Counting measure on toric hypersurface Calabi–Yau threefolds

Determine a mathematically well-defined counting measure on the set of toric hypersurface Calabi–Yau threefolds constructed from triangulations of four-dimensional reflexive polytopes in the Kreuzer–Skarke database, including an appropriate treatment of equivalences among triangulations, so that statistical fractions of models as a function of the Hodge number h^{1,1} can be rigorously defined and interpreted.

Background

The analysis in the paper relies on sampling Calabi–Yau threefolds realized as hypersurfaces in toric varieties using triangulations of reflexive polytopes from the Kreuzer–Skarke database. The authors compute fractions of models within experimental sensitivity ranges as a function of h^{1,1}, but emphasize that their sampling strategy may not correspond to a fair or canonical measure on the space of triangulations and inequivalent Calabi–Yau manifolds.

Establishing a precise counting measure over toric hypersurface Calabi–Yau threefolds would enable rigorous statistical statements about the distribution of axion properties across the landscape and would clarify how to weight different triangulations and equivalences when interpreting experimental constraints. The authors explicitly note that such a counting measure is currently unknown.