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

We construct representations of the deformed Shatashvili-Vafa vertex algebra $\mathrm{SV}_a$, 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 $\alpha'=0$ 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 $\mathrm{SV}_a$, and we provide a mathematical realization of this expectation by obtaining embeddings of $\mathrm{SV}_a$ 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.