Type IIA on a Compact Calabi-Yau and D=11 Supergravity Uplift of its Orientifold

TL;DR

This paper computes period integrals and monodromies for a specific compact Calabi-Yau manifold, explores its orientifold involution effects, and discusses implications for type IIA supergravity uplift and superpotential nullity.

Contribution

It introduces a novel method for obtaining period solutions in compact Calabi-Yau manifolds, including large and small complex structure limits, and analyzes the orientifold involution's impact on supergravity.

Findings

Derived Picard-Fuchs equations and Meijer basis for CY_3(3,243).

Evaluated monodromy matrices in different complex structure limits.

Conjectured the action of antiholomorphic involution on periods and cohomology.

Abstract

Using the prescription of [1] for defining period integrals in the Landau-Ginsburg theory for compact Calabi-Yau's, we obtain the Picard-Fuchs equation and the Meijer basis of solutions for the compact Calabi-Yau CY_3(3,243) expressed as a degree-24 Fermat hypersurface AFTER resolution of the orbifold singularities. This is similar in spirit to the method of obtaining Meijer basis of solutions in [2] for the case wherein one is away from the orbifold singularities, and one is considering the large-base limit of the Calabi-Yau. The importance of the method lies in the ease with which one can consider the large AND small complex structure limits, as well as the ability to get the "ln"-terms in the periods without having to parametrically differentiate infinite series. We consider in detail the evaluation of the monodromy matrix in the large and small complex structure limits. We also consider the action of the freely acting antiholomorphic involution of [2],[3] on D=11 supergravity compactified on CY_3(3,243) x S^1 [4] and obtain the Kaehler potential for the same in the limit of large volume of the Calabi-Yau. As a by-product, we also give a conjecture for the action of the orientation-reversing antiholomorphic involution on the periods, given its action on the cohomology, using a canonical (co)homology basis. Finally, we also consider showing a null superpotential on the orientifold of type IIA on CY_3(3,243), having taken care of the orbifold singularities, thereby completing the argument initiated in [2].