Topological Amplitudes in Heterotic Superstring Theory

TL;DR

This work extends the notion of topological amplitudes from Type II to heterotic string theory by identifying genus-$g$ topological data $F^g$ with $W^{2g}$ F-terms in four-dimensional $N=1$ supergravity. A key finding is that heterotic holomorphic anomalies do not close on $F^g$ alone; they couple to anti-chiral correlators, giving rise to higher-weight F-terms of the form $W^{2g}\Pi^n$ and revealing a supersymmetric Ward identity structure in the generating-function formalism. The paper provides a tree-level solution method, connects explicit heterotic amplitudes to topological correlators, and discusses orbifold examples illustrating these ideas in concrete settings. The framework offers insights into SUSY breaking and dualities in $N=1$ theories and highlights open questions regarding handle degenerations and charged-matter effects. Overall, it unifies worldsheet topological data with four-dimensional EFT in the heterotic context and charts directions for further nonperturbative and duality investigations.

Abstract

We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N=1 spacetime supersymmetric background. The genus g topological partition function $F^g$ corresponds to a term in the effective action of the form $W^{2g}$, where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form $W^{2g}Π^n$, where $Π$ is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N=1 spacetime supersymmetry Ward identity. We give also a solution of the tree level recursion relations and discuss orbifold examples.

Figures (2)

• Figure 1: Reducible contributions to the kinematical structure $({\bar{\chi}}^{\bar{\imath}} {\bar{\chi}}^{\bar{\jmath}})({\bar{\chi}}^{\bar{k}}{\bar{\chi}}^{\bar{\ell}})\partial_{\mu} z^{\bar{m}}\partial_{\mu}z^{\bar{n}}$.
• Figure 2: Reducible contributions to the kinematical structure $({\bar{\chi}}^{\bar{\imath}}{\not\!\partial} z^{\bar{m}}) ({\bar{\chi}}^{\bar{\jmath}}{\not\!\partial} z^{\bar{n}})({\bar{\chi}}^{\bar{k}} {\bar{\chi}}^{\bar{\ell}})$.