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.
Finiteness of volume of moduli spaces
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We give a ``physics proof'' of a conjecture made by the first author at Strings 2005, that the moduli spaces of certain conformal field theories are finite volume in the Zamolodchikov metric, using an RG flow argument.
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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.