Supergravity anomaly equations from modularity of Calabi--Yau threefolds

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.