$\mathrm{G}_2$-structures with torsion and the deformed Shatashvili-Vafa vertex algebra
Andoni De Arriba de La Hera, Mateo Galdeano, Mario Garcia-Fernandez
TL;DR
The paper establishes that the chiral de Rham complex of seven-manifolds with integrable $G_2$-structures and closed torsion supporting heterotic-type equations contains embeddings of the deformed Shatashvili--Vafa vertex algebra $SV_a$, with the deformation parameter $a$ governed by the scalar torsion class. It provides an explicit realization of these embeddings through left-equivariant superaffine algebras and the chiral de Rham complex, showing that $a$ is proportional to $\tau_0$ (via $a=-(1/\sqrt{k}) (7/6)\tau_0$ in the level-$k$ realization) and that the dilaton-like correction in the supersymmetric generator $G$ encodes $\tau_1$. The work analyzes five concrete $G_2$-structure families on $S^3\times T^4$ and $S^3\times S^3\times S^1$, demonstrating the conjectured link between torsion data and vertex algebra embeddings and aligning with semiclassical expectations that connect $a$ to the cosmological constant. Overall, it strengthens the bridge between heterotic $G_2$ backgrounds and algebraic structures in the chiral de Rham framework, with potential dilaton-corrected refinements awaiting future exploration.
Abstract
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 $α'=0$ 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, $a \sim τ_0$, as expected from previous work in the semi-classical limit by the second author, jointly with De la Ossa and Marchetto.