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350
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 346 (2 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
Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties
 J. Alg. Geom
, 1994
"... We consider families F(∆) consisting of complex (n − 1)dimensional projective algebraic compactifications of ∆regular affine hypersurfaces Zf defined by Laurent polynomials f with a fixed ndimensional Newton polyhedron ∆ in ndimensional algebraic torus T = (C ∗ ) n. If the family F(∆) defined by ..."
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Cited by 300 (19 self)
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We consider families F(∆) consisting of complex (n − 1)dimensional projective algebraic compactifications of ∆regular affine hypersurfaces Zf defined by Laurent polynomials f with a fixed ndimensional Newton polyhedron ∆ in ndimensional algebraic torus T = (C ∗ ) n. If the family F(∆) defined by a Newton polyhedron ∆ consists of (n − 1)dimensional CalabiYau varieties then the dual, or polar, polyhedron ∆ ∗ in the dual space defines another family F( ∆ ∗ ) of CalabiYau varieties, so that we obtain the remarkable duality between two different families of CalabiYau varieties. It is shown that the properties of this duality coincide with the properties of Mirror Symmetry discovered by physicists for CalabiYau 3folds. Our method allows to construct many new examples of CalabiYau 3folds and new candidates for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to CalabiYau varieties from two families F(∆) and F( ∆ ∗). 1
Fivebranes, Membranes And NonPerturbative String Theory
, 1995
"... Nonperturbative instanton corrections to the moduli space geometry of type IIA string theory compactified on a CalabiYau space are derived and found to contain order e \Gamma1=g s contributions, where g s is the string coupling. The computation reduces to a weighted sum of supersymmetric extrema ..."
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Cited by 296 (6 self)
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Nonperturbative instanton corrections to the moduli space geometry of type IIA string theory compactified on a CalabiYau space are derived and found to contain order e \Gamma1=g s contributions, where g s is the string coupling. The computation reduces to a weighted sum of supersymmetric extremal maps of strings, membranes and fivebranes into the CalabiYau space, all three of which enter on equal footing. It is shown that a supersymmetric 3cycle is one for which the pullback of the Kahler form vanishes and the pullback of the holomorphic threeform is a constant multiple of the volume element. Quantum mirror symmetry relates the sum in the IIA theory over supersymmetric, odddimensional cycles in the CalabiYau space to a sum in the IIB theory over supersymmetric, evendimensional cycles in the mirror.
CFT’s from CalabiYau Fourfolds
 Nucl. Phys. B584
"... We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a CalabiYau fourfold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated ..."
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Cited by 223 (13 self)
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We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a CalabiYau fourfold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain KazamaSuzuki coset models, such as the N = 2 minimal models. June
On the gauge theory/geometry correspondence
 Adv. Theor. Math. Phys
, 1999
"... The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exa ..."
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Cited by 203 (23 self)
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The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on both sides for arbitrary λ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative Dbrane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.
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 156 (10 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 ...
Mirror Symmetry is TDuality
, 1996
"... It is argued that every CalabiYau manifold X with a mirror Y admits a family of supersymmetric toroidal 3cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y . The mirror transformation is equivalent to Tduality on the 3cycles. The geomet ..."
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Cited by 152 (5 self)
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It is argued that every CalabiYau manifold X with a mirror Y admits a family of supersymmetric toroidal 3cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y . The mirror transformation is equivalent to Tduality on the 3cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed. y email: andy@denali.physics.ucsb.edu yy email: yau@abel.math.harvard.edu yyy email: zaslow@abel.math.harvard.edu 1. Introduction The discovery of mirror symmetry in string theory [1] has led to a number of mathematical surprises. Most investigations have focused on the implications of mirror symmetry of the geometry of CalabiYau moduli spaces. In this paper we shall consider the implications of mirror symmetry of the spectrum of BPS soliton states, which are associated to minimal cycles in the CalabiYau. New surprises will be found. The basic idea we will investigate is briefly as follows. Cons...
Distributions of flux vacua
 JHEP
"... Abstract: We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on CalabiYau manifolds. We compare this with related problems such as counting attractor points. Contents ..."
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Cited by 115 (17 self)
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Abstract: We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on CalabiYau manifolds. We compare this with related problems such as counting attractor points. Contents
Supergravity flows and Dbrane stability
, 2000
"... We investigate the correspondence between existence/stability of BPS states in type II string theory compactified on a CalabiYau manifold and BPS solutions of four dimensional N=2 supergravity. Some paradoxes emerge, and we propose a resolution by considering composite configurations. This in turn ..."
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Cited by 111 (9 self)
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We investigate the correspondence between existence/stability of BPS states in type II string theory compactified on a CalabiYau manifold and BPS solutions of four dimensional N=2 supergravity. Some paradoxes emerge, and we propose a resolution by considering composite configurations. This in turn gives a smooth effective field theory description of decay at marginal stability. We also discuss the connection with 3pronged strings, the Joyce transition of special Lagrangian submanifolds,
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 97 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.