Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau 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 Calabi-Yau 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

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 n-dimensional Newton polyhedron ∆ in n-dimensional algebraic torus T = (C ∗ ) n. If the family F(∆) defined by a Newton polyhedron ∆ consists of (n − 1)-dimensional Calabi-Yau varieties then the dual, or polar, polyhedron ∆ ∗ in the dual space defines another family F( ∆ ∗ ) of Calabi-Yau varieties, so that we obtain the remarkable duality between two different families of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of Mirror Symmetry discovered by physicists for Calabi-Yau 3-folds. Our method allows to construct many new examples of Calabi-Yau 3-folds 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 Calabi-Yau varieties from two families F(∆) and F( ∆ ∗). 1

Non-perturbative instanton corrections to the moduli space geometry of type IIA string theory compactified on a Calabi-Yau 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 Calabi-Yau space, all three of which enter on equal footing. It is shown that a supersymmetric 3-cycle is one for which the pullback of the Kahler form vanishes and the pullback of the holomorphic three-form is a constant multiple of the volume element. Quantum mirror symmetry relates the sum in the IIA theory over supersymmetric, odd-dimensional cycles in the Calabi-Yau space to a sum in the IIB theory over supersymmetric, even-dimensional cycles in the mirror.

We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, 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 Kazama-Suzuki coset models, such as the N = 2 minimal models. June

The ’t Hooft expansion of SU(N) Chern-Simons 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 B-field 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 D-brane 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.

We study the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau 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 D-branes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the Ramond-Ramond field configuration on a Calabi-Yau 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 T-duality are also discussed. Permanent address: Department of Particle Physics, Weizmann Institute of Science, 76100 Rehovot Israel. 1 Introduction D-branes in type II string theories have been identified as Ramond-Ramond charged BPS states [1]. In the presence of a D-brane, the boundary conditions for open strings are modified in such a way that Dirichlet boundary conditions ...

It is argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y . The mirror transformation is equivalent to T-duality on the 3-cycles. 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 Calabi-Yau 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 Calabi-Yau. New surprises will be found. The basic idea we will investigate is briefly as follows. Cons...

Abstract: We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points. Contents

We investigate the correspondence between existence/stability of BPS states in type II string theory compactified on a Calabi-Yau 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 3-pronged strings, the Joyce transition of special Lagrangian submanifolds, An old and common idea in physics is that a particle makes its presence manifest via excitation of fields. If one puts a lot of particles together, one gets a macroscopic object, well described by classical physics, and correspondingly one expects the field excitations to be well described by a classical field theory. In particular, it seems

this paper, we will find a third conformal field theory of the conifold -- call it the C model. In the C model, there is a Z 2 discrete torsion sitting at the "singularity," which is in fact not a singularity in the conformal field theory sense, but just a region in which stringy effects are essential. The C model has no marginal operators (in particular the analogs of H