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On (orientifold of) type IIA on a compact Calabi-Yau. (English) Zbl 1040.81077
Summary: We study the gauged sigma model and its mirror Landau-Ginzburg 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 [S. Cecotti, Int. J. Mod. Phys. A 6, No. 10, 1749–1813 (1991; Zbl 0743.57022); K. Hori and C. Vafa, Mirror symmetry, Rep. No. HUTP-00/A005, see also arXiv:hep-th/0002222], of the parent $$\mathcal N = 2$$ type IIA theory of S. Kachru and C. Vafa [Nucl. Phys. B 450, No. 1–2, 68–89 (1995; Zbl 0957.14509)]. We obtain the Meijer’s basis of solutions to the equation in the large and small complex structure limits (on the mirror Landau-Ginzburg side) of the above mentioned Calabi-Yau, and make some remarks about the monodromy properties associated based on [D. R. Morrison, Stud. Adv. Math. 9, 185–199 (1998; Zbl 0841.32013)], at the same and another mathematically interesting point. Based on a recently shown $$\mathcal N = 1$$ four-dimensional triality [A. Misra, An $${\mathcal N}=1$$ triality by spectrum matching, hep-th/0212054] between heterotic on the self-mirror Calabi-Yau $$CY_3(11,11)$$, M-theory on $$\frac{CY_3(3,243)\times S^1}{\mathbb{Z}_2}$$ and $$F$$-theory on an elliptically fibered $$CY_4$$ with the base given by $$\mathbb{C}\mathbb{P}^1 \times$$ Enriques surface, we firstgive 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 and Vafa [loc. cit.] for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau-Ginzburg model corresponding to the resolved Calabi-Yau as well.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 83E30 String and superstring theories in gravitational theory 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J81 Relationships between surfaces, higher-dimensional varieties, and physics
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