Constraints in the BV formalism: six-dimensional supersymmetry and its twists

TL;DR

This paper develops a perturbative formulation of the six-dimensional abelian (2,0) theory using a generalized BV formalism, computes its twists, and explores connections to lower-dimensional theories and dualities.

Contribution

It introduces a presymplectic generalization of the BV formalism for the (2,0) theory and computes its holomorphic and non-minimal twists at the perturbative level.

Findings

Formulation matches known results under dimensional reduction

Holomorphic twist involves symplectic-valued bosons and coclosed forms

Connections to Kodaira-Spencer theory and issues with electric-magnetic duality

Abstract

We formulate the abelian six-dimensional $\mathcal{N}=(2,0)$ theory perturbatively, in a generalization of the Batalin-Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an $L_\infty$ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the $\mathcal{N}=(1,0)$ hypermultiplet, together with a degenerate holomorphic theory of holomorphic coclosed one-forms from the $\mathcal{N}=(1,0)$ tensor multiplet, which can be interpreted as representing the intermediate Jacobian. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira-Spencer theory. Matching our formalism to five-dimensional Yang-Mills theory after reduction leads to some issues related to electric-magnetic duality; we offer some speculation on a nonperturbative resolution.