On Semi-Periods

TL;DR

This paper explores semi-periods on Calabi-Yau manifolds, extending the solution space of period equations via hypergeometric systems and symmetry considerations, with potential for complete solution generation.

Contribution

It introduces semi-periods as solutions from an extended hypergeometric system, linking chain integration to cycle construction and proposing a method to find all solutions.

Findings

Complete solutions obtained in simple examples

Extended system includes more solutions than traditional periods

Conjecture on a modified method generating full solution space

Abstract

The periods of the three-form on a Calabi-Yau manifold are found as solutions of the Picard-Fuchs equations; however, the toric varietal method leads to a generalized hypergeometric system of equations which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-periods are solutions of this extended system. They are obtained by integration of the three-form over chains; these chains can be used to construct cycles which, when integrated over, give periods. In simple examples we are able to obtain the complete set of solutions for the extended system. We also conjecture that a certain modification of the method will generate the full space of solutions in general.