Deformed Shatashvili-Vafa algebra for superstrings on AdS${}_3\times {\mathcal{M}}_7$

Abstract

String backgrounds of the form ${\mathbb{M}}_3\times {\mathcal{M}}_7$ where ${\mathbb{M}}_3$ denotes $3$-dimensional Minkowski space while ${\mathcal{M}}_7$ is a $7$-dimensional G${}_2$-manifold, are characterised by the property that the world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the generalisation of this statement to backgrounds where the Minkowski factor ${\mathbb{M}}_3$ is replaced by ${\mathrm{AdS}}_3$. We argue that in this case the world-sheet theory is characterised by a certain $\mathcal{N}=1$ superconformal $\mathcal{W}$-algebra that has the same spin spectrum as the SV algebra and also contains a tricritical Ising model $\mathcal{N}=1$ subalgebra. We determine the allowed representations of this $\mathcal{W}$-algebra, and analyse to which extent the special features of the SV algebra survive this generalisation.

arXiv comment42 pages

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