Showing papers on "Mirror symmetry published in 1995"
TL;DR: 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 zeros).
Abstract: Mirror symmetry (MS) 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 zeros). 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
$$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$
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1,510 citations
TL;DR: In this article, nonperturbative 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 − 1/g s contributions, where g s is the string coupling.
963 citations
TL;DR: In this paper, an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry is made. But this work is restricted to the case of a single curve.
Abstract: This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry.
712 citations
TL;DR: In this paper, nonperturbative 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 $e^{-1/g_s}$ contributions, where $g s$ is the string coupling.
Abstract: 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 $e^{-1/g_s}$ 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.
516 citations
TL;DR: Mirror symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed in this article for Calabi-Yau spaces with two and three moduli.
Abstract: Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.
488 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 ⊂ V.
411 citations
TL;DR: In this paper, the authors extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete intersections with higher dimensional moduli spaces.
382 citations
TL;DR: In this article, a duality relation between Type II theories on CalabiYau spaces and heterotic strings on K3×T 2, both of which have N = 2 spacetime supersymmetry, was established.
378 citations
TL;DR: A critical examination of the extensive literature about the effects on symmetry detection of several major factors such as the orientation of the symmetry axis, the location of the stimulus in the visual field, grouping, and perturbations is carried out.
Abstract: This paper reviews empirical evidence for the detection of visual symmetries and explanatory theories and models of symmetry detection. First, mirror symmetry is compared to other types of symmetry. The idea that symmetry detection is preattentive is then discussed and other roles that attention might play in symmetry detection are considered. The major part of the article consists of a critical examination of the extensive literature about the effects on symmetry detection of several major factors such as the orientation of the symmetry axis, the location of the stimulus in the visual field, grouping, and perturbations. Constraints on plausible models of symmetry detection are derived from this rich database and several proposals are evaluated against it. As a result of bringing this research together, open questions and remaining gaps to be filled by future research are identified.
323 citations
TL;DR: In this paper, the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion is considered, and examples of these models also give particularly simple and clear examples of mirror symmetry.
314 citations
01 Jan 1995
TL;DR: In this article, the authors review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are quasimodular forms for the mirror family; they can be computed by a summation over trivalent Feynman graphs.
Abstract: I review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: (1) counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are ‘quasimodular forms’ for the mirror family; (2) they can be computed by a summation over trivalent Feynman graphs.
01 Jan 1995
TL;DR: A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4.
Abstract: A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4. Hodge numbers of a nonsingular quintic are know to be: h p, p = 1, p = 0, 1, 2, 3 (Kahler form and its powers), h3, 0 = h0,3 = 1 (a quintic happens to bear a holomorphic volume form), h2,1 = h1, 2 = 101 = 126 - 25 (it is the dimension of the space of all quintics modulo projective transformations, and h2,1 is responsible here for infinitesimal variations of the complex structure) and all the other h p,q = 0.
TL;DR: In this article, 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.
TL;DR: In this paper, the authors compute the elliptic genus for arbitrary two-dimensional N = 2 Landau-Ginzburg orbifolds and search for possible mirror pairs of such models.
TL;DR: The moduli dependence of (2,2) superstring compactifications based on Calabi-Yau hypersurfaces in weighted projective space has been investigated for Fermat-type polynomial constraints in this paper.
TL;DR: In this paper, Batyrev's construction of the missing mirrors in the Calabi-Yau manifold set was used to show that many of these missing mirrors may be interpreted as non-transverse hypersurfaces in weighted P 4's, i.e. hypersurface for which dp vanishes at a point other than the origin.
TL;DR: The moduli dependence of superstring compactifications based on Calabi-Yau hypersurfaces in weighted projective space has been investigated in this paper for Fermat-type polynomial constraints.
Abstract: The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with $c=9$ whose potential is a sum of $A$-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at $c=9$. We use mirror symmetry to derive the dependence of the models on the complexified Kahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (``twisted'') deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.
TL;DR: In this paper, a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski in the context of toric geometry is presented, where the Grobner basis for the toric ideal determines a finite set of differential operators for the local solutions of the GKZ system.
Abstract: We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Grobner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up to $h^{1,1}=3$. We also find and analyze several non Landau-Ginzburg models which are related to singular models.
TL;DR: Symmetry detection was investigated for static and dynamic noise targets consisting of a field of approximately 0.3 million random dots on which was imposed a bilateral symmetry, implying that there is some neural mechanism performing full temporal integration of the symmetry information up to durations of a second or more.
Abstract: — Symmetry detection was investigated for static and dynamic noise targets consisting of a field of ~ 03 million random dots on which was imposed a bilateral symmetry The minimum duration for detection was 40 ms for static and 80 ms for dynamic symmetry The exponents of the psychometric functions averaged about 4 for both static and dynamic tasks, as opposed to the value of 1 expected for such suprathreshold tasks, implying that there is some neural mechanism performing full temporal integration of the symmetry information up to durations of a second or more Static symmetry was perceivable when information around the symmetry axis was masked up to 3 deg away from the symmetry axis, revealing extrafoveal symmetry detection in ~ 300 ms exposures The static data were fitted with a model consisting of three mechanisms with Gaussian spatial profiles and mutual inhibition (two mechanisms were sufficient for the dynamic data) The profile of the widest mechanism was 20 times wider for static than for dynamic symmetry
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01 Jan 1995
TL;DR: An introduction to the subject supergravity and Kahler geometry can be found in this paper, where the authors introduce the subject of Calabi-Yau manifolds, Gepner tensor products moduli spaces and topological field theories.
Abstract: An introduction to the subject supergravity and Kahler geometry Calabi-Yau manifolds N=2 field theories in two dimensions Gepner tensor products moduli spaces and special geometry topological field theories mirror symmetry Picard-Fuchs equations monodromy and duality groups
TL;DR: In this article, it was shown that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi-Yau string vacua.
Abstract: It is argued that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi--Yau string vacua. The condensate signals a smooth transition to a new Calabi--Yau space with different Euler characteristic and Hodge numbers. In this manner string theory unifies the moduli spaces of many or possibly all Calabi--Yau vacua. Elementary string states and black holes are smoothly interchanged under the transitions, and therefore cannot be invariantly distinguished. Furthermore, the transitions establish the existence of mirror symmetry for many or possibly all Calabi--Yau manifolds.
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TL;DR: In this paper, the authors considered the variant of mirror symmetry conjecture for K3 surfaces which relates the geometry of curves of a general member of a family of K3 with algebraic functions on the moduli of the mirror family.
Abstract: We consider the variant of Mirror Symmetry Conjecture for K3 surfaces which relates "geometry" of curves of a general member of a family of K3 with "algebraic functions" on the moduli of the mirror family Lorentzian Kac--Moody algebras are involved in this construction We give several examples when this conjecture is valid
TL;DR: In this paper, a duality relation between Type II string theory on a self-mirror Calabi-Yau space and a background of heterotic string theory was proposed.
Abstract: We propose and give strong evidence for a duality relating Type II theories on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual theories are derived from an analysis of the soliton string worldsheet in the context of $N=2$ orbifolds of dual $N=4$ compactifications of Type II and heterotic strings. In particular we construct a pairing between Type II string theory on a self-mirror Calabi-Yau space $X$ with $h^{11}= h^{21}= 11$ and a $(4,0)$ background of heterotic string theory on $K3\times T^2$. Under the duality transformation the usual first-quantized mirror symmetry of $X$ becomes a second-quantized mirror symmetry which determines nonperturbative quantum effects. This enables us to compute the exact quantum moduli space. Mirror symmetry of $X$ implies that the low-energy $N=2$ gauge theory is finite, even at enhanced symmetry points. This prediction is verified by direct computation on the heterotic side. Other branches of the moduli space, and corresponding dual pairs which are not finite $N=2$ theories, are connected to this one via black hole condensation.
TL;DR: In this article, duality and mirror symmetry properties of Landau-Ginzburg orbifolds were discussed considering their elliptic genera under the duality transform performed by orbifoldizing the LG model via some discrete group of superpotential, and the roles of the untwisted and twisted sectors are exchanged.
Abstract: We discuss duality and mirror symmetry phenomena of Landau-Ginzburg orbifolds considering their elliptic genera Under the duality (or mirror) transform performed by orbifoldizing the Landau-Ginzburg model via some discrete group of the superpotential we observe that the roles of the untwisted and twisted sectors are exchanged As explicit evidence detailed orbifold data are presented for $N=2$ minimal models, Arnold's exceptional singularities, $K3$ surfaces constructed from Arnold's singularities and Fermat hypersurfaces (To appear in the proceedings of the workshop, ``Quantum Field Theory, Integrable Models and Beyond'', Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 February 1994)
01 Apr 1995
TL;DR: In this paper, the authors review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ''constants of integration'' in the mirror map should be fixed.
Abstract: Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on the other. We review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ``constants of integration'' in the mirror map should be fixed. Such predictions can be useful for checking whether or not various conjectural constructions of mirror manifolds are producing reasonable answers.
TL;DR: In this article, the K3 surface and the complex torus were discussed from the viewpoint of mirror symmetry and a mirror map for the two-point correlation function and the prepotential was constructed.
Abstract: We discuss the K3 surface and the complex torus from the viewpoint of “mirror symmetry.” We calculate the periods of some K3 surface and construct a mirror map for the two-point correlation function and the prepotential. We find that there are no instanton corrections of the Yukawa coupling for K3 (also torus), which is expected from the viewpoint of algebraic geometry.
TL;DR: In this article, it is shown that the mirror transform based on fractional transformations allows an extension of the mirror map to conifold boundary points of the moduli space of weighted Calabi-Yau manifolds.
Abstract: Recent work initiated by Strominger has lead to a consistent physical interpretation of certain types of transitions between different string vacua. These transitions, discovered several years ago, involve singular conifold configurations which connect distinct Calabi-Yau manifolds. In this paper we discuss a number of aspects of conifold transitions pertinent to both worldsheet and spacetime mirror symmetry. It is shown that the mirror transform based on fractional transformations allows an extension of the mirror map to conifold boundary points of the moduli space of weighted Calabi-Yau manifolds. The conifold points encountered in the mirror context are not amenable to an analysis via the original splitting constructions. We describe the first examples of such nonsplitting conifold transitions, which turn out to connect the known web of Calabi-Yau spaces to new regions of the collective moduli space. We then generalize the splitting conifold transition to weighted manifolds and describe a class of connections between the webs of ordinary and weighted projective Calabi-Yau spaces. Combining these two constructions we find evidence for a dual analog of conifold transitions in heterotic N$=$2 compactifications on K3$\times $T$^2$ and in particular describe the first conifold transition of a Calabi-Yau manifold whose heterotic dual has been identified by Kachru and Vafa. We furthermore present a special type of conifold transition which, when applied to certain classes of Calabi-Yau K3 fibrations, preserves the fiber structure.
TL;DR: For a large class of N = 2 SCFTs, which includes minimal models and many σ models on Calabi-Yau manifolds, the mirror theory can be obtained as an orbifold as mentioned in this paper.
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TL;DR: In this article, the authors established a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for k3 surfaces, and computed various examples of mirror moduli families for the moduli space of degree 2n polarized K3 surface.
Abstract: We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n polarized K3 surfaces. It turns out to be related to the moduli space of elliptic curves with level n.
TL;DR: In this article, the authors complete the classification of (2, 2) vacua that can be constructed from Landau-Ginzburg models by Abelian twists with arbitrary discrete torsions.
Abstract: We complete the classification of (2, 2) vacua that can be constructed from Landau-Ginzburg models by Abelian twists with arbitrary discrete torsions. Compared to the case without torsion, the number of new spectra is surprisingly small. In contrast to a popular expectation mirror symmetry does not seem to be related to discrete torsion (at least not in the present compactification framework). The Berglund-Hubsch construction naturally extends to orbifolds with torsion; for more general potentials, on the other hand, the new spectra neither have nor provide mirror partners in our class of models.