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93
Topological string theory on compact CalabiYau: Modularity and boundary conditions
, 2006
"... The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact CalabiYau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the m ..."
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Cited by 83 (11 self)
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The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact CalabiYau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the moduli space M(M) along with preferred local coordinates. Modular properties of the sections Fg as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo’s theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.
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 72 (5 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 holomorphic and background independent partition function for matrix models and topological strings
, 2009
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Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
, 2008
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Effective superpotentials for compact D5brane CalabiYau geometries
, 2008
"... For compact CalabiYau geometries with D5branes we study N = 1 effective superpotentials depending on both open and closedstring fields. We develop methods to derive the open/closed PicardFuchs differential equations, which control D5brane deformations as well as complex structure deformations ..."
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Cited by 35 (6 self)
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For compact CalabiYau geometries with D5branes we study N = 1 effective superpotentials depending on both open and closedstring fields. We develop methods to derive the open/closed PicardFuchs differential equations, which control D5brane deformations as well as complex structure deformations of the compact CalabiYau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5brane CalabiYau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed coordinates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants form the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk GromovWitten invariants in the mirror geometry.
The Witten equation and its virtual fundamental cycle
, 2007
"... We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory. ..."
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Cited by 31 (5 self)
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We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory.
Holomorphic Anomaly and Matrix Models
, 2007
"... The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vaf ..."
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Cited by 29 (10 self)
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The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf–Vafa conjecture relating matrix models to type B topological strings on certain local Calabi–Yau threefolds.
The space of stability conditions on the local projective plane
 Duke Math. J
"... ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this conne ..."
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Cited by 27 (3 self)
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ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simplyconnected. We determine the group of autoequivalences preserving this connected component. Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with level structure, with the morphism given by solutions of PicardFuchs equations. This result is motivated by the notion of Πstability and by mirror symmetry. 1.