Showing papers on "Mirror symmetry published in 1993"

Holomorphic Anomalies in Topological Field Theories

Abstract: We study the stringy genus one partition function of $N=2$ SCFT's. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit of this partition function yields the partition function of topological theory coupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds including the quintic threefold. This may be viewed as the first application of mirror symmetry at the string quantum level.

Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

Abstract: We give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three- fold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the "mirror symmetry" phe- nomenon recently observed by string theorists. DEPARTMENT OF MATHEMATICS, DUKE UNIVERSITY, DURHAM, NORTH CAROLINA 27706 E-mail address: drm@math.duke.edu This content downloaded from 157.55.39.224 on Wed, 14 Dec 2016 04:59:36 UTC All use subject to http://about.jstor.org/terms

Intensity and edge-based symmetry detection with an application to car-following

Abstract: We present two methods for detecting mirror symmetry in images, one based directly on the intensity values and another one based on a discrete representation of local orientation. A symmetry finder has been developed which uses the intensity-based method to search an image for compact regions which display some degree of mirror symmetry due to intensity similarities across a straight axis. In a different approach, we look at symmetry as a bilateral relationship between local orientations. A symmetry-enhancing edge detector is presented which indicates edges dependent on the orientations at two different image positions. SEED, as we call it, is a detector element implemented by a feedforward network that holds the symmetry conditions. We use SEED to find the contours of symmetric objects of which we know the axis of symmetry from the intensity-based symmetry finder. The methods described in this paper have been developed and tested for the recognition and tracking of cars in a real-time system for automatic car-following and headway control on normal roads.

Compactifications of moduli spaces inspired by mirror symmetry

Abstract: We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the Kahler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]

Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

Abstract: We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

The monomial-divisor mirror map

Abstract: For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry, which we call the monomial-divisor mirror map. We indicate how this map can be interpreted as the differential of the expected mirror isomorphism between the moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise conjecture about the form of that mirror isomorphism, which when combined with some earlier conjectures of the third author would completely specify it. We then conclude that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi-Yau space should be connected by analytic continuation, and that further analytic continuation should lead to moduli spaces of other kinds of conformal field theories. (This last conclusion was first drawn by Witten.)

Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations

Abstract: A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.

Atomic structure of the (310) twin in niobium: Experimental determination and comparison with theoretical predictions.

Abstract: The atomic structure of the (310) twin in Nb was predicted using interatomic potentials derived from the embedded atom method (EAM), Finnis-Sinclair theory (FS), and the model generalized pseudopotential theory (MGPT). The EAM and FS predicted structures with crystal translations which break mirror symmetry. The MGPT predicted one stable structure which possessed mirror symmetry. This defect was experimentally determined to have mirror symmetry. These findings emphasize that the angular dependent interactions modeled by the MGPT are important for determining defect structures in bcc transition metals.

Critical superstring vacua from noncritical manifolds: A novel framework for string compactification and mirror symmetry.

Abstract: A new framework is found for the compactification of supersymmetric string theory. It is shown that the massless spectra of critical string vacua with central charge c=3${\mathit{D}}_{\mathrm{crit}}$ can be derived from manifolds of complex dimension ${\mathit{D}}_{\mathrm{crit}}$+2(Q-1),Q\ensuremath{\ge}1, whose first Chern class is quantized in a particular way. This new class is more general than that of Calabi-Yau manifolds because it contains spaces corresponding to vacua with no K\"ahler deformations, i.e., no antigenerations, thus providing mirrors of rigid Calabi-Yau manifolds. The constructions introduced here lead to new insights into the relation between Landau-Ginzburg vacua on the one hand and Calabi-Yau manifolds on the other.

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

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.

Mirror Symmetry of K3 and Torus

Abstract: We discuss a K3 and torus from view point of "mirror symmetry". We calculate the periods of the K3 surface and obtain the mirror map, the two-point correlation function, and the prepotential. Then we find there is no instanton correction on K3 (also torus), which is expected from view point of Algebraic geometry.

Rational curves on Calabi-Yau manifolds: verifying predictions of mirror symmetry

Abstract: Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy certain incidence properties are calculated, and shown to agree with the numbers predicted by Greene, Morrison, and Plesser using mirror symmetry in every instance. This increases the number of verified predictions from 3 to 65. Calculations are performed using the Maple package {\sc schubert} written by Katz and Str{\o}mme.

The Monomial-Divisor Mirror Map

Abstract: For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry, which we call the monomial-divisor mirror map. We indicate how this map can be interpreted as the differential of the expected mirror isomorphism between the moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise conjecture about the form of that mirror isomorphism, which when combined with some earlier conjectures of the third author would completely specify it. We then conclude that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi-Yau space should be connected by analytic continuation, and that further analytic continuation should lead to moduli spaces of other kinds of conformal field theories. (This last conclusion was first drawn by Witten.)

Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory

Abstract: We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kahler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kahler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space---the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kahler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kahler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.

Recent efforts in the computation of string couplings

Abstract: We review recent advances towards the computation of string couplings. Duality symmetry, mirror symmetry, Picard-Fuchs equations, etc. are some of the tools.

Spacetime Topology Change: The Physics of Calabi-Yau Moduli Space

Abstract: We review recent work which has significantly sharpened our geometric understanding and interpretation of the moduli space of certain $N$=2 superconformal field theories. 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.

Multiple Mirror Manifolds and Topology Change in String Theory

Abstract: We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the {\it quantum theories} based on certain nonlinear sigma models with topologically distinct target spaces can be smoothly connected even though classically a physical singularity would be encountered. We accomplish this by rephrasing the description of these nonlinear sigma models in terms of their mirror manifold partners--a description in which the full quantum theory can be described exactly using lowest order geometrical methods. We establish that, for the known class of mirror manifolds, the moduli space of the corresponding conformal field theory requires not just two but {\it numerous} topologically distinct Calabi-Yau manifolds for its geometric interpretation. A {\it single} family of continuously connected conformal theories thereby probes a host of topologically distinct geometrical spaces giving rise to {\it multiple mirror manifolds}.

Lines on Calabi Yau complete intersections, mirror symmetry, and Picard Fuchs equations

Abstract: A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.

Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings

Abstract: By means of toric geometry we study hypersurfaces in weighted projective space of dimension four. In particular we compute for a given manifold its intrinsic topological coupling. We find that the result agrees with the calculation of the corresponding coupling on the mirror model in the large complex structure limit.

Generalized Calabi-Yau Manifolds and the Mirror of a Rigid Manifold

Abstract: We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories with $c=9$ and so are perfectly good for compactifying the heterotic string to the four dimensions of space-time. As a check of mirror symmetry we compute the structure of the space of complex structures of the mirror and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kahler class parameters on the Z orbifold together with their instanton corrections.

Pathways to fundamental theories : proceedings of the Johns Hopkins Workshop on Current Problems in Particle Theory, 16, Göteborg, 1992 (June 8-10)

Abstract: Seventy relatives of the monster module, A.N. Schellekens on the relation between integrability and infinite-dimensional algebras, P. West remarks on noncompact symmetries, J.H. Schwartz interactions on modular invariance of string on curved manifolds, S. Hwang supersymmetric unification, G. Ross a new supersymmetric index, C. Vafa quantum aspects of black holes, A. Strominger canonical quantization of the Liouville theory, quantum group structures and correlation functions, G. Weigt on Kontsevich integrals, C. Itzykson on string field theory of less than or equal to 1, A. Marshakov a brief survey of mirror symmetry, B.R. Greene string theory with dynamical point-like structure, M.B. Green geometry and W-gravity, C. Hull physical states in the W3 string, C.N. Pope gravity and gauge theory at high energies, H. Verlinde.

Linear Structure on Calabi-Yau Moduli Spaces

Abstract: We show that the formal moduli space of a Calabi-Yau manifold $X^n$ carries a linear structure, as predicted by mirror symmetry. This linear structure is canonically associated to a splitting of the Hodge filtration on $H^n(X)$.