology and shows that boundary vertex operator algebras (VOAs) are enhanced by line operators. In the abelian case, the non-perturbative chiral algebra is identified as the rank-two Narain lattice VOA, with three equivalent descriptions provided, while non-abelian generalizations are conjectured.

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.

Motivation & Objective

• To understand the structure of chiral algebras arising from the reduction of 3D $σ=2$ supersymmetric gauge theories on an interval with $(0,2)$ boundary conditions.
• To compute the perturbative $̅{Q}_+$ cohomology of the reduced theory using the holomorphic-topological twist and $β\gamma$ system formalism.
• To identify the non-perturbative chiral algebra in the abelian case and provide multiple equivalent descriptions.
• To explore the role of line operators in enhancing boundary VOAs and to conjecture the structure in the non-abelian case.

Proposed method

• Employing the 3D holomorphic-topological twist to simplify the supersymmetry structure and focus on chiral operators.
• Using the $β\gamma$ system to describe the cohomological structure of the reduced 2D theory.
• Computing the perturbative $̅{Q}_+$ cohomology by analyzing the closed and exact forms in the twisted theory.
• Identifying the boundary vertex operator algebras as VOAs in the $̅{Q}_+$ cohomology, with line operators acting as bi-modules.
• Constructing the non-perturbative chiral algebra in the abelian case as the rank-two Narain lattice VOA via explicit cohomology computation.
• Providing three equivalent descriptions of the non-perturbative result: via the $β\gamma$ system, the lattice VOA, and the cohomology of differential forms on $\mathrm{SL}(2,\mathbb{C})$.

Experimental results

Research questions

• RQ1 What is the structure of the chiral algebra in the $̅{Q}_+$ cohomology of a 3D $σ=2$ theory compactified on an interval with $(0,2)$ boundary conditions?
• RQ2 How do line operators between the boundaries enhance the boundary vertex operator algebras?
• RQ3 What is the non-perturbative chiral algebra in the abelian case of the reduced theory?
• RQ4 How are the boundary VOAs and their line operator modules related to the geometry of $\mathrm{SL}(2,\mathbb{C})$?
• RQ5 What is the conjectured structure of the chiral algebra in the non-abelian case?

Key findings

• The perturbative $̅{Q}_+$ cohomology of the reduced 2D theory is computed explicitly using the $β\gamma$ system formalism.
• The boundary vertex operator algebras are enhanced by line operators, which act as bi-modules in the category of representations.
• In the abelian case, the non-perturbative chiral algebra is identified as the rank-two Narain lattice vertex operator algebra.
• Three equivalent descriptions of the non-perturbative chiral algebra are provided: via the $β\gamma$ system, the lattice VOA, and the cohomology of differential forms on $\mathrm{SL}(2,\mathbb{C})$.
• The non-perturbative result is shown to be isomorphic to $\mathcal{Z}_d(\Omega^{3,0}) / d\Omega^{2,0}$, with the generator $\mathrm{Tr}(g^{-1}dg)^3$ mapping to a Cech cocycle representing the holomorphic 2-form $\mu$.
• The paper provides a conjecture for the non-perturbative chiral algebra in the non-abelian case, based on the structure of line operators and boundary VOAs.