Local superconformal algebras

TL;DR

This paper develops a derived algebraic framework to study local superconformal algebras, encompassing deformations, symmetries, and twists in superspace geometry, with applications to supergravity and superconformal theories.

Contribution

It introduces a local super dg Lie algebra model for deformations of distributions on supermanifolds, unifying various superconformal structures and their twists in a derived geometric setting.

Findings

Constructed a derived model for the space of functions constant along distributions.

Reproduced known superconformal multiplets, including conformal supergravity.

Computed all twists of the stress tensor multiplet in superconformal theories.

Abstract

Given a supermanifold equipped with an odd distribution of maximal dimension and constant symbol, we construct the formal moduli problem of deformations of the distribution. This moduli problem is described by a local super dg Lie algebra that provides both a resolution of the structure-preserving vector fields on superspace and a derived enhancement of superconformal symmetry. Applying our construction in standard physical examples returns the conformal supergravity multiplet in every known example, in any dimension and with any amount of supersymmetry$\unicode{x2014}$whether or not a superconformal algebra exists. We discuss new examples related to twisted supergravity, higher Virasoro algebras, and exceptional super Lie algebras. The compatibility of our techniques with twisting also leads to a computation of every twist of the stress tensor multiplet of a superconformal theory, including universal operator product expansions. Our approach uses a derived model for the space of functions constant along the distribution, which is applicable even when the distribution is non-involutive; we construct other natural multiplets, such as K\"ahler differentials, that appear naturally through this lens on superspace geometry.