On (Orientifold of) type IIA on a Compact Calabi-Yau

We study the gauged sigma model and its mirror Landau-Ginsburg model corresponding to type IIA on the Fermat degree-24 hypersurface in WCP^4[1,1,2,8,12] (whose blow-up gives the smooth CY_3(3,243)) away from the orbifold singularities, and its orientifold by a freely-acting antiholomorphic involution. We derive the Picard-Fuchs equation obeyed by the period integral as defined in the work of Cecotti and Hori-Vafa, of the parent N=2 type IIA theory of Kachru and Vafa. We obtain the Meijer's basis of solutions to the equation in the large {\it and} small complex structure limits (on the mirror Landau-Ginsburg side) of the abovementioned Calabi-Yau, and make some remarks about the monodromy properties associated based on the work of Morrison, at the same and another MATHEMATICAlly interesting point. Based on a recently shown N=1 four-dimensional triality (hep-th/0212054) between Heterotic on the self-mirror Calabi-Yau CY_3(11,11), M theory on (CY_3(3,243) x S^1)/Z_2 and F-theory on an elliptically fibered CY_4 with the base given by CP^1 x Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY_3(3,243), and then explicitly verify the same using mirror symmetry formulation of Hori-Vafa for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau-Ginsburg model corresponding to the resolved Calabi-Yau as well.