I-Supergravity1-29.pdf
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Transcript of I-Supergravity1-29.pdf
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Lectures on Supergravity
Joaquim GomisBased on the SUGRA book of Dan Freedman
and Antoine Van Proeyen to appear in Cambridge University Press
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Public Material
Lectures on supergravity, Amsterdam-Brussels-Paris doctoral school, Paris 2009, October-November 2009: PDF-file.
http://itf.fys.kuleuven.be/~toine/SUGRA_DoctSchool.pdf
A. Van Proeyen, Tools for supersymmetry, hep-th 9910030
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History and overview ofSupergravity
60’ and 70’s. Yang Mills theories, Spontaneous symmetry breaking. Standard model
Supersymmetry
Yu. Gol’fand , E. Lichtman (1971)J.L. Gervais and B. Sakita (1971)A,Neveu, J. Schwarz, P.Ramond (1971)D. Volkov, V. Akulov (1972)J. Wess, B. Zumino (1974)
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History and overview ofSupergravity
• Yu. Gol’fand , E. Lichtman- Parity violation in QFT, 4d• J.L. Gervais and B. Sakita & A,Neveu, J. Schwarz String theory-Dual
models. Worls sheet supersymmetry 2d• D. Volkov, V. Akulov- Goldstone particles of spin ½? 4d• J. Wess, B. Zumino Supersymmetric field theory in 4d
Supergroup, superalgebra
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History and overview ofSupergravity
• Super Poincare
Translations
Spinor supercharge
Lorentz transformations
Massless multiplets contains spins (s, s-1/2), for s=1/2, 1, 2,
R symmetry
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History and overview ofSupergravity
Supergravity
Gauged supersymmetry was expected to be an extension of generalRelativity with a superpartner of the gravito call gravition
Multiplet (2,3/2)
S. Ferrara, D. Freedman, P. Van Nieuwenhuizen (1976)S. Deser, B. Zumino (1976)D. Volkov, V. Soroka (1973), massive gravitinos,..
Extensions with more supersymmetries and extension has beenconsidered, N=2 supergravity, special geometry. N=1 Supergravity in 11d
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Motivation for Supergravity
Supergravity (SUGRA) is an extension of Einstein's general relativity to include supersymmetry (SUSY). General relativity demands extensions since it has shortcomings including at least the following:
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Motivation for Supergravity• Space time singularities. The singularity theorems of Penrose,
Hawking and Geroch shows that general relativity is incomplete.
• Failure to unify gravity with the strong and electro weak forces.
• Einstein gravity is not power counting renormalizable. It is renormalizable as an effective theory. It is not a fundamental theory
• If we include supersymmetry in a theory of gravity. The simple example of divergences: zero point energy of the vacuum, can potentially be cancelled by super partners of ordinary particles
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The current status of supergravity
• A reliable approximation to M-theory.• An essential ingrediente for supersymmetric
phenomenology ( minimal supersymmetricestándar model coupled to N=1 supergravity).
• Applications in cosmology• An crucial part for the AdS/CFT correspondence
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Index
• Scalar field and its symmetries• The Dirac Field• Clifford algebras ans spinors• The Maxwell and Yang-Mills Gauge fields• Free Rarita-Schwinger field• Differential geometry• First and second order formulation of gravity• N=1 Global Supersymmetry in D=4
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Index
• N=1 pure supergravity in 4 dimensions• D=11 supergravity
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Scalar field
Noether symmetry leaves the action invariant
Symmetry transformations
Metric (-,+,+,+…+)
map solutions into solutions
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General internal symmetry
Infinitesimal transformations
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General internal symmetryCommutator of infinitesimal transformations
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Spacetime symmetries
Vector representation
Relations among Lorentz transformations
Lorentz condition
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Spacetime symmetries
Orbital part
Lorentz algebra
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Noether chargesInfinitesimal Noether symmetry
Noether current
Noether trick. Consider
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Noether charges
Hamiltonian formalism
For internal symmetries
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Noether charges
At quantum level
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The fundamental spinor representations
The transformation induces a Lorentz transformation
Properties
Hermitean matrix
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The Dirac Field
Applying the Dirac operator
Clifford algebra
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The Dirac FieldExplicit representation for D=4 in terms of
Finite Lorentz transformations
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The Dirac Field
Dirac action
Equation of motion for adjoint spinor
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Weyl spinors
Undotted components
Dotted components
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Weyl spinors
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Energy momentum tensor
where
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Clifford algebras and spinors
• Clifford algebras in general dimensions
Euclidean Clifford algebras
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Clifford algebras and spinors