String Modular Phases in Calabi-Yau Families

TL;DR

This paper explores the structure of singular Calabi-Yau varieties, linking their motivic L-series to modular forms and weighted Fermat hypersurfaces, implying potential string theory interpretations and phase transitions.

Contribution

It introduces a construction of deformed motives over moduli spaces of Calabi-Yau varieties and reveals their L-series correspondence with modular forms and Fermat hypersurfaces.

Findings

Motivic L-series match modular forms' L-series.

Singular Calabi-Yau spaces can have string worldsheet interpretations.

Degenerations lead to dimensional transmutation of motives.

Abstract

We investigate the structure of singular Calabi-Yau varieties in moduli spaces that contain a Brieskorn-Pham point. Our main tool is a construction of families of deformed motives over the parameter space. We analyze these motives for general fibers and explicitly compute the $L-$series for singular fibers for several families. We find that the resulting motivic $L-$functions agree with the $L-$series of modular forms whose weight depends both on the rank of the motive and the degree of the degeneration of the variety. Surprisingly, these motivic $L-$functions are identical in several cases to $L-$series derived from weighted Fermat hypersurfaces. This shows that singular Calabi-Yau spaces of non-conifold type can admit a string worldsheet interpretation, much like rational theories, and that the corresponding irrational conformal field theories inherit information from the Gepner conformal field theory of the weighted Fermat fiber of the family. These results suggest that phase transitions via non-conifold configurations are physically plausible. In the case of severe degenerations we find a dimensional transmutation of the motives. This suggests further that singular configurations with non-conifold singularities may facilitate transitions between Calabi-Yau varieties of different dimensions.