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47
Lagrangian Floer theory on compact toric manifolds: Survey
, 2010
"... This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed. ..."
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Cited by 79 (7 self)
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This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed.
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 42 (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
Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
 hepth/0404043 46 A. Iqbal and A.K. KashaniPoor, The Vertex on a Strip. hepth/0410174
"... Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of loc ..."
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Cited by 36 (2 self)
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Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel’fandKapranovZelevinski type.
Frobenius Manifolds: isomonodromic deformations and infinitesimal period mappings
"... this paper apply essentially to this kind of examples. The manifold is then the parameter space of a universal unfolding or a moduli space, which hence carries an affine structure. We owe it to K. Saito [30] to have developed general tools (infinitesimal period mapping and primitive forms) to show t ..."
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Cited by 20 (0 self)
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this paper apply essentially to this kind of examples. The manifold is then the parameter space of a universal unfolding or a moduli space, which hence carries an affine structure. We owe it to K. Saito [30] to have developed general tools (infinitesimal period mapping and primitive forms) to show the existence of such a structure in the base space of the miniversal unfolding of a holomorphic function with an isolated singularity. M. Saito [31, 32] has given complete arguments, using Hodge theory
WALLCROSSINGS IN TORIC GROMOV–WITTEN THEORY I: CREPANT EXAMPLES
, 2006
"... Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the rel ..."
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Cited by 19 (4 self)
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Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the relationship between the genuszero Gromov–Witten invariants of the weighted projective spaces P(1, 1, 2), P(1, 1, 1, 3) and those of their crepant resolutions. Our methods are applicable to other toric birational transformations. Our results verify the Crepant Resolution Conjecture when X = P(1, 1, 2) and suggest that it needs modification when
Extended affine root systems IV (SimplyLaced Elliptic Lie Algebras
 Publ. RIMS, Kyoto Univ
"... Abstract.. Let (R,G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram Γ(R,G) is simplylaced (see Sect. 2). We associate three Lie algebras, explained in 1), 2) and 3) below, to the elliptic root system, and show that all three are ..."
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Cited by 19 (0 self)
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Abstract.. Let (R,G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram Γ(R,G) is simplylaced (see Sect. 2). We associate three Lie algebras, explained in 1), 2) and 3) below, to the elliptic root system, and show that all three are isomorphic. The isomorphism class is called the elliptic algebra. 1) The first one is the subalgebra g̃(R) generated by the vacuum eα for α ∈ R of the quotient Lie algebra VQ(R)/DVQ(R) of the lattice vertex algebra (studied by Borcherds) attached to the elliptic root lattice Q(R). This algebra is isomorphic to the 2toroidal algebra and to the intersection matrix algebra proposed by Slodowy. 2) The second algebra ẽ(Γ(R,G)) is presented by Chevalley generators and generalized Serre relations attached to the elliptic Dynkin diagram Γ(R,G). Since the Cartan matrix for the elliptic diagram has some positive off diagonal entries, the algebra is defined not only by KacMoody type relations but some others. 3) The third algebra h̃Zaf ∗ gaf is defined as an amalgamation of a Heisenberg algebra and an affine KacMoody algebra, where the amalgamation relations between the two algebras are explicitly given. This algebra admits a sort of triangular decomposition in a generalized sense. The first algebra g̃(R) does not depend on a choice of the marking G whereas the second ẽ(Γ(R,G)) and the third h̃Zaf ∗ gaf do. This means the isomorphism depend on the choice of the marking i.e. on a choice of an element of PSL(2,Z).
Unfoldings of meromorphic connections and a construction of Frobenius manifolds
 ASPECTS OF MATHEMATICS
, 2003
"... The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin’s theorem on semisimple Frobenius manifolds. Another special case starts with variations of Ho ..."
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Cited by 19 (2 self)
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The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin’s theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov–Kontsevich construction. For homogeneous polynomials which give Calabi–Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic.
UNIFORMIZATION OF THE ORBIFOLD OF A FINITE REFLECTION GROUP
"... We try to understand the relationship between the K(π, 1)property of the complexified regular orbit space of a finite reflection group and the flat structure on the orbit space via the uniformization equation attached to the flat structure. ..."
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Cited by 15 (4 self)
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We try to understand the relationship between the K(π, 1)property of the complexified regular orbit space of a finite reflection group and the flat structure on the orbit space via the uniformization equation attached to the flat structure.