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50
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 73 (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].
A New supersymmetric index
 Nucl. Phys. B
, 1992
"... We show that Tr(−1) FFe −βH is an index for N=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry’s curvature) and satisfies an exact differential equation as a function o ..."
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Cited by 43 (7 self)
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We show that Tr(−1) FFe −βH is an index for N=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry’s curvature) and satisfies an exact differential equation as a function of β. For integrable theories we can also compute the index thermodynamically, using the exact Smatrix. The equivalence of these two results implies a highly nontrivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations. 4/92
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
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 17 (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
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
SOME CONSTRAINTS ON FROBENIUS MANIFOLDS WITH A TT*STRUCTURE
, 904
"... Abstract. The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDVstructure. We show that there exists a positive definite CDVstructure on any semisimple Frobenius manifold. We also compare three natural connections on a CDVstructure and conclude that the under ..."
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Cited by 4 (1 self)
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Abstract. The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDVstructure. We show that there exists a positive definite CDVstructure on any semisimple Frobenius manifold. We also compare three natural connections on a CDVstructure and conclude that the underlying Hermitian manifold of a nontrivial CDVstructure is not a Kähler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifolds. Cecotti and Vafa [1] [4] considered moduli spaces of N = 2 supersymmetric quantum field theories and introduced a geometry on them which is governed by the tt*equations. By the work of K. Saito and M. Saito, it was previously known that the base space of a semiuniversal
Multidimensional Toda type systems
"... On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed. 1 ..."
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Cited by 3 (2 self)
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On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed. 1