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21
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 25 (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 21 (1 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.
Lagrangian Floer theory on compact toric manifolds I
, 2009
"... The present authors introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function PO purely in terms of Amodel data in [FOOO3]. In this paper, we carry out explicit calculations involving PO on toric manifolds and study the relationship between this ..."
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Cited by 21 (3 self)
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The present authors introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function PO purely in terms of Amodel data in [FOOO3]. In this paper, we carry out explicit calculations involving PO on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the LandauGinzburg superpotential. Combining this study with the results from [FOOO3], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to EntovPolterovich’s
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 14 (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
Unfoldings of meromorphic connections and a construction of Frobenius manifolds, Frobenius manifolds (Quantum cohomology and singularities
 Aspects of Mathematics
"... Abstract. 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 variat ..."
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Cited by 12 (1 self)
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Abstract. 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. 1.
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 10 (2 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 system IV (Simplylaced elliptic Lie algebras)
"... Let (R; G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram 0 (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 isomorph ..."
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Cited by 10 (0 self)
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Let (R; G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram 0 (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.
Uniformization of the Orbifold of a Finite Reflection Group
"... We try to understand the relationship between the K(; 1)property of the complexi ed regular orbit space of a nite reection group and the at structure on the orbit space via the uniformization equation attached to the at structure. ..."
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Cited by 8 (2 self)
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We try to understand the relationship between the K(; 1)property of the complexi ed regular orbit space of a nite reection group and the at structure on the orbit space via the uniformization equation attached to the at structure.
Traces on the Sklyanin algebra and correlation functions of the eightvertex
"... Abstract. We propose a conjectural formula for correlation functions of the Zinvariant (inhomogeneous) eightvertex model. We refer to this conjecture as Ansatz. It states that correlation functions are linear combinations of products of three transcendental functions, with theta functions and deri ..."
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Cited by 6 (3 self)
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Abstract. We propose a conjectural formula for correlation functions of the Zinvariant (inhomogeneous) eightvertex model. We refer to this conjecture as Ansatz. It states that correlation functions are linear combinations of products of three transcendental functions, with theta functions and derivatives as coefficients. The transcendental functions are essentially logarithmic derivatives of the partition function per site. The coefficients are given in terms of a linear functional Trλ on the Sklyanin algebra, which interpolates the usual trace on finite dimensional representations. We establish the existence of Trλ and discuss the connection to the geometry of the classical limit. We also conjecture that the Ansatz satisfies the reduced qKZ equation. As a nontrivial example of the Ansatz, we present a new formula for the nextnearest neighbor correlation functions. 1.