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Topic: Calabi Yau manifolds

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Ricci-flat manifold - RecipeFacts In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci tensor vanishes. Ricci-flat manifolds are special cases of Einstein manifolds. Ricci-flat manifolds, in general, have restricted holonomy groups. www.recipeland.com /encyclopaedia/index.php/Ricci_flat   (77 words)

Fields Institute - Calabi-Yau Varieties and Mirror Symmetry Calabi-Yau manifolds appear in the theory because in passing from the 10-dimensional space time to a physically realistic description in four dimension, string theory requires that the additional 6-dimensional space is to be a Calabi-Yau manifold. Also special classes of Calabi-Yau manifolds, e.g., of Fermat type hypersurfaces, or their deformations pertinent to mirror symmetry, offer promising testing grounds for physical predictions as well as rigorous mathematical analysis and computations. on the computation of the zeta-functions of Calabi-Yau manifolds over finite fields reveal a surprising connection of mirror symmetry to p-adic L-functions (which are the essential ingredients in Iwasawa theory). www.fields.utoronto.ca /programs/scientific/01-02/cyms   (1320 words)

Calabi-Yau compactifications The drawback of compactifications on Calabi Yau manifolds is that they are highly nontrivial spaces and we cannot describe the strings on such manifolds, contrary to what we did in the case of free theories such as tori and orbifolds. Larger classes of manifolds can be constructed by considering intersections of hypersurfaces in higher dimensional projective spaces, the so-called complete intersection Calabi-Yau manifolds (CICY). Actually, it is known in the mathematical literature, that all Calabi-Yau spaces can be defined as (intersection of) hypersurfaces in weighted projective spaces. fisica.usac.edu.gt /public/curccaf_proc/quevedo1/node6.html   (1320 words)

Andrejewski Vorlesung: Gang Tian In this lecture, we discuss methods of computating quantum cohomology for special algebraic manifolds, such as Calabi-Yau spaces and Fano manifolds. This lecture will concern special geometric properties of Calabi-Yau manifolds. These include geometry of moduli spaces and special Lagrangian submanifolds. www.uni-math.gwdg.de /andrej/tian.html   (1320 words)

34a For a general construction of tropical Calabi-Yau manifolds arising from degenerations of genuine Calabi-Yau manifolds, see Gross' recent paper with Siebert. References: Thinking about Calabi-Yau manifolds in a tropical sort of way first arose in Kontsevich's and Soibelman's paper from 2000. An affine manifold is a real manifold with coordinate charts whose transition maps are in www.aimath.org /WWN/amoebas/articles/html/34a   (594 words)

Open Questions: Mathematics and Physics Fiber bundles are a mathematical construct that can be applied in the differential geometry of manifolds to generalize the notion of curvature. Donaldson's work also led to the construction of new "invariants" of 4-manifolds, which are numeric or algebraic objects that help classify manifolds. The absence of dispersion in solitons is a result of nonlinearities in the differential equations that govern them. www.openquestions.com /oq-ma006.htm   (1929 words)

Calabi-Yau Home Page Three classes of Calabi-Yau manifolds have been constructed completely at this point. The list of 3,284 (39kb) theories with more than five variables define higher-dimensional manifolds, so-called Special Fano Varieties or Generalized Calabi-Yau Manifolds. This page (and its mirror) is intended to become a resource for people to post all kinds of information about every Calabi-Yau manifold that anyone would care about, as well as information about the physical theories that they define. www.th.physik.uni-bonn.de /th/People/netah/cy.html   (505 words)

IngentaConnect Gauge Theoretical Construction of Non-compact Calabi-Yau Manifold... We construct the non-compact Calabi–Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. We construct the non-compact Calabi–Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. IngentaConnect Gauge Theoretical Construction of Non-compact Calabi-Yau Manifold... api.ingentaconnect.com /content/ap/ph/2002/00000296/00000002/art06226   (222 words)

Calabi-Yau manifold - Wikipedia, the free encyclopedia The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau in 1977 and became Yau's theorem. Essentially, Calabi-Yau manifolds are shapes that satisfy the requirement of space for the six "unseen" spatial dimensions of string theory, which must be contained in a space smaller than our currently observable lengths because they have not yet been detected. In mathematics, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class. en.wikipedia.org /wiki/Calabi-Yau_manifold   (384 words)

Calabi-Yau compactifications The drawback of compactifications on Calabi Yau manifolds is that they are highly nontrivial spaces and we cannot describe the strings on such manifolds, contrary to what we did in the case of free theories such as tori and orbifolds. In particular we can not compute explicitly the couplings in the effective theory, except for the simplest renormalizable Yukawa couplings. They were actually the first standard Kaluza-Klein compactification considered in string theory, leading to chiral 4D models and generically gauge group fisica.usac.edu.gt /public/curccaf_proc/quevedo1/node6.html   (844 words)

math lessons - Hyperkähler manifold Hyperkähler manifolds are special classes of Kähler manifolds. In mathematics, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group Sp(k). Every hyperkähler manifold M has a 2-sphere of complex structures with respect to which the metric is Kähler. www.mathdaily.com /lessons/Hyperk%E4hler_manifold   (138 words)

Conifold - Wikipedia, the free encyclopedia It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions". Unlike manifolds, a conifold can (or should) contain conical singularities i.e. In mathematics, a conifold is a generalization of the notion of a manifold. en.wikipedia.org /wiki/Conifold   (305 words)

Introduction and summary Now we see that our deformation arguments are not limited to a single Calabi--Yau moduli space, but rather extend to all Calabi--Yau manifolds connected by conifold transitions; the latter includes essentially all known Calabi--Yau manifolds. The singularities of the conifolds which glue together Calabi--Yau moduli spaces are more complicated than the simple type analyzed in [ 12 ]. It has long been known in the mathematics literature [ 3, 4, 5, 6 ] that it is possible to travel from one Calabi--Yau to another by degenerating certain three-cycles and then blowing them back up as two-cycles, thereby changing the Hodge numbers. www.cgtp.duke.edu /~drm/condensation/node1.html   (305 words)

Digitale Hochschulschriften der LMU - D-branes on Calabi-Yau Spaces Among them are manifolds admitting elliptic and K3-fibrations and manifolds whose moduli space can be embedded into the moduli space of another manifold. At the other points the size of the manifold is smaller than its quantum fluctuations such that the classical geometry looses its meaning and has to be replaced by a conformal field theory. One of these points corresponds to a manifold in the large volume limit on which the D-branes are described by classical geometry of vector bundles. edoc.ub.uni-muenchen.de /archive/00000445   (403 words)

edutainment the (0,2) exactly solvable structure of landau-ginzburg theories, chiral rings and calabi-yau manifolds, hep-th/9510055 a new construction of a 3generation calabi-yau manifold, pl b193(1987)175 critical superstring vacua from noncritical manifolds: a novel framework for string compactification and mirror symmetry, hep-th/9210062 www.th.physik.uni-bonn.de /th/People/netah/papers/edutainment.html   (308 words)

Calabi-Yau Home Page TESS, a code which computes the Hodge number of complete intersection Calabi-Yau manifolds embedded in products of ordinary projective spaces. Three classes of Calabi-Yau manifolds have been constructed completely at this point. A number of different types of Calabi-Yau theories is contained in the class Landau-Ginzburg theories at c=9 ( 116kb) whose complete construction in two independent ways has been described in the following LG references. www.th.physik.uni-bonn.de /th/People/netah/cy.html   (308 words)

Digitale Hochschulschriften der LMU - D-branes on Calabi-Yau Spaces In this thesis the properties of D-branes on Calabi–Yau spaces are investigated. Among them are manifolds admitting elliptic and K3-fibrations and manifolds whose moduli space can be embedded into the moduli space of another manifold. One main focus is on D4-branes, in particular on the dimension of their moduli space. edoc.ub.uni-muenchen.de /archive/00000445   (308 words)

Calabi-Yau spaces Calabi-Yau manifolds of (complex) dimension three appear in (Supersymmetric[?]) String theory compactifications to four dimensions. A Calabi-Yau manifold is a compact Kahler[?] manifold with a vanishing first Chern class[?], and consequently is Ricci flat. The text of this article is licensed under the GFDL. www.ebroadcast.com.au /lookup/encyclopedia/ca/Calabi-Yau_spaces.html   (54 words)

Kähler manifold - Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry. A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle $TM\; \backslash otimes\; \backslash mathbf\; C$ satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transport gives rise to complex-linear mappings on the tangent spaces). In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. www.grohol.com /psypsych/Kahler_manifold   (457 words)

Mirror symmetry Recall that in [ 14 ] a construction of pairs of mirror manifolds was presented which relied crucially on the existence of special points in moduli space at which the associated Calabi--Yau has enhanced discrete symmetries. More precisely, this isomorphism leads one to define, in the context of type II string theory, two Calabi--Yau spaces as constituting a mirror pair if the type IIA string on the first is isomorphic to the type IIB string on the second. For this reason, such deformation arguments have only been used to establish mirror symmetry for a continuously connected (in the sense of conformal field theory) family of Calabi--Yau spaces containing at least one point at which the explicit construction of [ 14 ] could be applied. www.cgtp.duke.edu /~drm/condensation/node6.html   (457 words)

Calabi-Yau manifold - Wikipedia, the free encyclopedia Consequently, a Calabi-Yau manifold can also be defined as a compact Ricci-flat Kähler manifold. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n -fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a en.wikipedia.org /wiki/Calabi-Yau_spaces   (457 words)

Moduli space of Calabi-Yau manifolds A Calabi-Yau manifold and therefore also the (2,2)-superconformal field theory defined on this space, is determined by giving the Ricci-flat Kähler metric and the B -field. In the mathematical literature this is a well known moduli problem and is referred to just as the moduli problem of the Calabi-Yau. As harmonic 2-forms are annihilated by the natural Laplacian on forms, which is the same as the Lichnerowicz Laplacian on forms, we can add B and G and identify the tangent space to the moduli space as the kernel of the Lichnerowicz Laplacian. www.phys.uu.nl /~hofman/scriptie/duality/node40.html   (457 words)

CALABI-YAU MANIFOLDS This is the first systematic exposition in book form of the material on Calabi-Yau spaces, related mathematics and the physics application, otherwise scattered through research articles in journals and conference proceedings. In the main part of the Book, collected and reviewed are relevant results on (1) several major techniques of constructing such spaces and (2) computation of physically relevant quantities such as massless field spectra and their Yukawa interactions. Issues of (3) stringy corrections and (4) moduli space and its geometry are still in the stage of rapid and continuing development, whence there is more emphasis on open problems here. www.worldscibooks.com /physics/1410.html   (457 words)

Open Questions: Geometry and Topology Calabi studied such classes in the 1950s and formulated what became known as the "Calabi conjecture", which states that there exists one and only one Kähler form on a compact complex manifold that is in the same Chern class as the Ricci tensor. Moduli spaces are topological spaces that are not quite manifolds, because they have singularities (such as kinks and creases), so they are not everywhere differentiable, although (as topological spaces) they have a well-defined finite dimension away from the singularties. Given the proof of the Calabi conjecture, an equivalent condition for Ricci-flatness is that the Chern class be zero. www.openquestions.com /oq-ma003.htm   (457 words)

MATH / PHYSICS Research Group Research Together with his student Markman, Donagi exhibited an integrable structure on the moduli space of Calabi-Yau manifolds, a basic ingredient of the conjectural mirror symmetry. The second is algebro-geometric: abelian gerbes bounded by a subgroup emerge as the natural habitat of Fourier-Mukai transforms of sheaves on genus one fibered varieties and thus allow for major improvements in the explicit construction of vector bundles on higher dimensional spaces. It was realized that the geometric techniques for analyzing moduli spaces of bundles were precisely the needed tools for understanding the brane-world scenarios. dept.physics.upenn.edu /mprg/resprograms.html   (457 words)

week134 Because of this, a curve in one moduli space of Calabi-Yau manifolds can be physically equivalent to a curve in some other moduli space, which sometimes lets you continue the curve beyond a singularity in the first moduli space. The product of Minkowski spacetime with a fixed Calabi-Yau manifold is a solution of the 10-dimensional Einstein equations, and this is part of why this kind of spacetime serves as a good background for string theory. However, this is probably good when one is writing to an audience of philosophers: one should explain the problems instead of trying to sell them on a particular claimed solution, because the proposed solutions come and go rather rapidly, while the problems remain. math.ucr.edu /home/baez/week134.html   (2591 words)

Brian Greene Brian Greene is one of the fathers of mirror symmetry relating two different Calabi-Yau manifolds (concretely, relating the conifold to one of its orbifold s) The book talks about and opens an argument on how Calabi-Yau manifold s, as the multi-dimensional (11D, 16D, 26D) points, comprise our space-time. He is the author of, a popularization of superstring and M-theory, and winner of The Aventis Prizes for Science Books in 2000. www.serebella.com /encyclopedia/article-Brian_Greene.html   (943 words)

cynk2.tex We shall verify the modularity conjecture for all rigid Calabi--Yau manifolds constructed in the paper. If $D$ is smooth then $X$ is a (smooth) Calabi--Yau manifold, if $D$ is singular then $X$ is also singular, and the singularities of $X$ are in one--to--one correspondence with the singularities of $D$. Those deformations correspond to ruled surfaces in the Calabi--Yau manifold, their geometry is explained in \cite{Wilson} (see also \cite{sze, sze2}). www.uni-essen.de /~mat903/preprints/cynk2.tex   (2729 words)

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