Notes on correlation functions in (0,2) theories

TL;DR

This work develops a mathematical framework for computing correlation functions in perturbative heterotic (0,2) theories by generalizing A-model rational curve counting to include holomorphic vector bundles $(X,\mathcal{E})$. It shows that correlators can be expressed as integrals over the worldsheet instanton moduli space $\mathcal{M}$ of top forms built from induced bundles $\mathcal{F}$, $\mathcal{F}_1$, and, when necessary, the obstruction sheaf $\mathrm{Obs}$, with a key identity $\Lambda^{top}\mathcal{F}^\vee \otimes \Lambda^{top}\mathcal{F}_1 \otimes \Lambda^{top}(\mathrm{Obs})^\vee \cong K_{\mathcal{M}}$ ensuring top-form integrands. Gauged linear sigma models provide natural compactifications of $\mathcal{M}$ and canonical extensions of the relevant sheaves, preserving the correct cohomological structure and enabling explicit computations. On the (2,2) locus, the construction reduces to the familiar A-model obstruction theory and ordinary mirror symmetry, while in the general (0,2) setting it incorporates charged-state couplings and new rational curve corrections. The note summarizes the formal translation and the GLSM-based framework, with detailed verification and computations reported in the cited work.

Abstract

In this note we shall review recent work on generalizing rational curve counting to perturbative heterotic theories.