2003/11/30 by A. Misra, Aalok Misra
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds #hep-th
paper · pdf · doi:10.1002/prop.200310170
published as Fortsch.Phys. 52 (2004) 831-870 · 45 pages, LaTex, 2 ps figures; v3: (Fort. der. Physik) Journal version
arxiv created 2004/03/16 · openalex publication_date 2004/08/09 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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) × S 1 [4] and obtain the Kähler 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].