Gauss-Bonnet-Chern theorem on moduli space

TL;DR

This paper proves the Gauss-Bonnet-Chern theorem on moduli spaces of polarized Kähler manifolds, establishing the rationality of Chern-Weil forms and their finiteness in string theory flux compactifications.

Contribution

It extends the Gauss-Bonnet-Chern theorem to moduli spaces and demonstrates the rationality and finiteness of related integrals in Calabi-Yau moduli and string theory.

Findings

Gauss-Bonnet-Chern theorem holds on moduli space of polarized Kähler manifolds.

Chern-Weil forms are rational with respect to the Weil-Petersson metric.

Integrals of Chern-Weil forms in flux compactifications are finite and rational.

Abstract

In this paper, we proved the Gauss-Bonnet-Chern theorem on moduli space of polarized Kahler manifolds. Using our results, we proved the rationality of the Chern-Weil forms (with respect to the Weil-Petersson metric) on CY moduli. As an application in physics, by the Ashok-Douglas theory, counting the number of flux compactifications of the type IIb string on a Calabi-Yau threefold is related to the integrations of various Chern-Weil forms. We proved that all these integrals are finite (and also rational).