1. Centro de Estudios Científicos CECS-Valdivia-Chile

2. In collaboration with: José Edelstein José Edelstein Mokhtar Hassaine Mokhtar Hassaine Ricardo Troncoso 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] (Super) Gravities of a different sort Constrained Dynamics and Quantum Gravity QG05 Sardegna, September 2005 J. Zanelli CECS - Valdivia

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 Einstein’s 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. Gravity is a funny interaction that forces all bodies to move in the same orbits. [Galileo, Newton]

5. TxTxTxTx T x+dx : : Spacetime is a differentiable manifold endowed with a tangent space T x, identical to Minkowski space, at each point. The freedom to perform independent Lorentz rotations at each point in spacetime is a gauge symmetry [Utiyama (1955), Kibble (1961)...]. Local orthonormal frames Tangent spaces In this way, General Relativity generalizes Special Relativity. Equivalence principle

6. The local Lorentz rotations in spacetime endows spacetime with a TxTxTxTx This is a genuine gauge symmetry. At each point there is a Lie This is a genuine gauge symmetry. At each point there is a Lie algebra, algebra, SO(D-1,1) Fibers fiber bundle structure. General coordinate transformations do not form a Lie algebra General coordinate transformations do not form a Lie algebra and do not define a gauge symmetry with fiber bundle structure. and do not define a gauge symmetry with fiber bundle structure.

7. Even worse, they are functions of the dynamical field, Even worse, they are functions of the dynamical field, which limits the practical usefulness of this symmetry. which limits the practical usefulness of this symmetry. a)It should be possible (and desirable) to eliminate all reference to coordinates in physics, and b)GCT can have no physically observable consequences. Invariance under GCT must be a feature of any correct physical Invariance under GCT must be a feature of any correct physical theory, since coordinates are an invention of our culture and not theory, since coordinates are an invention of our culture and not features of nature. Hence, features of nature. Hence, The are not constant (open algebra). The are not constant (open algebra). General coordinate transformations The diffeomorphism algebra takes the form: The diffeomorphism algebra takes the form: The first order formalism achieves this.

8. e1e1e1e1 TxTxTxTx e2e2e2e2 e3e3e3e3 Isomorphism between between the local orthonormal frame on the tangent space and the coordinate basis Vielbein: Local frames (vielbein) This induces a metric structure for spacetime:

9. TxTxTxTx T x+dx Parallelism The notion of parallel transport needed for differentiability is provided by the connection that relates local frames at nearby points, Lorentz (spin) connection:

10. D - dimensional gravity Local symmetry: Local Lorentz transformations, Dynamical fields Vielbein: ( = composite ) Lorentz (spin) connection:. Lorentz (spin) connection:.

11. Manifestly Lorentz invariant theory 2. Manifestly Lorentz invariant theory Coordinate independence is achieved naturally using coordinate- Coordinate independence is achieved naturally using coordinate- free fields (1-forms). free fields (1-forms). A D-form constructed out of,, their exterior derivatives, A D-form constructed out of,, their exterior derivatives, and the Lorentz invariant tensors and, and the Lorentz invariant tensors and, The Lagrangian for D-dimensional gravity is assumed to be: (Quasi*-) invariant under local Lorentz transformations, in (Quasi*-) invariant under local Lorentz transformations, in order to ensure Lorentz covariance of the field equations. order to ensure Lorentz covariance of the field equations. (* Up to total derivatives)

12. Using only exterior derivatives limits the number of options: Taking exterior derivatives of and, two tensors are found : This limits the number of ingredients in L to: Zero-forms one-forms two-forms Torsion Torsion 2-forms 2-forms Curvature Curvature Bianchi identities

13. Three series are found: are found: A: Generalization of E-H for all D, parity even. B: Some D only, vanish for T a =0 or  =0, parity odd. C: For D=4k-1only, parity odd. A. Lovelock B. Torsion C. Lorentz C-S Euler classes Pontryagin classes (Exotic gravities)

14. Also, L p = continuation of the Euler density Also, L p = continuation of the Euler density from 2p to D dimensions from 2p to D dimensions A. Lovelock series (*) (* Wedge product is understood) In even dimensions, the last term is the Euler density, (D=2n) (D=2n) [Lovelock, 1970] If, the Lagrangian is. If, the Lagrangian is.

15. Cosmological constant term (volume) Einstein-Hilbert Gauss-Bonnet Gauss-Bonnet Poincaré invariant term (odd D) [see below] This series includes: Euler density (even D) [see above]......

16. These traces are related to the Pontryagin form (Chern class), B. Torsion series If, more invariant terms are allowed: / whose integral is a topological invariant in 4k dimensions...and products thereof.

17. They are also related to the Pontryagin forms: They are also related to the Pontryagin forms: C. Lorentz Chern-Simons These are quasi-invariant terms involving explicitly: They do not vanish if or if. They do not vanish if or if. Products of these and torsional terms are also acceptable Products of these and torsional terms are also acceptable Lagrangians, provided they are D-forms. Lagrangians, provided they are D-forms. and in general, Fixed numbers series series

18. Arbitrary coefficients! The Action,with where,],[ 12212 21 1... ]2/[ 0 G eeRRL L L LLeI D ppp D D a aaa aa a ap p D p p D Lovelock M D Exotic D Lovelock           The most general action for gravity in D dimensions is The exotic part contains torsional terms and a Lorentz C-S term, with more arbitrary constant coefficients, C-S term, with more arbitrary constant coefficients,

19. 1 st order field equations for e, ω. 1 st order field equations for e, ω. Diffeomorphic invariant by construction Diffeomorphic invariant by construction Invariant under local Lorentz transformations Invariant under local Lorentz transformations Features Einstein-Hilbert theory is the only option for D=4 Einstein-Hilbert theory is the only option for D=4 Torsion-free sector : Second order theory for g μν Second order theory for g μν Same degrees of freedom as ordinary gravity Same degrees of freedom as ordinary gravity General action for a D - dimensional geometry.

20. Puzzles / problems How could this be so if gravity descends from a How could this be so if gravity descends from a fundamental renormalizable or finite theory? fundamental renormalizable or finite theory? Dynamics: Non-uniform phase space ( ??) Dynamics: Non-uniform phase space ( ??) Violently different behavior for different choices of α p,  q. Violently different behavior for different choices of α p,  q. Hopeless quantum scenario: α p,  q dimensionful and Hopeless quantum scenario: α p,  q dimensionful and unprotected from renormalization. unprotected from renormalization. The action has a number of cosmological constants The action has a number of cosmological constants whose values are not fixed (the cosmological constant whose values are not fixed (the cosmological constant problem raised to a large number) problem raised to a large number)

21. At the point in parameter space where all the cosmological constants coincide, : All the dimensionful coupling constants can be absorbed by a redefinition of the vielbein redefinition of the vielbein In odd dimensions, the action describes a gauge theory for the corresponding group. Symmetry enhancement and no the corresponding group. Symmetry enhancement and no dimensionful coupling constants! dimensionful coupling constants! The vielbein and the spin connection can be combined as components of a single connection 1-form, components of a single connection 1-form, where Lorentz generator generators of generators of (A)dS boosts/translations (A)dS boosts/translations (A)dS/Poincaré algebra a miracle occurs ???

22. the Lorentz group in a larger one the Lorentz group in a larger one anti-de Sitter de Sitter D-dim. Lorentz Poincaré Its is only necessary to decide the signature of the last component, AdS dS 3. Embedding This is viewed combining e, ω ab in a single array: a, b= 1,…, D A,B=1,…, D+1 which is now a connection of the larger group.

23. The is + AdS dS dS _ The Euler () and Pontryagin () invariants can be constructed with F AB. These invariants are closed: The Euler ( E 2n ) and Pontryagin ( P 4k ) invariants can be constructed with F AB. These invariants are closed: These are the combinations we’re looking for!! C E 2n-1, C P 4k-1 are functions of e, ω ab and their derivatives. E 2n, P 4k invariant under (A)dS, so are the C ’s. Hence, they can be (locally) written as the exterior derivative of Hence, they can be (locally) written as the exterior derivative of “something else”. “something else”. new curvature

24. Lovelock series For D =2n+1, choosing the coefficients in the Lovelock series as produces the particular form of the Lovelock lagrangian with an enhanced symmetry, Lorentz AdS(*). The secret of the enhancement: SO(D-1,2) invariant invariant (*) Alternate signs for dS. 3a. Special choices ( )

25. SO(D-1,1) ISO(D-1,1) with the enhancement One can also take the limit. The result is a invariant theory (that is not Einstein-Hilbert!) Poincaré- N.B. The Einstein-Hilbert action, is not Poincaré - invariant.

26. In D=4k-1, there is a particular combination of the torsion series invariant under (A)dS, related to the Pontryagin class. torsion series = P 4k Pontryagin invariant of 4k dimensions (The explicit form of is not very interesting and it involves torsion explicitly.) 3b. Special choices ( ) In the limit, the only surviving term in is the Lorentz Chern-Simons term In the limit, the only surviving term in is the Lorentz Chern-Simons term

27. The most general AdS invariant action for 2n-1– dimensional gravity is a linear combination of 2 types of Chern-Simons forms: A similar construction is not possible in D=2n. A similar construction is not possible in D=2n. The invariance of the Euler and Pontryagin forms under local The invariance of the Euler and Pontryagin forms under local AdS is inherited by the corresponding lagrangians. AdS is inherited by the corresponding lagrangians. The vanishing cosmological constant limit is not the E-H theory. The vanishing cosmological constant limit is not the E-H theory. These lagrangians have no free parameters and no dimensionful These lagrangians have no free parameters and no dimensionful coupling constants. coupling constants. Exotic (D=4k-1) where In sum:

28. The AdS invariant theory can be expressed in more standard form in terms of its Lie algebra valued connection and curvature: Both the Euler and Pontryagin Chern-Simons can be written as Appropriate multilinear invariant product (trace) AdS Theory as Chern-Simons

29. For D ≤3 all CS theories are topological: No propagating degrees of freedom. For D =2n+1 ≥5, a CS theory for a gauge algebra with N generators can as many as ∆ = Nn-N-n propagating d. of f. Phase space is not homogeneous: Phase space is not homogeneous: Nn-N-n generic configurations (most states, no symmetry) Nn-N-n generic configurations (most states, no symmetry) ∆ = 0 maximally degenerate (measure zero, max. symmetric) 4. Dynamics Degeneracy: The system may evolve from an initial state with ∆ 1 d.o.f., reching a state with ∆ 2 < ∆ 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. (generic C-S case)

30. Dynamics of C-S Gravity Around a flat background,,, Chern-Simons theories behave as the standard Einstein-Hilbert theory. (same d.o. f., propagators, etc. But, D-dimensional Minkowski space is not a classical solution !! But, D-dimensional Minkowski space is not a classical solution !! The natural vacuum is, D-dimensional AdS space (max. symmetric). The natural vacuum is, D-dimensional AdS space (max. symmetric). But around D-dimensional AdS space the theory has no d.o.f. !! But around D-dimensional AdS space the theory has no d.o.f. !! Black holes are particularly sensitive to the theory: Black holes are particularly sensitive to the theory: T m Chern-Simons theory Ideal gas? Generic Lovelock theory T m T stable

31. Ansatz: warped product Example of vacuum solution Consider the simplest case: Poincaré invariant theory Consider the simplest case: Poincaré invariant theory. d-dimensional spacetime Compact symmetric space Field equations (spacetime geometry) [Torsion-free ok.]

32. The field equations take the form 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. Curvature components d-dimensional spacetime Compact symmetric space

33. is more interesting… is more interesting… The effective theory has a 4 dim. de Sitter ( ) vacuum. The effective theory has a 4 dim. de Sitter ( ) vacuum. The perturbations obey the linearized 4 dim. The perturbations obey the linearized 4 dim. Einstein equations along the worldsheet. Einstein equations along the worldsheet. Propagation in the transverse direction is Propagation in the transverse direction is strongly suppressed (mass gap). strongly suppressed (mass gap). Propagation in the “internal” directions are Propagation in the “internal” directions are Kaluza Klein modes with discrete spectrum. Kaluza Klein modes with discrete spectrum. Einstein’s equations for 4 dim. spacetime metric provided Einstein’s equations for 4 dim. spacetime metric provided the D-d-1 extra dimensions are nondegenerate ( ). the D-d-1 extra dimensions are nondegenerate ( ). For d = 4 life

34. New ingredient: Gravitino This is also a 1-form, which suggests a new connection, Necessary to close the algebra. Depends on the dimension. Supersymmetry 5. Supersymmetry Supersymmetry is the only nontrivial extension of the spacetime symmetry [Lorentz, Poincaré, (A)dS]. The generators should form a (super) algebra.

35. Supersymmetry Tentative fermionic Lagrangian etc. … eventually this yields a supersymmetric lagrangian. transformations

36. The construction is straightforward and the resulting theories are Chern-Simons systems. The field content and the for the first cases are D AlgebraField content Standard Supergravity 5 su(2,2|1) 7 osp(2|8) 11 osp(1|32) Local SUSY (off-shell algebra): This is an exact local symmetry: no auxiliary fields, no on-shell conditions, no assumptions on the background (, etc.) superalgebras

37. Starting from the 11D CS Lagrangians for gravity in Example in 11D Osp(32|1): Super Poincaré: M-algebra:

38. The C-S lagrangian: where Invariant under the M-algebra, which includes Poincaré, local SUSY and the abelian transformations 11D M-algebra C-S Lagrangian

39. . /prospects : Renormalizability? Finite theory? Renormalizability? Finite theory? [Power counting, Adler-Bardeen theorem] [Power counting, Adler-Bardeen theorem] Relation with standard SUGRAs? Relation with standard SUGRAs? [Background R ab =0 yields EH+C.C.] [Background R ab =0 yields EH+C.C.] How is D=4 recovered? How is D=4 recovered? [Dynamical compactifications, branes, intersections thereof] [Dynamical compactifications, branes, intersections thereof] Interactions? Interactions? [Couplings to membranes] [Couplings to membranes] Dynamics? Dynamics? [Degeneracies, spontaneous compactifications] [Degeneracies, spontaneous compactifications] Are Chern-Simons theories free theories in unusual form? Are Chern-Simons theories free theories in unusual form? [Black hole thermodynamics] [Black hole thermodynamics] 6. Open questions

40. Summary Summary The equivalence principle leads to a geometric construction where The equivalence principle leads to a geometric construction where the vielbein and the spin connection are the dynamical fields. the vielbein and the spin connection are the dynamical fields. These theories are invariant under local Lorentz transformations These theories are invariant under local Lorentz transformations and invariant (by construction) under diffeomorphisms. and invariant (by construction) under diffeomorphisms. In odd dimensions and for a particular choice of these parameters, the In odd dimensions and for a particular choice of these parameters, the theory is dimension-free and its symmetry is enhanced to (A)dS. 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 The resulting action is a Chern-Simons form for the (A)dS or Poincaré groups. Poincaré groups. Around a flat background (R ab =0=T a ) these theories behave like Around a flat background (R ab =0=T a ) these theories behave like ordinary GR (Einstein-Hilbert). ordinary GR (Einstein-Hilbert). However, this is just one of the many sectors of the theory. However, this is just one of the many sectors of the theory. They give rise to 2 nd order field eqs. for, and spin They give rise to 2 nd order field eqs. for, and spin These theories possess a number of indeterminate dimensionful These theories possess a number of indeterminate dimensionful parameters. parameters.

41. This shows that the standard supergravity of Cremmer, Julia and This shows that the standard supergravity of Cremmer, Julia and Sherk is not the only consistent supersymmetric field theory Sherk is not the only consistent supersymmetric field theory containing gravity in 11D. There exist at least 3 more: containing gravity in 11D. There exist at least 3 more: Sholud we add new cusps in the diagram? The classical evolution can take from one sector to another with The classical evolution can take from one sector to another with fewer degrees of freedom in an irreversible manner. fewer degrees of freedom in an irreversible manner. The supersymmetric extension only exist for special combinations The supersymmetric extension only exist for special combinations of AdS or Poincaré invariant gravitational actions. of AdS or Poincaré invariant gravitational actions. The maximally (super)symmetric configuration is a vacuum state The maximally (super)symmetric configuration is a vacuum state that has no propagating degrees of freedom around it. At that point that has no propagating degrees of freedom around it. At that point the theory is topological. the theory is topological. - Super-AdS [Osp(32|1)] - Super-Poincaré - M-Algebra