Showing papers on "Mirror symmetry published in 1996"
TL;DR: In this paper, it was argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles and that the moduli space of such cycles together with their flat connections is precisely the space Y.
1,607 citations
TL;DR: In this paper, the authors discuss non-trivial fixed points of the renormalization group with dual descriptions in N = 4 gauge theories in three dimensions and show that small E8 instantons in string theory are described by a local quantum field theory.
868 citations
495 citations
TL;DR: In particular, the authors showed that in three dimensions small $E_8$ instantons in string theory are described by a local quantum field theory, which acts as mirror symmetry, exchanging the Higgs and Coulomb branches of the theories.
Abstract: We discuss non-trivial fixed points of the renormalization group with dual descriptions in $N=4$ gauge theories in three dimensions. This new duality acts as mirror symmetry, exchanging the Higgs and Coulomb branches of the theories. Quantum effects on the Coulomb branch arise classically on the Higgs branch of the dual theory. We present examples of dual theories whose Higgs/Coulomb branch are the ALE spaces and whose Coulomb/Higgs branches are the moduli space of instantons of the corresponding $ADE$ gauge group. In particular, we show that in three dimensions small $E_8$ instantons in string theory are described by a local quantum field theory.
463 citations
TL;DR: In this paper, the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau manifold were studied and the geometric data on the cycles were encoded in the boundary state.
457 citations
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.
259 citations
TL;DR: This study is an attempt to survey all existing hypotheses containing mirror symmetry retained in the world of chiral molecules and find out whether this phenomenon is possible to account for without going beyond conventional concepts of the kinetics of enantioselective processes.
Abstract: Reasoning from two basic principles of molecular physics, P invariance of electromagnetic interaction and the second law of thermodynamics, one would conclude that mirror symmetry retained in the world of chiral molecules. This inference is fully consistent with what is observed in inorganic nature. However, in the bioorganic world, the reverse is true. Mirror symmetry there is definitely broken. Is it possible to account for this phenomenon without going beyond conventional concepts of the kinetics of enantioselective processes? This study is an attempt to survey all existing hypotheses containing this phenomenon.
184 citations
TL;DR: In this article, 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 toh
1,1=3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.
153 citations
TL;DR: In this paper, the necessary and sufficient conditions for supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau four-folds were derived in the SCFT and low energy effective action frameworks.
146 citations
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TL;DR: A survey of recent developments in the theory of toric varieties can be found in this article, including new constructions of Toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.
Abstract: This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.
TL;DR: In this paper, the BPS states of non-critical strings which arise for zero-size instantons of exceptional groups were studied by using Ftheory and M-theory duals and employing mirror symmetry to compute the degeneracy of membranes wrapped around 2-cycles of the Calabi-Yau threefold.
Abstract: We study the BPS states of non-critical strings which arise for zero size instantons of exceptional groups. This is accomplished by using F-theory and M-theory duals and by employing mirror symmetry to compute the degeneracy of membranes wrapped around 2-cycles of the Calabi-Yau threefold. We find evidence for a number of novel physical phenomena, including having infinitely many light states with the first lightest state including a nearly massless gravitino.
01 Mar 1996
TL;DR: In this paper, mirror symmetry for superconformal sigma models with Calabi-Yau target spaces described as complete intersection subvarieties in toric varieties is formulated as a duality in the abelian gauge theory.
Abstract: Superconformal sigma models with Calabi–Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi–Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture.
TL;DR: In this article, the necessary and sufficient conditions for supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds were derived in the SCFT and low energy effective action frameworks.
Abstract: We derive in the SCFT and low energy effective action frameworks the necessary and sufficient conditions for supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds. We show that the Cayley cycles in $Spin(7)$ holonomy eight-manifolds and the associative and coassociative cycles in $G_2$ holonomy seven-manifolds preserve half of the space-time supersymmetry. We find that while the holomorphic and special Lagrangian cycles in Calabi-Yau 4-folds preserve half of the space-time supersymmetry, the Cayley submanifolds are novel as they preserve only one quarter of it. We present some simple examples. Finally, we discuss the implications of these supersymmetric cycles on mirror symmetry in higher dimensions.
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TL;DR: In this article, the recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners.
Abstract: The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners. The geometric description---that one Calabi-Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi-Yau manifold---is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the `mirror' Calabi-Yau manifolds should be related to one another, and how appropriate subspaces of the homology groups of those manifolds could be related. We treat the case of K3 surfaces in some detail.
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TL;DR: A-model and B-model correlation functions on a Calabi-Yau manifold and a precise mathematical conjecture relating these for a pair of mirror manifolds are given in this article.
Abstract: Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The focus of these lectures is on giving a mathematical description of the A-model and B-model correlation functions on a Calabi--Yau manifold, and a precise mathematical conjecture relating these for a pair of mirror manifolds.
TL;DR: In this paper, the authors considered 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.
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.
26 Sep 1996
TL;DR: In this paper, the authors generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of compactifications derived from (2, 2) vacua.
Abstract: We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.
18 Jun 1996
TL;DR: It is argued that several sets of symmetry specific invariants are derived that can be used in different situations, depending on the a priori assumptions made, and that all the results directly apply to the case of perspectively skewed point symmetry.
Abstract: Over recent years, symmetry research has shifted from the detection of affinely to perspectively skewed mirror symmetry. Also, links between invariance research and symmetry-specific geometric constraints have been established. The paper aims to contribute to both strands. Several sets of symmetry specific invariants are derived, that can be used in different situations, depending on the a priori assumptions made. It is also argued that all the results directly apply to the case of perspectively skewed point symmetry.
TL;DR: In this paper, it was shown that the massless spectra of (2, 2)-supersymmetric vacua of central charge ĉ=Dcrit can be derived from special Fano varieties of complex dimension Dcrit+2(Q−1), Q>1, and that in certain circumstances it is even possible to embed Calabi-Yau manifolds into such higher dimensional spaces.
Abstract: Because of the existence of rigid Calabi-Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kahler manifolds with positive first Chern class which generalize Calabi-Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class comprises Fano manifolds of a special type which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of (2, 2)-supersymmetric vacua of central charge ĉ=Dcrit can be derived from special Fano varieties of complex dimension Dcrit+2(Q−1), Q>1, and that in certain circumstances it is even possible to embed Calabi-Yau manifolds into such higher dimensional spaces. The constructions described here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and their corresponding Calabi-Yau manifolds on the other. Furthermore it is shown that Witten’s formulation of the Landau-Ginzburg/Calabi-Yau relation can be applied to the present framework as well.
TL;DR: In this paper, a local geometric realization of quantum field theories together with a local application of mirror symmetry is proposed to reduce non-trivial quantum field theory results to T-dualities of type II strings.
Abstract: Using the recent advances in our understanding of non-perturbative aspects of type II strings we show how non-trivial exact results for $N=2$ quantum field theories can be reduced to T-dualities of string theory. This is done by constructing a local geometric realization of quantum field theories together with a local application of mirror symmetry. This construction is not based on any duality conjecture and thus reduces non-trivial quantum field theory results to much better understood T-dualities of type II strings. Moreover it can be used in principle to construct in a systematic way the vacuum structure for arbitrary $N=2$ gauge theories with matter representations.
TL;DR: In this paper, the counting functions of rational curves on complete intesection Calabi-Yau manifolds were studied in terms of generalized hypergeometric differential systems, which can be viewed as a generalization of the Schwarzian equation.
Abstract: We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.
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TL;DR: In this paper, the authors give a survey on old and new results concerning Arnold's strange duality and show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author.
Abstract: We give a survey on old and new results concerning Arnold’s strange duality. We show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.
TL;DR: In this article, an algebra on the space of perturbative BPS states in toroidal compactification of the heterotic string is defined, which is closely related to a generalized Kac-Moody algebra.
Abstract: We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.
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TL;DR: In this article, the authors considered mirror symmetry for Calabi-Yau threefolds of the type considered by Voisin and Borcea, of the form SxE/involution where S is a K3 surface with involution, and E is an elliptic curve.
Abstract: We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3 surfaces in light of this construction. We then consider the example of mirror symmetry for Calabi-Yau threefolds of the type considered by Voisin and Borcea, of the form SxE/involution where S is a K3 surface with involution, and E is an elliptic curve. We show how dualizing a family of special Lagrangian real 3-tori does actually produce the mirrors in these examples.
TL;DR: In this paper, a dual N=4 supersymmetric gauge theory with unitary and symplectic gauge groups was constructed and analyzed. But the authors focused on the classical and quantum moduli spaces of the theories and constructed an explicit mirror map between the mass parameters and the Fayet-Iliopoulos parameters of the dual.
Abstract: We construct and analyze dual N=4 supersymmetric gauge theories in three dimensions with unitary and symplectic gauge groups. The gauge groups and the field content of the theories are encoded in quiver diagrams. The duality exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters. We analyze the classical and the quantum moduli spaces of the theories and construct an explicit mirror map between the mass parameters and the the Fayet-Iliopoulos parameters of the dual. The results generalize the relation between ALE spaces and moduli spaces of SU(n) and SO(2n) instantons. We interpret some of these results from the string theory viewpoint, for SU(n) by analyzing T-duality and extremal transitions in type II string compactifications, for SO(2n) by using D-branes as probes. Finally, we make a proposal for the moduli space of vacua of these theories in the absence of matter.
TL;DR: In this article, the authors introduced two classes of Calabi-Yau manifolds fibrated by K3 surfaces with some special Picard lattices, which are related with automorphic forms on IV type domains which they studied in their papers.
Abstract: We extend our variant of mirror symmetry for K3 surfaces \cite{GN3} and clarify its relation with mirror symmetry for Calabi-Yau manifolds. We introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers \cite{GN1}-\cite{GN6}. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi-Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi-Yau manifolds. Our papers \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi-Yau manifolds.
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TL;DR: In this article, the authors analyzed GKZ hypergeometric systems and applied them to study the quantum cohomology rings of Calabi-Yau manifolds and related properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi Yau hypersurface.
Abstract: We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995)