Showing papers on "Mirror symmetry published in 1994"
TL;DR: In this article, the authors describe the space of the two Kahler parameters of the Calabi-Yau manifold P4(1, 1,1,6,9) by exploiting mirror symmetry.
469 citations
TL;DR: In this paper, the moduli spaces of Calabi-yau three-folds and their associated conformally invariant nonlinear σ-models are analyzed, and they are described by an unexpectedly rich geometrical structure.
418 citations
TL;DR: In this article, the authors use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi-Yau hypersurface $M \subset V$.
Abstract: We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.
279 citations
24 Apr 1994
TL;DR: The moduli space of N = 4,4 string theories with a K3 target space is determined in this article, and it is shown that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20).
Abstract: The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces.
249 citations
TL;DR: In this article, the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion is considered, and examples of stringy regions that from a classical point of view are singularities that are to be neither resolved nor blown up are given.
Abstract: We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these models also give particularly simple and clear examples of mirror symmetry.
202 citations
Posted Content•
TL;DR: In this paper, a special class of convex rational polyhedral cones which allow to construct generalized Calabi-Yau varieties of dimension (d + 2(r-1)) is introduced, where r is a positive integer and d is the dimension of critical string vacua.
Abstract: We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with central chatge $c = 3d$ It is conjectured that the natural combinatorial duality satisfies by these cones corresponds to the mirror involution Using the theory of toric varieties, we show that our conjecture includes as special cases all already known examples of mirror pairs proposed by physicists and agrees with previous conjectures of the authors concerning explicit constructions of mirror manifolds In particular we obtain a mathematical framework which explains the construction of mirrors of rigid Calabi-Yau manifolds
170 citations
TL;DR: In this article, mirror symmetry of strings on Calabi-Yau manifolds is introduced, with an emphasis on its applications e.g. for the computation of Yukawa couplings.
Abstract: We give an introduction to mirror symmetry of strings on Calabi-Yau manifolds with an emphasis on its applications e.g. for the computation of Yukawa couplings. We introduce all necessary concepts and tools such as the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc. and illustrate all of this on a non-trivial example.
163 citations
TL;DR: In this paper, the authors analyze global aspects of the moduli space of Kahler forms for N=(2,2) con-formal σ-models and find strong evidence that, in the robust setting of quantum Calabi-Yau moduli spaces, string theory restricts the set of possible kahler forms by enforcing "minimal length" scales, provided that topology change is properly taken into account.
163 citations
Posted Content•
TL;DR: In this article, the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories.
Abstract: In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C^d/G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C^2/Z_n is analyzed explicitly. This is largely a review paper to appear in "Essays on Mirror Manifolds, II".
122 citations
Posted Content•
TL;DR: In this paper, the authors studied compactification down to 3 on a 7-dimensional manifold of $G_2$ holonomy, and down to 2 on an 8-dimensional manifolds of $Spin(7)-holonomy.
Abstract: The condition of having an $N=1$ spacetime supersymmetry for heterotic string leads to 4 distinct possibilities for compactifications namely compactifications down to 6,4,3 and 2 dimensions. Compactifications to 6 and 4 dimensions have been studied extensively before (corresponding to $K3$ and a Calabi-Yau threefold respectively). Here we complete the study of the other two cases corresponding to compactification down to 3 on a 7 dimensional manifold of $G_2$ holonomy and compactification down to 2 on an 8 dimensional manifold of $Spin(7)$ holonomy. We study the extended chiral algebra and find the space of exactly marginal deformations. It turns out that the role the $U(1)$ current plays in the $N=2$ superconformal theories, is played by tri-critical Ising model in the case of $G_2$ and Ising model in the case of $Spin(7)$ manifolds.
Certain generalizations of mirror symmetry are found for these two cases. We also discuss a topological twisting in each case.
108 citations
TL;DR: In this paper, it was shown that the elusive mirror of a rigid manifold is actually a supermanifold and that mirror symmetry should be viewed as a relation among super-varieties rather than bosonic varieties.
Posted Content•
TL;DR: In this article, a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott is presented, which is consistent with predictions made from mirror symmetry computations.
Abstract: We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.
TL;DR: In this article, it was shown that the elusive mirror of a rigid manifold is actually a supermanifold and that mirror symmetry should be viewed as a relation among super-varieties rather than bosonic varieties.
Abstract: By providing a general correspondence between Landau-Ginzburg orbifolds and non-linear sigma models, we find that the elusive mirror of a rigid manifold is actually a supermanifold. We also discuss when sigma models with super-target spaces are conformally invariant and describe their chiral rings. Both supermanifolds with and without Kahler moduli are considered. This work leads us to conclude that mirror symmetry should be viewed as a relation among super-varieties rather than bosonic varieties.
TL;DR: In this article, the chiral ring may be identical for different associated conformal field theories in terms of both A-model and B-model language, and this ambiguity is explained by both the A-and B-models.
TL;DR: In this paper, the idea of minimum distance, familiar from R ↔ 1 R duality when the string target space is a circle, is analyzed for less trivial geometries.
TL;DR: Batyrev as mentioned in this paper showed that many of these missing mirrors may be interpreted as non-transverse hypersurfaces for which dp vanishes at a point other than the origin.
Abstract: Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.
TL;DR: In this article, the moduli space of N = 2 superconformal field theories is reviewed and a review of recent work which has significantly honed the geometric understanding and interpretation of the modulus space of certain N=2 superconforming field theories are presented.
Abstract: Recent work which has significantly honed the geometric understanding and interpretation of the moduli space of certain N=2 superconformal field theories is reviewed. This has resolved some important issues in mirror symmetry and has also established that string theory admits physically smooth processes which can result in a change in topology of the spatial universe. Recent work which illuminates some properties of physically related theories associated with singular spaces such as orbifolds is described.
TL;DR: In this paper, the authors considered an (N-2)-dimensional Calabi-Yau manifold, which is defined as the zero locus of the polynomial of degree N (of Fermat type) in CP^{N-1} and its mirror manifold.
Abstract: We consider an (N-2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of Fermat type) in CP^{N-1} and its mirror manifold. We introduce the (N-2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and also by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of Algebraic geometry...
TL;DR: It was shown in this paper that mirror duality is a Weyl transformation in the moduli space of N = 2 backgrounds on group manifolds, and conjecture on the possible generalization to other backgrounds, such as Calabi-Yau manifolds.
01 Mar 1994
TL;DR: In this article, the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories.
Abstract: In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C^d/G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C^2/Z_n is analyzed explicitly. This is largely a review paper to appear in "Essays on Mirror Manifolds, II".
Posted Content•
TL;DR: The moduli space of N = 4,4 string theories with a K3 target space is determined in this article, and it is shown that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20).
Abstract: The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces.
TL;DR: In this paper, mirror symmetry of strings on Calabi-Yau manifolds is introduced with an emphasis on its applications e.g. for the computation of Yukawa couplings, and the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc.
Abstract: We give an introduction to mirror symmetry of strings on Calabi-Yau manifolds with an emphasis on its applications e.g. for the computation of Yukawa couplings. We introduce all necessary concepts and tools such as the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc. and illustrate all of this on a non-trivial example. Extended version of a lecture given at the Third Baltic Student Seminar, Helsinki September 1993
Posted Content•
TL;DR: Some aspects of mirror symmetry are reviewed in this paper, with an emphasis on more recent results extending mirror transform to higher genus Riemann surfaces and its relation to the Kodaira-Spencer theory of gravity.
Abstract: Some aspects of Mirror symmetry are reviewed, with an emphasis on more recent results extending mirror transform to higher genus Riemann surfaces and its relation to the Kodaira-Spencer theory of gravity (talk given in the Geometry and Topology Conference, April 93, Harvard, in honor of Raoul Bott)
Posted Content•
TL;DR: In this article, a new higher dimensional version of the McKay correspondence is proposed, which enables us to understand the "Hodge numbers" assigned to singular Gorenstein varieties by physicists, leading to the conjecture that string theory indicates the existence of some new cohomology theory H st ∗ (X) for algebraic varieties with gird singularities.
Abstract: We propose a new higher dimensional version of the McKay correspondence which enables us to understand the “Hodge numbers” assigned to singular Gorenstein varieties by physicists. Our results lead to the conjecture that string theory indicates the existence of some new cohomology theory H st ∗ (X) for algebraic varieties with Gorenstein singularities. We give a formal mathematical definition of the Hodge numbers hstp,q(X) inspired from the consideration of strings on orbifolds and from this new (conjectural) version of the McKay correspondence. The numbers hstp,q(X) are expected to give the spectrum of orbifoldized Landau-Ginzburg models and mirror duality relations for higher dimensional Calabi-Yau varieties with Gorenstein toroidal or quotient singularities.
TL;DR: In this paper, the authors studied the N = 2 coset models in their formulation as supersymmetric gauged Wess-Zumino-Witten models and showed that a model based on the coset G/H is invariant under a symmetry group isomorphic to Zk+Q where k is the level of the model and Q is dual Coxeter number of G.
TL;DR: In this article, the authors describe in detail the space of the two Kahler parameters of the Calabi-Yau manifold by exploiting mirror symmetry, and compute instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree.
Abstract: We describe in detail the space of the two K\"ahler parameters of the Calabi--Yau manifold $\P_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.
01 Jun 1994
TL;DR: In this paper, the elliptic genus for arbitrary two dimensional Landau-Ginzburg orbifolds is computed and a search for possible mirror pairs of such models is performed.
Abstract: The elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds is computed. This is used to search for possible mirror pairs of such models. An important aspect of this work is that there is no restriction to theories for which the conformal anomaly is $\hat c\in\ZZ$, nor are the results only valid at the conformal fixed point.
Posted Content•
TL;DR: In this paper, the authors review various constructions of mirror symmetry in terms of Landau-Ginzburg orbifolds for arbitrary central charge $c$ and hypersurfaces and complete intersections in toric varieties.
Abstract: We review various constructions of mirror symmetry in terms of Landau-Ginzburg orbifolds for arbitrary central charge $c$ and \CY\ hypersurfaces and complete intersections in toric varieties. In particular it is shown how the different techniques are related
TL;DR: In this paper, the authors compute the elliptic genus for arbitrary two-dimensional Landau-Ginzburg orbifolds and search for possible mirror pairs of such models, and give a sufficient (and possibly necessary) condition for two models to be conjugate, and show that it is satisfied by the mirror pairs.
Abstract: We compute the elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds. This is used to search for possible mirror pairs of such models. We show that if two Landau-Ginzburg models are conjugate to each other in a certain sense, then to every orbifold of the first theory corresponds an orbifold of the second theory with the same elliptic genus (up to a sign) and with the roles of the chiral and anti-chiral rings interchanged. These orbifolds thus constitute a possible mirror pair. Furthermore, new pairs of conjugate models may be obtained by taking the product of old ones. We also give a sufficient (and possibly necessary) condition for two models to be conjugate, and show that it is satisfied by the mirror pairs proposed by one of the authors and~Hubsch.
TL;DR: In this article, it is conjectured that the nodal surface of the first excited state of a convex 3D quantum billiard intersects the billiard surface in a single simple closed curve.
Abstract: Stemming from known properties of one‐dimensional (1‐D) and 2‐D quantum billiards, it is conjectured that the nodal surface of the first‐excited state of the convex 3‐D quantum billiard intersects the billiard surface in a single simple closed curve. Examples of the validity of this conjecture are given for a number of elementary 3‐D billiard configurations. From these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation. Two characteristic displacement parameters are defined which are labeled an axis length, L, and diameter, a. It is conjectured that a parameter κ≊1, exists, whose exact value depends on the properties of the billiard, such that for L≳κa (‘‘prolatelike’’) the nodal surface of the first‐excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half. For L<κa (‘‘oblatelike’’) the nodal surface of the first‐excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half. Arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards, termed ‘‘spherical‐like.’’ It is hypothesized that the nodal surface of the first‐excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron.