Gauged Supergravities in Different Frames
Dr. Mario Trigiante (Politecnico di Torino)
F.Cordaro, P.Frè, L.Gualtieri, P.Termonia, M.T. 9804056F.Cordaro, P.Frè, L.Gualtieri, P.Termonia, M.T. 9804056
Wit, Samtleben, M.T. 0311224Wit, Samtleben, M.T. 0311224; Dall’Agata, Inverso, M.T. 1209.0760Dall’Agata, Inverso, M.T. 1209.0760
Plan of the Talk
• Overview and Motivations: Gauged Supergravity and string/M-theory compactifications.
• Embedding tensor formulation of D=4 gauged SUGRAs and duality
• Conclusions
• Relevance of symplectic frames: New N=8 SUGRAs with SO(8) local symmetry
Introduction
• D=4 Supergravity from Superstring/M-theory:
SuperstringM-theory
M1,3 x MRicci flat
Flux = 0
D=4 ungaugedSupergravity
Global symmetriesDualities
•Minimal coupl.•mass. def.•V()
D=4 gaugedSupergravity
M1,3 x M
Flux 0
Embedding tensor
• Mass deformations: spontaneous SUSY breaking
•Scalar potential: moduli stabilization in Minkoswki, dS or AdS vacua
Ungauged (extended) Supergravities
• Electric-magnetic duality symmetry of Maxwell equations now must also involve the scalar fields (Gaillard-Zumino)
G = Isom(Mscal)Non-linear action on
Linear actionF
G
g¢F
G
Sp(2 nv, R) E/M duality promotes
G to global sym. of f.eqs. E B. ids.
g = 2 G A B
C D• Smaller symmetry of the action:
• Scalar fields (described by a non-lin. Sigma-model) are non-minimally coupled to the vector ones
The Issue of Symplectic Frames
• Different symplectic frames (SF) may yield inequivalent actions with different global symmetry groups Ge but same physics
• In the SUGRA description of string/M-theory compactifications, SF fixed by the resulting scalar-vector couplings
• Coupling of scalar fields to vectors is fixed up to a symplectic transfomation on F and G (Symplectic FrameSymplectic Frame)
Parity as an anti-Symplectic Duality
• Split total scalars so that:
isometry
• is an invariance of the theory
• is realized on the vector fields and their magnetic duals by an anti-symplectic duality transformation
• Distinction between the scalar/pseudo-scalar fields depends on the choice of the symplectic frame
Gauging• Gauging consists in promoting a group G ½ Ge ½ G from global to local symmetry of the action. Different SF ) different choices for G.
• Local invariance w.r.t. G
• Description of gauging which is independent of the SF:
E symplectic 2nv x 2nv matrix
• All information about the gauging encoded in a G-tensor: the embedding tensor
[Cordaro, Frè, Gualtieri, Termonia, M.T. 9804056; Nicolai, Samtleben 0010076; de Wit, Samtleben, M.T. 0311224 ]
• Restore SUSY of the action:
Mass terms:
Scalar potential:
Fermion shifts:
Closure:
Locality
Linear:
• String/M-theory origin:
[D’Auria, Gargiulo, Ferrara, M.T., Vaulà 0303049; Angelantonj, Ferrara, M.T. 0306185; de Wit, Samtleben, M.T. 0311224…]
• Manifestly G-covariant formulation de Wit, Samtleben, M.T. 0507289
• Emb. tensor from E11 and tensor hiearachies [de Wit, Samtleben 0501243; Riccioni, West 0705.0752; de Wit, Nicolai, Samtleben, 0801.1294]
N=8, D=4 SUGRA
• Scalar fields in non-linear -model with target space
(1) g
A
(28) AAB
ABC
ABCD
gravitational
A,B 2 8 of SU(8)R
Mscal = =
32 supercharges
Linear constraints
Quad. constraints
• Looking for SO(8):
• First gauging:[de Wit, Nicolai ’82]
Original dWN gauging
Hull’s CSO(p,q,r)-gaugings
Same groups gauged by the magnetic vectors
Gaugings defined by
• Take generic
• Quadratic constraints
• Gauge connection:
Choice corresponds to an SO(8)-gauging in a different SF in which A’IJ
are electric
• Features of E: it centralizes so(8) in Sp(56) and is NOT in E7(7) for generic angle:
but not in SU(28) for generic
• analogue of de Roo-Wagemann’s angle in N=4, N=2: parametrizes inequivalent theories.
• Vacua of original dWN theory =0studied by Warner and recently by Fischbacher (found several critical points, not complete yet)
• Studied vacua with a G2 residual symmetry: suffices to restrict to G2 singlets
• Scalar potential:
where and de Wit, Samtleben, M.T. 0705.2101
Dall’Agata, Inverso, M.T. 1209.0760Borghese, Guarino, Roest, 1209.3003
• Discrete symmetries of Veff:
(Parity) (SO(8) Triality)
originate from non trivial symmetries of the whole theory
• does not affect action terms up to second order in the fluctuations about the N=8 vacuum (mass spectrum).
• Possible relation to compactifiation of D=11 SUGRA on with torsion () (ABJ) [Aharony, Bergman, Jafferis, 0807.4924]
• Inequivalent theories only for
Conclusions
• Showed in a given example how initial choice of SF
determines, after gauging, physical properties of the model
• Study vacua of the new family of SO(8)-gauged maximal SUGRAS
• RG flow from new N=0 G2 vacuum to N=8 SO(8) one (both stable AdS4)