Generalized Calabi-Yau manifolds and the mirror of a rigid manifold

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 he...

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Autores principales: Candelas, P., Derrick, E., Parkes, L.
Lenguaje:eng
Publicado: 1993
Materias:
Particle Physics - Theory
General Theoretical Physics
Acceso en línea:https://dx.doi.org/10.1016/0550-3213(93)90276-U
http://cds.cern.ch/record/248864
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author Candelas, P.
Derrick, E.
Parkes, L.
author_facet Candelas, P.
Derrick, E.
Parkes, L.
author_sort Candelas, P.
collection CERN
description 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.
id cern-248864
institution Organización Europea para la Investigación Nuclear
language eng
publishDate 1993
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spelling cern-2488642023-03-14T19:27:37Zdoi:10.1016/0550-3213(93)90276-Uhttp://cds.cern.ch/record/248864engCandelas, P.Derrick, E.Parkes, L.Generalized Calabi-Yau manifolds and the mirror of a rigid manifoldParticle Physics - TheoryGeneral Theoretical PhysicsWe 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.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.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.The Z ̃ manifold is a Calabi-Yau manifold with b 21 = 0. At first sight it seems to provide a counter-example to the mirror hypothesis since its mirror would have b 11 = 0 and hence could not be Kähler. However, by identifying the Z ̃ manifold with the Gepner model 1 9 we are able to ascribe a geometrical interpretation to the mirror, Z ̃ , as a certain seven-dimensional manifold. The mirror manifold Z ̃ is a representative of a class of generalized Calabi-Yau manifolds, which we describe, that can be realized as manifolds of dimension five and seven. Despite their dimension these generalized Calabi-Yau manifolds 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 Z ̃ and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kähler class parameters on the Z ̃ orbifold together with their instanton corrections. In addition to reproducing known results we can calculate the periods of the manifold to arbitrary order in the blowing up parameters. This provides a means of calculating the Yukawa couplings and metric as functions also to arbitrary order in the blowing up parameters, which is difficult to do by traditional methods.hep-th/9304045CERN-TH-6831-93UTTG-24-92CERN-TH-6831-93oai:cds.cern.ch:2488641993
spellingShingle Particle Physics - Theory
General Theoretical Physics
Candelas, P.
Derrick, E.
Parkes, L.
Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title_full Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title_fullStr Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title_full_unstemmed Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title_short Generalized Calabi-Yau manifolds and the mirror of a rigid manifold
title_sort generalized calabi-yau manifolds and the mirror of a rigid manifold
topic Particle Physics - Theory
General Theoretical Physics
url https://dx.doi.org/10.1016/0550-3213(93)90276-U
http://cds.cern.ch/record/248864
work_keys_str_mv AT candelasp generalizedcalabiyaumanifoldsandthemirrorofarigidmanifold
AT derricke generalizedcalabiyaumanifoldsandthemirrorofarigidmanifold
AT parkesl generalizedcalabiyaumanifoldsandthemirrorofarigidmanifold

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