${\mathrm{G}}_2$-structures with torsion and the deformed Shatashvili-Vafa vertex algebra

We construct representations of the deformed Shatashvili-Vafa vertex algebra ${\mathrm{S}\mathrm{V}}_a$, with parameter $a\in \mathbb{C}$, as recently proposed in the physics literature by Fiset and Gaberdiel. The geometric input for our construction are integrable ${\mathrm{G}}_2$-structures with closed torsion, solving the heterotic ${\mathrm{G}}_2$ system with ${\alpha }^{\mathrm{\prime }}=0$ on the group manifolds $S^3\times T^4$ and $S^3\times S^3\times S^1$. From considerations in string theory, one expects the chiral algebra of these backgrounds to include ${\mathrm{S}\mathrm{V}}_a$, and we provide a mathematical realization of this expectation by obtaining embeddings of ${\mathrm{S}\mathrm{V}}_a$ in the corresponding superaffine vertex algebra and the chiral de Rham complex. In our examples, the parameter $a$ is proportional to the scalar torsion class of the ${\mathrm{G}}_2$ structure, $a\sim {\tau }_0$, as expected from previous work in the semi-classical limit by the second author, jointly with De la Ossa and Marchetto.