$\mathcal{S}\mathcal{W}$-algebras and strings with torsion

Abstract

We explore the connection between super $\mathcal{W}$-algebras ($\mathcal{S}\mathcal{W}$-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.

46 pages plus appendices, 1 figure, 6 tables; v2: typos fixed, matches published version

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