Spectrum bounds in geometry

Filipazzi, Hacon, and Svaldi proved that there are only finitely many topological types of elliptically fibered Calabi–Yau threefolds. We explore the implications of their results on the boundedness of the geometric quantities in the massless spectrum of the F-theory Calabi–Yau compactifications. A key ingredient is what we call the geometric anomaly equation, and the extension of the gravitational anomaly cancellation in physics, also to singular spaces. We review and extend the dictionary between geometry and physics. We conclude with explicit bounds.