Solvable Lie Algebras in Supergravity and Superstrings

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Solvable Lie Algebras in Supergravity and
Superstrings

• Pietro Fré
• Bonn February 2002

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In D lt 10 the structure of Superstring
Theory is governed...

• The geometry of the scalar manifold M
• M G/H is mostly a non compact coset manifold
• Non compact cosets admit an algebraic description
  in terms of solvable Lie algebras

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Before the gauging
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Two ways to determine G/H or anyhow the scalar
manifold

• By compactification from higher dimensions. In
  this case the scalar manifold is identified as
  the moduli space of the internal compact manifold
• By direct construction of each supergravity in
  the chosen dimension. In this case one uses the a
  priori constraints provided by supersymmetry. In
  particular holonomy and the need to reconcile p1
  forms with scalars

DUALITIES Special Geometries
The second method is more general, the first
knows more about superstrings, but the two must
be consistent
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The scalar manifold of supergravities is
necessarily a non compact G/H, except
In the exceptional cases the scalar coset is not
necessarily but can be chosen to be a non
compact coset. Namely Special Geometries include
classes of non compact coset manifolds
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Scalar cosets in d4
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Scalar manifolds by dimensions in maximal
supergravities
Rather then by number of supersymmetries we can
go by dimensions at fixed number of supercharges.
This is what we have done above for the maximal
number of susy charges, i.e. 32. These scalar
geometries can be derived by sequential toroidal
compactifications.
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How to determine the scalar cosets G/H from
supersymmetry
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.....and symplectic or pseudorthogonal
representations
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How to retrieve the D4 table
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Essentials of Duality Rotations
The scalar potential V(f) is introduced by the
gauging. Prior to that we have invariance under
duality rotations of electric and magnetic field
strengths
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Duality Rotation Groups
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The symplectic or pseudorthogonal embedding in
D2r
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.......continued
D4,8
D6,10
This embedding is the key point in the
construction of N-extended supergravity
lagrangians in even dimensions. It determines
the form of the kinetic matrix of the self-dual
p1 forms and later controls the gauging
procedures.
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The symplectic caseD4,8
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The Gaillard and Zumino master formula
We have
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Summarizing

• The scalar sector of supergravities is mostly a
  non compact coset U/H
• The isometry group U acts as a duality group on
  vector fields or p-forms
• U includes target space T-duality and strong/weak
  coupling S-duality.
• For non compact U/H we have a general
  mathematical theory that describes them in terms
  of solvable Lie algebras.....

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Solvable Lie algebra description...
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Differential Geometry Algebra
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Maximal Susy implies Er1 series
Scalar fields are associated with positive roots
or Cartan generators
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The relevant Theorem
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How to build the solvable algebra
Given the Real form of the algebra U, for each
positive root there is an appropriate step
operator belonging to such a real form
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String interpretation of scalar fields
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...in the sequential toroidal compactification
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Sequential Embeddings of Subalgebras and
Superstrings
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The type IIA chain of subalgebras
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Type IIA versus Type IIB decomposition of the
Dynkin diagram
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The Type IIB chain of subalgebras
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A decomposition that mixes NS and RR states ( U
duality)
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The electric subalgebra
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Understanding type IIA / type IIB T-duality
algebraically
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The two realizations of E7(7)
The two realizations correspond to attaching the
7th root in two different positions
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Dilaton and radii are in the CSA
The extra dimensions are compactified on circles
of various radii
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Outer Automorphisms of the S T subalgebra
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An explicit example of T duality
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The Maximal Abelian Ideal
From
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Conclusions

• (thanks to my collaborator, Mario Trigiante)
• The Solvable Lie algebra representation has many
  applications, in particular
• Classification of p-brane and black hole
  solutions
• Classification and construction of supergravity
  gaugings
• Study of supersymmetry breaking patterns and
  super Higgs phenomena