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 ⁴1,1,2,8,12] (whose blow‐up gives the smooth CY₃(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 1, 2], of the parent 𝒩 = 2 type IIA theory of 3]. We obtain the Meijer's basis of solutions to the equation in the large 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 4], at the same and another MATHEMATICAlly interesting point. Based on a recently shown 𝒩 = 1 four‐dimensional triality 6] between Heterotic on the self‐mirror Calabi‐Yau CY₃(11,11), M theory on and F‐theory on an elliptically fibered CY₄ with the base given by CP ¹ × Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY₃(3,243), and then explicitly verify the same using mirror symmetry formulation of 2] 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.