Supergravity anomaly equations from modularity of Calabi--Yau threefolds

TL;DR

This paper explores how anomaly cancellation in six-dimensional supergravity theories from F-theory compactifications is encoded in the modular properties of automorphic forms derived from Calabi--Yau threefolds, linking geometry, string theory, and automorphic forms.

Contribution

It establishes a novel connection between supergravity anomaly equations and the modularity of automorphic forms from Calabi--Yau threefolds, revealing how anomaly cancellation is automatically satisfied.

Findings

Anomaly coefficients are encoded in the modular properties of automorphic forms.

Green--Schwarz counterterms are related to Fourier--Mukai transforms and T-duality.

Anomaly cancellation follows from holomorphic anomaly equations in topological string theory.

Abstract

F-theory compactifications on elliptically fibered Calabi--Yau threefolds yield consistent six-dimensional $\mathcal{N}=(1,0)$ supergravity theories, for which the cancellation of gravitational, gauge and mixed anomalies imposes non-trivial algebraic relations between classical intersection data and enumerative geometry invariants of curves in the fiber. In this work, we capture the spectrum of such theories via meromorphic quasi-Jacobi forms of index zero whose Fourier coefficients determine the genus zero Gromov--Witten theory restricted to curve classes in the fiber. We find that the one-loop anomaly coefficients of the effective six-dimensional theories are encoded in the modular properties of these automorphic forms, while the Green--Schwarz counterterms are made manifest by the Fourier--Mukai transform action on zero- and two-branes associated with double T-duality along the elliptic fiber. Moreover, we show that the anomaly cancellation conditions are automatically satisfied in this class of string compactifications as a consequence of the holomorphic anomaly equations of topological string theory.