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17
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 74 (3 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].
N = 1 special geometry, mixed Hodge variations and toric geometry
, 2002
"... We study the superpotential of a certain class of N = 1 supersymmetric type II compactifications with fluxes and Dbranes. We show that it has an important twodimensional meaning in terms of a chiral ring of the topologically twisted theory on the worldsheet. In the openclosed string Bmodel, this ..."
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Cited by 48 (8 self)
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We study the superpotential of a certain class of N = 1 supersymmetric type II compactifications with fluxes and Dbranes. We show that it has an important twodimensional meaning in terms of a chiral ring of the topologically twisted theory on the worldsheet. In the openclosed string Bmodel, this chiral ring is isomorphic to a certain relative cohomology group V, which is the appropriate mathematical concept to deal with both the open and closed string sectors. The family of mixed Hodge structures on V then implies for the superpotential to have a certain geometric structure. This structure represents a holomorphic, N = 1 supersymmetric generalization of the wellknown N = 2 special geometry. It defines an integrable connection on the topological family of openclosed Bmodels, and a set of special coordinates on the space M of vev’s in N = 1 chiral multiplets. We show that it can be given a very concrete and simple realization for linear sigma models, which leads to a powerful and systematic method for computing the exact nonperturbative N = 1 superpotentials for a broad class of toric Dbrane geometries.
Homogeneous paraKáhler Einstein manifolds
 Russian Math. Surveys
"... Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract. A paraKähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skewsymmetric paracomplex structures K, i.e. a parallel field of skewsymmetric endomorphisms with K 2 = Id or, equivalently, as a sym ..."
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Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract. A paraKähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skewsymmetric paracomplex structures K, i.e. a parallel field of skewsymmetric endomorphisms with K 2 = Id or, equivalently, as a symplectic manifold (M, ω) with a biLagrangian structure L ± , i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie group G admits an invariant paraKähler structure (g, K) if and only if it is a covering of the adjoint orbit AdGh of a semisimple element h. We give a description of all invariant paraKähler structures (g,K) on a such homogeneous manifold. Using a paracomplex analogue of basic formulas of Kähler geometry, we prove that any invariant paracomplex structure K on M = G/H defines a unique paraKähler Einstein structure (g,K) with given nonzero scalar curvature. An explicit formula for the Einstein metric g is given. A survey of recent results on paracomplex geometry is included. Contents
Proofs of Two Conjectures Related To The Thermodynamic Bethe Ansatz
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1996
"... We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel p(u(θ)+u(θ'}} I. ..."
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Cited by 15 (0 self)
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We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel p(u(θ)+u(θ'}} I.
The tt ∗ structure of the quantum cohomology of CP 1 from the viewpoint of differential geometry
"... The quantum cohomology of CP 1 provides a distinguished solution of the third Painlevé (PIII) equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint (see [4], [5]). We shall derive this from a differential geometric viewpoint, using the theory of harmonic maps ..."
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Cited by 9 (6 self)
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The quantum cohomology of CP 1 provides a distinguished solution of the third Painlevé (PIII) equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint (see [4], [5]). We shall derive this from a differential geometric viewpoint, using the theory of harmonic maps
RUAN’S CONJECTURE AND INTEGRAL STRUCTURES IN QUANTUM COHOMOLOGY
, 2008
"... This is an expository article on the recent studies [23, 24, 44, 19] of Ruan’s crepant resolution/flop conjecture [59, 60] and its possible relations to the Ktheory integral structure [44, 50] in quantum cohomology. ..."
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Cited by 8 (3 self)
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This is an expository article on the recent studies [23, 24, 44, 19] of Ruan’s crepant resolution/flop conjecture [59, 60] and its possible relations to the Ktheory integral structure [44, 50] in quantum cohomology.
Frobenius modules and Hodge asymptotics
 Also, arXiv:math.AG/0207279. STRUCTURES FOR ORBIFOLD COHOMOLOGY 15
"... Abstract. We exhibit a direct correspondence between the potential defining the H 1,1 small quantum module structure on the cohomology of a CalabiYau manifold and the asymptotic data of the Amodel variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge s ..."
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Cited by 3 (2 self)
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Abstract. We exhibit a direct correspondence between the potential defining the H 1,1 small quantum module structure on the cohomology of a CalabiYau manifold and the asymptotic data of the Amodel variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules. 1.
EXAMPLES OF NONCOMMUTATIVE HODGE STRUCTURES
"... We show that, under a condition called minimality, if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semidefinite Hermitian form, then the associated integrable twistor structure (or TERP structure, or noncommutat ..."
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We show that, under a condition called minimality, if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semidefinite Hermitian form, then the associated integrable twistor structure (or TERP structure, or noncommutative Hodge structure) is pure and polarized.