11D Supergravity as a Gauge Theory for the MAlgebra

1
Centro de Estudios Científicos CECS-Valdivia-Chile
2
(Super) Gravities of a different sort Constrained
Dynamics and Quantum Gravity QG05 Sardegna,
September 2005
J. Zanelli CECS - Valdivia

• In collaboration with
• José Edelstein
• Mokhtar Hassaine
• Ricardo Troncoso
• Lecture notes on C-S SuGras
  hep-th/0502193
• - Phys. Lett. B596 (2004) 132
  hep-th/0306258
• - Phys. Rev. D58 (1998) 101703 hep-th/9710180
• - Phys. Rev. D54 (1996) 2605 gr-qc /9601003

3
Outline
1. Gravity and gauge invariance 2. Manifestly
Lorentz invariant theory 3. Special Choices
Euler, Pontryagin, Poincaré 4. Dynamics 5.
Supersymmetry 6. Open questions summary
4
1. Gravity and Gauge invariance
Gravity is a funny interaction that forces all
bodies to move in the same orbits. Galileo,
Newton
Einsteins insight Gravity is an effect of the
geometry of spacetime.
This is the first nonabelian gauge theory that we
know of.
... but this symmetry has nothing to do with the
freedom to perform general coordinate
transformations.
5
Equivalence principle
Spacetime
is a differentiable manifold endowed with a
tangent space Tx, identical to Minkowski space,
at each point.
In this way, General Relativity generalizes
Special Relativity.
The freedom to perform independent Lorentz
rotations at each point in spacetime is a gauge
symmetry Utiyama (1955), Kibble (1961)....
6
fiber bundle structure.
The local Lorentz rotations in spacetime endows
spacetime with a

• General coordinate transformations do not form a
  Lie algebra
• and do not define a gauge symmetry with fiber
  bundle structure.

7
General coordinate transformations

• Invariance under GCT must be a feature of any
  correct physical
• theory, since coordinates are an invention of
  our culture and not
• features of nature. Hence,

• It should be possible (and desirable) to
  eliminate all reference to coordinates in
  physics, and
• GCT can have no physically observable
  consequences.

The first order formalism achieves this.
8
Local frames (vielbein)
9
Parallelism
10
D - dimensional gravity
11
2. Manifestly Lorentz invariant theory
The Lagrangian for D-dimensional gravity is
assumed to be
12
Bianchi identities
13
Three series
are found
14
A. Lovelock series ()

• Also, Lp continuation of the Euler density
• from 2p to D dimensions

( Wedge product is understood)
15
This series includes
Cosmological constant term (volume)
Einstein-Hilbert
Gauss-Bonnet
Poincaré invariant term (odd D) see below
Euler density (even D) see above
16
B. Torsion series
These traces are related to the Pontryagin form
(Chern class),
whose integral is a topological invariant in 4k
dimensions.
17
C. Lorentz Chern-Simons
series

• Products of these and torsional terms are also
  acceptable
• Lagrangians, provided they are D-forms .

18
The Action
The most general action for gravity in D
dimensions is
19
Features
General action for a D - dimensional geometry.

• 1st order field equations for e, ?.

• Diffeomorphic invariant by construction

• Invariant under local Lorentz transformations

• Einstein-Hilbert theory is the only option for
  D4

20
Puzzles / problems

• Violently different behavior for different
  choices of ap, ?q.

• Hopeless quantum scenario ap, ?q dimensionful
  and
• unprotected from renormalization.

• How could this be so if gravity descends from a
• fundamental renormalizable or finite theory?

• Dynamics Non-uniform phase space (??)

21
a miracle occurs
At the point in parameter space where all the
cosmological constants coincide,

• In odd dimensions, the action describes a gauge
  theory for
• the corresponding group. Symmetry enhancement
  and no
• dimensionful coupling constants!

???
22
3. Embedding
the Lorentz group in a
larger one
23
new curvature
The Euler (E2n) and Pontryagin (P4k) invariants
can be constructed with FAB. These invariants
are closed

• Hence, they can be (locally) written as the
  exterior derivative of
• something else.

These are the combinations were looking for!!
24
Lovelock series
3a. Special choices ( )
() Alternate signs for dS.
25
Poincaré-
26
torsion series
3b. Special choices ( )
In D4k-1, there is a particular combination of
the torsion series invariant under (A)dS, related
to the Pontryagin class.
27
In sum
The most general AdS invariant action for 2n-1
dimensional gravity is a linear combination of 2
types of Chern-Simons forms

• The invariance of the Euler and Pontryagin forms
  under local
• AdS is inherited by the corresponding
  lagrangians.

• The vanishing cosmological constant limit is not
  the E-H theory.

• These lagrangians have no free parameters and no
  dimensionful
• coupling constants.

• A similar construction is not possible in D2n.

28
AdS Theory as Chern-Simons
The AdS invariant theory can be expressed in more
standard form in terms of its Lie algebra valued
connection and curvature
29
4. Dynamics
(generic C-S case)
For D 3 all CS theories are topological No
propagating degrees of freedom.
For D 2n1 5, a CS theory for a gauge algebra
with N generators can as many as ? Nn-N-n
propagating d. of f.
Degeneracy The system may evolve from an initial
state with ?1 d.o.f., reching a state with ?2 lt
?1 d.o.f. in a finite time. This is an
irreversible process in which there is no record
of the initial conditions in the final state.
30
Dynamics of C-S Gravity

• But, D-dimensional Minkowski space is not a
  classical solution!!

• The natural vacuum is, D-dimensional AdS space
  (max. symmetric).

• But around D-dimensional AdS space the theory
  has no d.o.f.!!

• Black holes are particularly sensitive to the
  theory

31
Example of vacuum solution
32
Curvature components
Existence of propagating modes in the
d-dimensional spacetime requires k 1. This
implies d 3 or 4. Otherwise, all fluctuations of
the metric become zero modes (gauge directions)
unobservable.
33
For d 4 life
is more interesting
34
5. Supersymmetry
Supersymmetry is the only nontrivial extension of
the spacetime symmetry Lorentz, Poincaré, (A)dS.
This is also a 1-form, which suggests a new
connection,
35
transformations
Supersymmetry
eventually this yields a supersymmetric
lagrangian.
36
superalgebras
The construction is straightforward and the
resulting theories are Chern-Simons systems . The
field content and the for the first cases are
37
Example in 11D
Starting from the 11D CS Lagrangians for gravity
in
38
11D M-algebra C-S Lagrangian
The C-S lagrangian
Invariant under the M-algebra, which includes
Poincaré, local SUSY and the abelian
transformations
39
6. Open questions

• . /prospects
• Renormalizability? Finite theory?
• Power counting, Adler-Bardeen theorem
• Relation with standard SUGRAs?
• Background Rab0 yields EHC.C.
• How is D4 recovered?
• Dynamical compactifications, branes,
  intersections thereof
• Interactions?
• Couplings to membranes
• Dynamics?
• Degeneracies, spontaneous compactifications
• Are Chern-Simons theories free theories in
  unusual form?
• Black hole thermodynamics

40
Summary

• The equivalence principle leads to a geometric
  construction where
• the vielbein and the spin connection are the
  dynamical fields.

• These theories are invariant under local Lorentz
  transformations
• and invariant (by construction) under
  diffeomorphisms.

• Around a flat background (Rab0Ta) these
  theories behave like
• ordinary GR (Einstein-Hilbert).

• However, this is just one of the many sectors of
  the theory.

• In odd dimensions and for a particular choice of
  these parameters, the
• theory is dimension-free and its symmetry is
  enhanced to (A)dS.

• The resulting action is a Chern-Simons form for
  the (A)dS or
• Poincaré groups.

41

• The classical evolution can take from one sector
  to another with
• fewer degrees of freedom in an irreversible
  manner.

• The supersymmetric extension only exist for
  special combinations
• of AdS or Poincaré invariant gravitational
  actions.

• The maximally (super)symmetric configuration is
  a vacuum state
• that has no propagating degrees of freedom
  around it. At that point
• the theory is topological.

• This shows that the standard supergravity of
  Cremmer, Julia and
• Sherk is not the only consistent supersymmetric
  field theory
• containing gravity in 11D. There exist at
  least 3 more

- Super-AdS Osp(321) - Super-Poincaré -
M-Algebra