Heterotic Quantum Cohomology

We reexamine the massless spectrum of a heterotic string vacuum at large radius and present two results. The first result is to construct a vector bundle $\mathcal{Q}$ and operator $\overline{\mathcal{D}}$ whose kernel amounts to deformations solving `F-term' type equations. This resolves a dilemma in previous works in which the spin connection is treated as an independent degree of freedom, something that is not the case in string theory. The second result is to utilise the moduli space metric, constructed in previous work, to define an adjoint operator $\overline{\mathcal{D}}^\dag$. The kernel of $\overline{\mathcal{D}}^\dag$ amounts to deformations solving `D-term' type equations. Put together, we show there is a vector bundle $\mathcal{Q}$ with a metric, a $\overline{\mathcal{D}}$ operator and a gauge fixing (holomorphic gauge) in which the massless spectrum are harmonic representatives of $\overline{\mathcal{D}}$. This is remarkable as previous work indicated the Hodge decomposition of massless deformations were complicated and in particular not harmonic except at the standard embedding.