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

String backgrounds of the form $\mathbb{M}_3 \times {\cal M}_7$ where $\mathbb{M}_3$ denotes $3$-dimensional Minkowski space while ${\cal M}_7$ 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 ${\rm AdS}_3$. We argue that in this case the world-sheet theory is characterised by a certain ${\cal N}=1$ superconformal ${\cal W}$-algebra that has the same spin spectrum as the SV algebra and also contains a tricritical Ising model ${\cal N}=1$ subalgebra. We determine the allowed representations of this ${\cal W}$-algebra, and analyse to which extent the special features of the SV algebra survive this generalisation.

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