Results 1  10
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17
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 28 (6 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.
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 18 (1 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].
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 3 (2 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
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.
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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Cited by 3 (0 self)
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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
, 1995
"... 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 I. ..."
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Cited by 2 (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 I.
FOURIERLAPLACE TRANSFORM OF FLAT UNITARY CONNECTIONS AND TERP STRUCTURES by
"... Abstract. We show that 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 ..."
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Abstract. We show that 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. Contents Introduction........................................................ 1 1. Polarized pure twistor structure attached to a flat unitary bundle 3 2. Stokes filtration and Stokes data................................ 8 3. Natural operations on Stokes filtrations and Stokes data........ 14
Communications IΠ Mathematical Physics © SpringerVerlag 1996 Proofs of Two Conjectures Related
, 1995
"... Abstract: 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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Abstract: 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.
Symmetry, Integrability and Geometry: Methods and Applications Clifford Algebra Derivations of TauFunctions for TwoDimensional Integrable Models with Positive and Negative Flows ⋆
"... Abstract. We use a Grassmannian framework to define multicomponent tau functions as expectation values of certain multicomponent Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy ..."
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Abstract. We use a Grassmannian framework to define multicomponent tau functions as expectation values of certain multicomponent Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the 2ncomponent KP hierarchy. The hierarchy equations can be formulated in terms of pseudodifferential equations for n × n matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato– Wilson relations. A reduction process leads to the AKNS, twocomponent Camassa–Holm and Cecotti–Vafa models and the formalism provides simple formulas for their solutions. Key words: Clifford algebra; taufunctions; Kac–Moody algebras; loop groups; Camassa– Holm equation; Cecotti–Vafa equations; AKNS hierarchy