Derived deformation theory of heterotic $G_2$ systems near the standard embedding

Abstract

We study heterotic $G_2$ structures near a torsion-free standard embedding, to first order in ${\alpha }^{\prime }$ and to all orders in formal deformations. The tangent connection is the Hull connection determined by the metric and flux. We construct a locally invertible family of differential operators relating the supersymmetry and instanton $BPS$ equations to the Euler-Lagrange equations of the heterotic superpotential. Using this comparison and the physical gauge symmetries, we construct a homological vector field locally formally conjugate to the Hamiltonian vector field of the superpotential. On a closed manifold in the fixed gerbe sector, the torsion equation implies vanishing of the leading flux. We prove ellipticity of the linearized deformation complex with mixed differential-order weights and construct a finite-dimensional cyclic $L_{\infty }$ model with a formal effective potential.

arXiv comment44 pages. Major v2 revision: recognising the redundancy of the leading-flux condition led to a substantially simpler and shorter construction