Quantum D-modules and generalized mirror transformations

By H. Iritani. 2004

61 citations recorded by the source corpus.

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].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4fc73b145624a246224d33ef238469ddde6c972e

How researchers see this paper

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In Geometry and Topology

1. Helge Ruddat
2. L. Katzarkov
3. Christopher Woodward
4. Mark Gross
5. D. Auroux
6. Bernd S Siebert
7. P. Hacking
8. M. Abouzaid

In Mathematical Physics

1. L. Katzarkov
2. Helge Ruddat
3. Christopher Woodward
4. Mark Gross
5. D. Auroux
6. Bernd S Siebert
7. I. Smith
8. M. Abouzaid

Citations in the corpus

• QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, methodology): “In physics language, the quantum Kirwan map relates correlators of a (possibly non-linear, non-abelian) gauged sigma model with those of the sigma model of the git quotient.”
• Gromov–Witten Invariants of Toric Fibrations (2009, unknown): “Iritani [9] for toric manifolds is obtained.”
• Towards Mirror Symmetry for Varieties of General Type (2012, methodology): “given by Uˇ = (C∗)n+2, wˇ = x 1 +···+xn+2 + xd n+2 x1 ·····xn+1 . While such a mirror leads to oscillatory integrals which correctly describe the genus zero Gromov-Witten theory of Sd (see [CG07] and [Ir08]), we wish to focus on a more geometric form of mirror symmetry. In particular, from the point”