Showing papers on "Calabi–Yau manifold published in 1993"

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.

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.

A Finiteness Theorem for Elliptic Calabi-Yau Threefolds

Abstract: We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.

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.

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.

A remark on the global indices of -Calabi-Yau 3-folds

Abstract: It is well known that the so-called Beauville number B = 25.33.52.7.11.13.17.19 is a universal bound of the global indices of ℚ–Calabi–Yau 3-folds, but it has been unknown whether this number is best possible or not.

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.

Periods for Calabi--Yau and Landau--Ginzburg Vacua

Abstract: The complete structure of the moduli space of \cys\ and the associated Landau-Ginzburg theories, and hence also of the corresponding low-energy effective theory that results from (2,2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. These periods are shown to be readily calculable for a great many such models. We illustrate this by computing the periods explicitly for a number of classes of \cys. We also point out that it is possible to read off from the periods certain important information relating to the mirror manifolds.

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.

Non-Compact Calabi-Yau Spaces and Other Non-Trivial Backgrounds for 4-D Superstrings

Abstract: A large class of new 4-D superstring vacua with non-trivial/singular geometries, spacetime supersymmetry and other background fields (axion, dilaton) are found. Killing symmetries are generic and are associated with non-trivial dilaton and antisymmetric tensor fields. Duality symmetries preserving N=2 superconformal invariance are employed to generate a large class of explicit metrics for non-compact 4-D Calabi-Yau manifolds with Killing symmetries.

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.

K\"ahler manifolds with positive first Chern class and mirrors of rigid Calabi-Yau manifolds

Abstract: Recent work on the relation between a special class of K\"ahler manifolds with positive first Chern class and critical N$=$2 string vacua with c$=$9 is reviewed and extended. (Based on a talk presented at the International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY 93), Boston, MA, March 1993)

Non-compact Calabi-Yau Spaces and other Non-Trivial Backgrounds for Four-dimensional Superstrings

Abstract: A large class of new 4-D superstring vacua with non-trivial/singular geometries, spacetime supersymmetry and other background fields (axion, dilaton) are found. Killing symmetries are generic and are associated with non-trivial dilaton and antisymmetric tensor fields. Duality symmetries preserving N=2 superconformal invariance are employed to generate a large class of explicit metrics for non-compact 4-D Calabi-Yau manifolds with Killing symmetries.

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.

Unobstructedness of Calabi-Yau OrbiKleinfolds

Abstract: We show that Calabi-Yau spaces with certain types of hypersurface- quotient singularities have unobstructed deformations. This applies in particular to all Calabi-Yau orbifolds nonsingular in codimension 2.

Noncritical Dimensions for Critical String Theory: Life Beyond the Calabi-Yau Frontier

Abstract: A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension 2k + D Crit, k ≥ 1, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string ground states. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi-Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi-Yau manifolds because it contains spaces which correspond to critical string vacua with no Kahler deformations, i.e. no antigenerations, thus providing mirrors of rigid Calabi-Yau manifolds. The constructions reviewed here lead to new insights into the relation between exactly solvable models and their mean field theories on the one hand and Calabi-Yau manifolds on the other, leading, for instance, to a modification of Gepner’s conjecture. They also raise fundamental questions about the Kaluza-Klein concept of string compactification, in particular regarding the role played by the dimension of the internal theories.

Thickening Calabi-Yau moduli spaces

Abstract: We describe a kind of deformation of the anti-DeRham algebra on a Calabi-Yau manifold $X$. These are in 1-1 correspondence with the total cohomology $\oplus H^i (X, \C)$.

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 $H^n(X)$.