Chern-Simons Forms and Supergravities in D = 11

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1 Chern-Simons Forms and Supergravities in D = 11 Fernando Izaurieta 1, Alfredo Pérez 2, Eduardo Rodríguez 1, Patricio Salgado 2. 1 Applied Mathematics and Physics Departament, Universidad Católica de la Santísima Concepción 2 Physics Department, Universidad de Concepción Marcel Grossmann Meeting XII, Paris, July

2 Contents 1 Motivation: Cremmer Julia Scherk & Chern Simons Supergravities 2 : Is there a link? 3

3 Contents 1 Motivation: Cremmer Julia Scherk & Chern Simons Supergravities 2 : Is there a link? 3

4 Motivation: CJS SUGRA and M Theory

5 Motivation: CJS SUGRA and M Theory

6 Chern Simons SUGRA The Connection Let us start with osp (32 1) as a nice symmetry for Chern Simons SUGRA in d = 11, { Q α, Q β } = with ( [Γ a ] α β P a 1 [ Γ ab ] α 2 β J ab + 1 ) 5! [Γa 1 a 5 ] α β Z a 1 a 5 P a : AdS Boosts, J ab : Lorentz, Z a1 a 5 : Extra bosonic generators (not central charge!).

7 Chern Simons Supergravity Lucky us, 11 is an odd number!! osp (32 1) Chern Simons Lagrangian in d = 11 t=1 L (11) CS = 6k dt A ( tda + t 2 A 2) 5 t=0 A : 1-form connection field A = 1 l ea P a ωab J ab + 1 5!l ba 1 a 5 Z a1 a l ψq. (A. H. Chamseddine, 1990)

8 Chern Simons Supergravity Lucky us, 11 is an odd number!! osp (32 1) Chern Simons Lagrangian in d = 11 t=1 L (11) CS = 6k dt A ( tda + t 2 A 2) 5 t=0 A : 1-form connection field A = 1 l ea P a ωab J ab + 1 5!l ba 1 a 5 Z a1 a l ψq. (A. H. Chamseddine, 1990)

9 Chern Simons Supergravity Lucky us, 11 is an odd number!! osp (32 1) Chern Simons Lagrangian in d = 11 t=1 L (11) CS = 6k dt A ( tda + t 2 A 2) 5 t=0 A : 1-form connection field A = 1 l ea P a ωab J ab + 1 5!l ba 1 a 5 Z a1 a l ψq. (A. H. Chamseddine, 1990)

10 Chern Simons Supergravity Lucky us, 11 is an odd number!! osp (32 1) Chern Simons Lagrangian in d = 11 t=1 L (11) CS = 6k dt A ( tda + t 2 A 2) 5 t=0 A : 1-form connection field A = 1 l e + ω + 1 l b + 1 l ψ. (A. H. Chamseddine, 1990)

11 Chern Simons Supergravity CS = M? It has been suggested that this kind of theory could correspond to M Theory! Oldest References: [1] A. H. Chamseddine, Topological gravity and supergravity in various dimensions, Nucl. Phys. B 346: ,1990. [2] R. Troncoso, J. Zanelli, New gauge Supergravity in Seven and Eleven Dimensions. Phys. Rev. D 58 (1998) [arxiv:hep-th/ ]. [3] P. Hořava, M Theory as a Holographic Field Theory. Phys. Rev. D 59 (1999) [arxiv:hep-th/ ]. [4] M. Bañados, The Linear spectrum of OSp (32 1) Chern Simons supergravity in eleven-dimensions, Phys. Rev. Lett. 88:031301, [arxiv:hep-th/ ]. [5] H. Nastase, Towards a Chern Simons M Theory of Osp (32 1) Osp (32 1) [arxiv:hep-th/ ]. Why?

12 Chern Simons Supergravity Features Features of Chern Simons SUGRA It is a pure gauge theory, in the fiber bundle sense. It is off-shell invariant, without auxiliary fields. It has only one dimensionful free parameter (quantized). It seems to be renormalizable! It has no ghosts. It has no higher spin fields. It does have local propagating degrees of freedom in d = 11. It is background-free. This beautiful theory seems to have everything we love in a QFT! However...

13 Chern Simons Supergravity Features...if CS SUGRA in d = 11 does correspond to (a low energy limit of) M Theory, then it should be possible to recover CJS SUGRA from CS SUGRA somehow!! Until now, this point remains as an important test for the theory!

14 Chern Simons Supergravity Features Dynamics of both theories seem to be quite different (Chern Simons is highly non-lineal!) Could CJS SUGRA correspond just to a sector of the Chern Simons Lagrangian??

15 Contents 1 Motivation: Cremmer Julia Scherk & Chern Simons Supergravities 2 : Is there a link? 3

16 CJS v/s CS

17 CJS SUGRA Lagrangian First Order Formalism with L (11) CJS = 1 4 9! ɛ abc 1 c 9 R ab e c1 e c i 2 ψγ (8) Dψ + i ( T a i ) 8 4 ψγ a ψ e a ψγ(6) ψ+ 1 2 F F + ( F + b) (da 3 a) ab 1 3 A 3dA 3 da 3. Γ (n) = 1 n! Γ a 1 a n e a1 e an a = i 4 ψγ (2) ψ b = i 4 ψγ (5) ψ

18 CS SUGRA Lagrangian d = 11, osp (32 1) superalgebra where L (11) CS = 6k t=1 t=0 dt A ( tda + t 2 A 2) 5 A = 1 l ea P a ωab J ab + 1 5!l ba 1 a 5 Z a1 a l ψq.

19 Comparing both Gravity Sectors It seems trivial: let s write down the CS Lagrangian and see if standard CS-SUGRA is inside or not!

20 WARNING!!! CS SUGRA is a hugely complicated Lagrangian! ( terms!) Jumping to compare both of them would be hazardous for mental health! DANGER!!

21 WARNING!!! The way of finding the needle in the haystack is: 1 (carefully) using the Cartan s Homotopy Formula. [hep-th/ ] 2 Expanding CS as a polinomial in l. Let s see the results!! The Needle in the Haystack

22 Order O ( l 9) in CS SUGRA From the terms, the eye of the needle in the haystack seems to be L (11) CS [ ( O l 9 )] b=0 = 1 6! 1 l 9 er ( e 2) ! 1 4! 9 l 9 et 2 ( e 2) 3 + 2!3! ! 1 l 9 ψd ω ψ ( e 2) ! 1 4! 2 l 9 ψ 3! ! 1 ψ l 9 [e, ψ] 3 ( e 2) 2 3!2! 4 (T + 23 ψ2 ) [e, ψ] ( e 2) 3 +

23 Order O ( l 9) in CS SUGRA L (11) CS [ ( O l 9 )] b=0 = 1 6! 1 l 9 er ( e 2) ! 1 4! 9 l 9 et 2 ( e 2) 3 + 2!3! ! 1 l 9 ψd ω ψ ( e 2) ! 1 4! 2 l 9 ψ 3! ! 1 ψ l 9 [e, ψ] 3 ( e 2) 2 3!2! 4 (T + 23 ψ2 ) [e, ψ] ( e 2) 3 + L (11) CJS = 1 4 9! ɛ abc 1 c 9 R ab e c1 e c i 2 ψγ (8) Dψ + i ( T a i ) 8 4 ψγ a ψ e a ψγ(6) ψ+ 1 2 F F + ( F + b) (da 3 a) ab 1 3 A 3dA 3 da 3.

24 A 3? Motivation: CJS & CS SUGRA It seems to work, but what about A 3...?

25 A 3? Motivation: CJS & CS SUGRA Since A 3 l 3, when we consider it as a composed field in this context, the only possible ansatz is of the form, A 3 = A abc e a e b e c + A a1 a 5 bcb a 1 a 5 e b e c + + A a1 a 5 b 1 b 5 cb a 1 a 5 b b 1 b 5 e c + + A a1 a 5 b 1 b 5 c 1 c 5 b a 1 a 5 b b 1 b 5 b c 1 c 5

26 A 3? Motivation: CJS & CS SUGRA Using that ansatz for A 3, it is possible to look for terms in L (11) CS which could be related to A 3 da 3 da 3... They seem to be inside L (11) b!!!

27 Contents 1 Motivation: Cremmer Julia Scherk & Chern Simons Supergravities 2 : Is there a link? 3

28 It seems that Cremmer-Julia-Scherk could be considered as a sector of a Chern Simons theory! To solve this problem is important to know the possible rôle of CS SUGRA in the context of M Theory! Important: Even if CJS SUGRA is contained in an important sector of CS SUGRA, it doesn t mean that the CS dynamics should be related to the one of CJS!!

29 Questions Is it possible to include fermionic fields inside A 3 using other algebras? (e.g., expansions) If CJS SUGRA is contained as sector of CS SUGRA, what would that mean in the context of String Theory? Is Chern Simons SUGRA in d = 11 related to SUGRA IIA & IIB in d = 10?