$\mathcal{SW}$-algebras and strings with torsion

TL;DR

This paper investigates the relationship between super W-algebras and G-structures with torsion in string theory, revealing how torsion deforms symmetry algebras at first order in string length scale.

Contribution

It establishes a connection between super W-algebras and G-structures with torsion, analyzing their quantum and classical limits in string backgrounds with flux.

Findings

Torsion deforms OPE coefficients in symmetry algebras.

Comparison of quantum and classical symmetry algebras.

First-order torsion effects relate to scalar torsion classes.

Abstract

We explore the connection between super $\mathcal{W}$-algebras ($\mathcal{SW}$-algebras) and $\mathrm{G}$-structures with torsion. The former are realised as symmetry algebras of strings with $\mathcal{N}=(1,0)$ supersymmetry on the worldsheet, while the latter are associated with generic string backgrounds with non-trivial Neveu-Schwarz flux $H$. In particular, we focus on manifolds featuring $\mathrm{Spin}(7)$, $\mathrm{G}_2$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$-structures. We compare the full quantum algebras with their classical limits, obtained by studying the commutators of superconformal and $\mathcal{W}$-symmetry transformations, which preserve the action of the $(1,0)$ non-linear $\sigma$-model. We show that, at first order in the string length scale $\ell_s$, the torsion deforms some of the OPE coefficients corresponding to special holonomy through a scalar torsion class.