Supergravity anomaly equations from modularity of Calabi--Yau threefolds

TL;DR

The work shows that six-dimensional ${ m N}=(1,0)$ anomaly cancellation in F-theory compactifications on elliptically fibered Calabi–Yau threefolds can be understood as a consequence of modular properties of topological-string data. By encoding the spectrum and anomaly coefficients in meromorphic quasi-Jacobi forms whose coefficients are fiber-restricted Gromov–Witten invariants, and by employing Fourier–Mukai monodromies together with holomorphic anomaly equations, the authors derive the Green–Schwarz counterterms and anomaly constraints from first principles. They present a detailed base-degree-zero analysis, connect three-point couplings to quasi-Jacobi forms, and substantiate the framework with explicit U(1) and SU(3) examples, illustrating the geometric origin of anomaly cancellation and KK towers. The results illuminate a unifying automorphic perspective on consistency conditions in F-theory, with potential implications for related dualities and conjectures like the tower WGC.

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.

Figures (1)

• Figure 1: Schematic representation of Green--Schwarz mechanism and equation (\ref{['eqn:modAnomlay']}). We associate $\tau$ with external graviton insertions and $z^A$ with external gauge boson insertions. The topological four-point couplings $C_{ABCD}$ correspond to the pure gauge anomalies, $C_{\tau\tau AB}$ to the mixed anomalies and $C_{\tau\tau\tau\tau}$ to the pure gravitational anomaly.