Notes on Certain (0,2) Correlation Functions

TL;DR

The article develops a heterotic analogue of quantum cohomology by computing worldsheet instanton corrections to correlation functions defined by products of sheaf cohomology classes $H^q(X, \Lambda^p {\cal E}^{\vee})$ in (0,2) theories. It introduces the half-twisted framework, analyzes massless states, and derives both classical and quantum selection rules under the anomaly and bundle conditions ${\Lambda^{top}}{\cal E}^{\vee} \cong K_X$ and ${\rm ch}_2({\cal E})={\rm ch}_2(TX)$. A central methodological advance is the use of linear sigma-model (GLSM) compactifications to extend the relevant bundles over moduli space compactifications, constructing the induced sheaves $\mathcal F$ and $\mathcal F_1$ and addressing obstructions via the proposed generalized obstruction-sheaf framework. The authors verify predictions of Adams-Basu-Sethi in explicit examples on $\mathbb P^1\times\mathbb P^1$ with deformed tangent bundles, reproducing deformed quantum cohomology relations and validating the computational approach, while also outlining important open mathematical questions related to extensions, excess zero modes, and higher-genus generalizations.

Abstract

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.