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Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 523 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 474 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that
Black Hole Entropy Function, Attractors and Precision Counting of Microstates
, 2007
"... In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N = 4 supersymmetric strin ..."
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Cited by 324 (28 self)
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In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N = 4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multicentered black holes as well.
Superstrings and topological strings at large
 N”, J. Math. Phys
"... We embed the large N ChernSimons/topological string duality in ordinary superstrings. This corresponds to a large N duality between generalized gauge systems with N = 1 supersymmetry in 4 dimensions and superstrings propagating on noncompact CalabiYau manifolds with certain fluxes turned on. We a ..."
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Cited by 254 (27 self)
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We embed the large N ChernSimons/topological string duality in ordinary superstrings. This corresponds to a large N duality between generalized gauge systems with N = 1 supersymmetry in 4 dimensions and superstrings propagating on noncompact CalabiYau manifolds with certain fluxes turned on. We also show that in a particular limit of the N = 1 gauge theory system, certain superpotential terms in the N = 1 system (including deformations if spacetime is noncommutative) are captured to all orders in 1/N by the amplitudes of noncritical bosonic strings propagating on a circle with selfdual radius. We also consider Dbrane/antiDbrane system wrapped over vanishing cycles of compact CalabiYau manifolds and argue that at large N they induce a shift in the background to a topologically distinct CalabiYau, which we identify as the ground state system of the Brane/antiBrane system. August
N=4 topological strings
 Nucl. Phys. B
, 1995
"... We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applicat ..."
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Cited by 225 (23 self)
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We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applications for this new topological theory include: 1) Proving the vanishing to all orders of all scattering amplitudes for the selfdual N = 2 string with flat background, with the exception of the threepoint function and the closedstring partition function; 2) Showing that the topological partition function of the N = 2 string on the K3 background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and 3) Providing a new prescription for calculating superstring amplitudes which appears to be free of totalderivative ambiguities. July
Mirror symmetry, D–branes and counting holomorphic discs
, 2000
"... We consider a class of special Lagrangian subspaces of CalabiYau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian subman ..."
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Cited by 202 (25 self)
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We consider a class of special Lagrangian subspaces of CalabiYau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical AbelJacobi map on the mirror. We recover some results already anticipated as well as obtain some highly nontrivial new predictions.
DBranes on CalabiYau Spaces and Their Mirrors
, 1996
"... We study the boundary states of Dbranes wrapped around supersymmetric cycles in a general CalabiYau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms Dbranes, and we verify tha ..."
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Cited by 185 (11 self)
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We study the boundary states of Dbranes wrapped around supersymmetric cycles in a general CalabiYau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms Dbranes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the RamondRamond field configuration on a CalabiYau manifold. This also enables us to study open string worldsheet instanton corrections and relate them to closed string instanton counting. The cases when the mirror symmetry is realized as Tduality are also discussed. Permanent address: Department of Particle Physics, Weizmann Institute of Science, 76100 Rehovot Israel. 1 Introduction Dbranes in type II string theories have been identified as RamondRamond charged BPS states [1]. In the presence of a Dbrane, the boundary conditions for open strings are modified in such a way that Dirichlet boundary conditions ...
The topological vertex
, 2003
"... We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the th ..."
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Cited by 167 (25 self)
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We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the Bmodel mirror which is the quantum KodairaSpencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the Bbranes on the mirror Riemann