Quantum [formula omitted]-modules and generalized mirror transformations

Abstract

In the previous paper [Hiroshi Iritani, Quantum D -modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D -module after a mirror transformation when the first Chern class c 1 ( M ) of the tangent bundle is nef. In this paper, even when c 1 ( M ) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D -module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].