Notes on Certain (0,2) Correlation Functions

TL;DR

This paper explores advanced correlation function computations in heterotic string theory, extending quantum cohomology concepts to sheaf cohomology and involving moduli space compactifications and gauge bundle considerations.

Contribution

It introduces a heterotic generalization of quantum cohomology calculations using sheaf cohomology and moduli space compactifications, expanding the mathematical framework of string correlation functions.

Findings

Heterotic correlation functions involve sheaf cohomology elements.

Moduli space compactification is essential for these computations.

Euler classes of obstruction bundles are generalized in this context.

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.