Finiteness of quantum gravity amplitudes implies moduli spaces have at most Euclidean volume growth, which in turn implies duality groups act semisimply on charge lattices.
On the Weil-Petersson volume and the first Chern Class of the moduli space of Calabi-Yau manifolds
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In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.
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Finiteness and the Emergence of Dualities
Finiteness of quantum gravity amplitudes implies moduli spaces have at most Euclidean volume growth, which in turn implies duality groups act semisimply on charge lattices.