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16
Geometry and analytic theory of Frobenius manifolds
, 1998
"... Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifol ..."
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Cited by 37 (3 self)
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Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.
Frobenius manifolds and Virasoro constraints
 Selecta Math. (N.S
, 1999
"... For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.S.Xiong and of S.Katz is prove ..."
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Cited by 30 (4 self)
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For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology. 1
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 27 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
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 19 (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].
Mirror symmetry in two steps: AIB
"... Abstract. We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I ..."
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Cited by 5 (0 self)
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Abstract. We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted LandauGinzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the LandauGinzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.
tt∗GEOMETRY IN QUANTUM COHOMOLOGY
, 2009
"... We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt ∗geometry near the large radius limit. We compute an example of P¹ which is pure and ..."
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We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt ∗geometry near the large radius limit. We compute an example of P¹ which is pure and polarized over the whole Kähler moduli space H²(P¹, C∗).
BCOV RING AND HOLOMORPHIC ANOMALY EQUATION
, 810
"... Abstract. We study certain differential rings over the moduli space of CalabiYau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasimodular forms due to KanekoZagier[KZ]. 1. ..."
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Abstract. We study certain differential rings over the moduli space of CalabiYau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasimodular forms due to KanekoZagier[KZ]. 1.
BCOV RING AND HOLOMORPHIC ANOMALY EQUATION
, 810
"... Abstract. We study certain differential rings over the moduli space of CalabiYau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasimodular forms due to KanekoZagier[23]. 1. ..."
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Abstract. We study certain differential rings over the moduli space of CalabiYau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasimodular forms due to KanekoZagier[23]. 1.