11D Supergravity as a Gauge Theory for the MAlgebra

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Pre-Strings 2007 Granada, España June 2007
The Universe as a Topological Defect J. Zanelli,
CECS Valdivia (Chile)
Based on joint work with Andrés Anabalón and
Steve Willison hep-th/0610136 hep-th/0702192 .
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Summary
The (metric-free) 6D Euler class on a bounded
manifold defines a CS theory of gravity on the 5D
boundary.
If instead of a 5D boundary, the manifold has a
4D defect, the result is Einsteins gravity plus
matter at the defect.
This is a dimensional reduction mechanism
generates a metric alternative to the
Kaluza-Klein idea.
The mechanism at work here is the symmetry
breaking SO(4,2)?SO(3,1) produced by the 4D
defect analogous to the one that gives rise to
the 21 black hole.
The 4D geometry originates from a 6D gauge
connection. Goldstone fields describe 4D matter.
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Outline
1. Review of Chern-Simons theories. 2. Odd D
gravity as a CS theory for the Euler class 3. The
effect of a defect Transgressions and gWZW
actions 4. Life in a 4-dimensional defect 5.
Conclusions
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1. Chern-Simons theories
Action (three dimensions)
Although this is an explicit functional of the
connection, it is invariant (up to boundary
terms) under gauge transformations
C-S theories are more economical than YM no
invertible spacetime metric (g) or Killing form
in the Lie algebra (?) required.
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The coefficients c1, cn are rational numbers
determined by Weils combinatorial formula,
A CS form is a Lagrangian that describes a gauge
theory for a connection A in a spacetime of
dimension D2n-1.
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All the interesting features of CS forms
originate from their relation with characteristic
classes.
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From an aesthetic point of view, Chern-Simons
theories are superior to Yang-Mills

• They have no arbitrary dimensionful parameters
• They use fewer ingredients
• Same gauge invariance as YM
• They require no a-priori metric spacetime
  manifold
• Excellent pedigree - fiber bundle structure
• -
  characteristic classes

However, they dont seem to be Gods favorites

• They exist only in odd dimensions
• Dimensional reduction needed
• How does gravity fit in?

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2. Gravity
G R is the oldest known non-abelian gauge theory
(1915)

• Equivalence Principle
• The laws of physics in a small spacetime
  neighbohood are
• indistinguishable form those in flat
  spacetime.

• In particular, spacetime is locally approximated
  by and has the
• symmetries of Minkowski space.

• Spacetime is equipped with a bundle of tangent
  spaces, invariant
• under local Lorentz rotations.

• GR is the best gravitation theory available
• Precision better than one in 105 (solar).
• None of its predictions have been refuted.

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Gravity actions
For dimensions D4, the gravitational action is
the integral of a D-form built from the vielbein
and Lorentz (spin) connection 1-forms,
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Chern-Simons gravity
This is equivalent to a gauge theory with a
Chern-Simons action. Achúcarro
Townsend, 1986
Gauge invariant under the de Sitter (?gt0), anti
de Sitter (?lt0), or Poincaré (?0) groups.
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Chern-Simons gravity
This is equivalent to a gauge theory with a
Chern-Simons action. Achúcarro
Townsend, 1986
Gauge invariant under the de Sitter (?gt0), anti
de Sitter (?lt0), or Poincaré (?0) groups.
Hence, it is finite and exactly soluble.
Witten, 1988
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Chern-Simons gravity (contd.)
A CS action can now be defined to describe a
gauge theory for the AdS group. The Lagrangian is
the Chern-Simons form,
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Although the ingredients belong to
representations of the Lorentz group, the
resulting theory has a larger symmetry.
In terms of the geometric variables, the
Lagrangian in D2n1 dimensions reads
The constant coefficient l (AdS radius) can be
absorbed by rescaling the vielbein
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Chern-Simons gravity for D2n-1

• Has no dimensionful constants (its scale
  invariant)
• Has no arbitrary renormalizable constants
• Admits asymptotically AdS solutions
• Possesses black hole solutions
• Admits and
  limit
• Admits SUSY extensions for and any odd
  D, and
• yields field content with spins 2 only
• Gives rise to acceptable D4 effective theories

All these features arise from the same key fact
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Consequently, the entire local SO(2n) invariance
of the Euler form is inherited by the CS
lagrangian one dimension below.
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Living at the boundary
In general, a characteristic class in a bounded
2n-dimensional manifold defines a CS action on
the 2n-1-dimensional boundary.
where P2n is a topological invariant density in
2n dimensions.
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This is a good lagrangian for 5D gravity and has
SO(4,2) gauge symmetry. It has interesting BH
solutions, supersymmetric extensions, etc.
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For example, integrating the Euler density on a
6D bounded manifold yields a theory of gravity in
5D,
This is a good lagrangian for 5D gravity and has
SO(4,2) gauge symmetry. It has interesting BH
solutions, supersymmetric extensions, etc.
unfortunately, we seem to live in 4D.
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3. The effect of a defect
Consider now a 6D spacetime with a 4D topological
defect
What happens with the action now?
We need to define what we mean by defect
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A 4D topological defect in a 6D spacetime results
from identifying
Defect M4
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Action
Gauge invariance
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This looks very much like a transgression
A transgression is uniquely defined by two basic
features
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And the final expression is.
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4. Life in a 4D defect
The resulting action, is a functional of a
SO(4,2) connection A(x), and a gauge element,
h(x).
The identification that generates the defect
breaks SO(4,2) invariance. It is best to write
everything in terms of field that transform
irreducibly under the remaining symmetry,
SO(3,1)xSo(1,1).
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and the field equations are
It is natural to interpret the field ca as the
vielbein, and µ as a (positive) effective
cosmological constant in the 4D defect.
The field equations for the Goldstone fields
defined by the fluctuations of h around h0, are
extremely messy and we have not completely
analyzed them.
The configuration h h0, is a classical solution
but we dont know whether it is stable or not
(no positive energy theorems yet).
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5. Summary

• CS forms for the AdS group SO(2n-2,2) define
  gravity
• theories in 2n-1 dimensions

• They can be viewed as boundary theories coming
  from
• integrating the Euler density on a bounded, D
  2n manifold.

• They have no free adjustable parameters and the
  metric is
• not a fundamental field.

• Integrating the Euler density on a 2nD manifold
  with a
• topological defect, generates a gWZW for the
  defect.

• Tthe resulting classical dynamics is Einstein 4D
  gravity
• with positive cosmological constant.