Exact Gravitational Threshold Correction in the FHSV Model

TL;DR

This paper derives a nonperturbative automorphic form formula for gravitational threshold corrections in a specific N=2 supersymmetric heterotic/IIA duality model, linking string theory, automorphic forms, and algebraic geometry.

Contribution

It provides the full nonperturbative formula for the gravitational threshold correction $F_1$ in the FHSV model using automorphic forms related to the fake Monster Lie superalgebra.

Findings

Derived the automorphic form formula for $F_1$ in the FHSV model.

Connected automorphic forms to determinants of $ar ext{d}$ operators on K3 surfaces.

Reinterpreted a known result in algebraic geometry through string theory automorphic forms.

Abstract

We consider the automorphic forms which govern the gravitational threshold correction $F_1$ in models of heterotic/IIA duality with N=2 supersymmetry in four dimensions. In particular we derive the full nonperturbative formula for $F_1$ for the dual pair originally considered by Ferrara, Harvey, Strominger and Vafa (FHSV). The answer involves an interesting automorphic product constructed by Borcherds which is associated to the ``fake Monster Lie superalgebra.'' As an application of this result we rederive a result of Jorgenson & Todorov on determinants of $\bar \partial$ operators on $K3$ surfaces.