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)