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Sasaki-Einstein Manifolds and Volume Minimisation

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian...

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Detalles Bibliográficos
Autores principales: Martelli, D, Sparks, J, Yau, S T
Lenguaje:eng
Publicado: 2006
Materias:
Acceso en línea:http://cds.cern.ch/record/933171
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author Martelli, D
Sparks, J
Yau, S T
author_facet Martelli, D
Sparks, J
Yau, S T
author_sort Martelli, D
collection CERN
description We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.
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institution Organización Europea para la Investigación Nuclear
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spelling cern-9331712019-09-30T06:29:59Zhttp://cds.cern.ch/record/933171engMartelli, DSparks, JYau, S TSasaki-Einstein Manifolds and Volume MinimisationParticle Physics - TheoryWe study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.hep-th/0603021CERN-PH-TH-2006-039HUTP-2006-A-0002oai:cds.cern.ch:9331712006-03-03
spellingShingle Particle Physics - Theory
Martelli, D
Sparks, J
Yau, S T
Sasaki-Einstein Manifolds and Volume Minimisation
title Sasaki-Einstein Manifolds and Volume Minimisation
title_full Sasaki-Einstein Manifolds and Volume Minimisation
title_fullStr Sasaki-Einstein Manifolds and Volume Minimisation
title_full_unstemmed Sasaki-Einstein Manifolds and Volume Minimisation
title_short Sasaki-Einstein Manifolds and Volume Minimisation
title_sort sasaki-einstein manifolds and volume minimisation
topic Particle Physics - Theory
url http://cds.cern.ch/record/933171
work_keys_str_mv AT martellid sasakieinsteinmanifoldsandvolumeminimisation
AT sparksj sasakieinsteinmanifoldsandvolumeminimisation
AT yaust sasakieinsteinmanifoldsandvolumeminimisation