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53
AN INTEGRAL STRUCTURE IN QUANTUM COHOMOLOGY AND MIRROR SYMMETRY FOR TORIC ORBIFOLDS
, 2009
"... We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the ..."
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Cited by 72 (5 self)
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We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].
Symplectic geometry of Frobenius structures
"... The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in twodimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expan ..."
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Cited by 52 (4 self)
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The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in twodimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GLN of hidden symmetries. We try to keep the text introductory and nontechnical. In particular, we supply details of some simple results from the axiomatic theory, including a severalline proof of the genus 0 Virasoro constraints not mentioned elsewhere, but merely quote and refer to the literature for a number of less trivial applications, such as the quantum Hirzebruch–Riemann–Roch theorem in the theory of cobordismvalued Gromov–Witten invariants. The latter is our joint work in progress with Tom Coates, and we would like to thank him for numerous discussions of the subject.
tt* Geometry, Frobenius manifolds, their connections, and the construction for singularities
, 2002
"... ..."
Semiinfinite Hodge structures and mirror symmetry for projective spaces, preprint
, 10
"... Abstract. We express total set of rational GromovWitten invariants of CP n via periods of variations of semiinfinite Hodge structure associated with their mirror partners. For this explicit example we give detailed description of general construction of solutions to WDVVequation from variations o ..."
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Cited by 40 (2 self)
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Abstract. We express total set of rational GromovWitten invariants of CP n via periods of variations of semiinfinite Hodge structure associated with their mirror partners. For this explicit example we give detailed description of general construction of solutions to WDVVequation from variations of semiinfinite Hodge structures of CalabiYau type which was suggested in a proposition from our previous paper ([B2] proposition 6.5). Contents
Topological strings in generalized complex space
, 2006
"... A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically ..."
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Cited by 37 (1 self)
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A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically closes offshell, the model transparently depends only on J, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N = 2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector β and recover holomorphic noncommutative Kontsevich ∗product.
Quantum RiemannRoch, Lefschetz and Serre
 OF MATH
"... Given a holomorphic vector bundle E: EX → X over a compact Kähler manifold, one introduces twisted GWinvariants of X replacing virtual fundamental cycles of moduli spaces of stable maps f: Σ → X by their capproduct with a chosen multiplicative characteristic class of H 0 (Σ, f ∗ E) − H 1 (Σ, f ∗ E ..."
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Cited by 25 (3 self)
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Given a holomorphic vector bundle E: EX → X over a compact Kähler manifold, one introduces twisted GWinvariants of X replacing virtual fundamental cycles of moduli spaces of stable maps f: Σ → X by their capproduct with a chosen multiplicative characteristic class of H 0 (Σ, f ∗ E) − H 1 (Σ, f ∗ E). Using the formalism [17] of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for X. The result (Theorem 1) is a consequence of Mumford’s Riemann – Roch – Grothendieck formula [31, 13] applied to the universal stable map. When E is concave, and the inverse C ×equivariant Euler class is chosen, the twisted theory yields GWinvariants of EX. The “nonlinear Serre duality principle ” [19, 20] expresses GWinvariants of EX via those of the supermanifold ΠE ∗ X, where the Euler class and E ∗ replace the inverse Euler class and E. We derive from Theorem 1 the nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex, and a submanifold Y ⊂ X is defined by
QUANTUM DMODULES AND GENERALIZED MIRROR TRANSFORMATIONS
, 2004
"... In the previous paper [Iri1], we constructed equivariant Floer cohomology for complete intersections in toric variety and showed that it is isomorphic to the small quantum Dmodule after a mirror transformation when the first Chern class of the tangent bundle is nef. In this paper, we show that in ..."
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Cited by 24 (7 self)
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In the previous paper [Iri1], we constructed equivariant Floer cohomology for complete intersections in toric variety and showed that it is isomorphic to the small quantum Dmodule after a mirror transformation when the first Chern class of the tangent bundle is nef. In this paper, we show that in nonnef case, equivariant Floer cohomology reconstructs the big quantum Dmodule using mirror theorem by Coates and Givental [CG]. This reconstruction procedure gives a generalized mirror transformation first observed by Jinzenji in low degrees [Jin1, Jin2].