IX Simposio Latino Americano de Fısica de Altas Energias SILAFAE 2012

N = 8 gauged supergravity and Extended field theoryJose Alejandro Rosabal Rodriguez (in collaboration with G. Aldazabal, M. Grana, D. Marques)Instituto Balseiro - Universidad Nacional de Cuyo - Centro Atomico Bariloche - CONICET, Bariloche, Argentina

AbstractWe present a unified description of the bosonic sector of N = 8gauged supergravity (GSG) in four dimensions. The theory isbased on an extended manifold of 1 + 3 + 56 dimensions. Thebosonic degrees of freedom are unified in a generalized metric.The diffeomorphism and gauge symmetries are promoted togeneralized diffeomorphism Diff (4)× E7(7) × R+.

MotivationDifferent attempts to promote T-duality, a distinctive symmetryof string theory, to a symmetry of a field theory, can be groupedinto the name of Double Field theory (DFT) [1]. Interestinglyenough, the Scherk-Schwarz (SS) reduction of (bosonic) DFTleads to an action that can be identified with the bosonic sectorof N = 4 gauged supergravities [2]. The aim of the presentwork is to extend the above ideas to incorporate the full stringyU-duality symmetry group in four dimensions. We name itExtended Field Theory (EFT).

Generalized diffeomorphisms on E7(7) × R+

We start by defining the generalized diffeomorphism in theE7(7) × R+ group compatible with supersymmetry.Generalized Lie derivative over an E7(7) frame

(LEAEB)M = E PA ∂PE M

B − 12PM QP R∂QE R

A E PB +

ω

2∂PE PA E M

B

(LEAEB)M = F CAB E M

C ; F CAB = X C

AB + T CAB

P : 56× 56→ 133 = Adj(E7(7)) ; ω conformal weight of R+

Linear constraintsT C

AB = −ϑAδCB + 8ϑK P K C

A B ; P C MB NX N

AM = X CAB

X BAB = X A

AB = 0 ; XA[BC ] = 0

X(ABC) = 0 ; X CAB ∈ 912 ; ϑA ∈ 56

Closure condition (quadratic constraints)Generalized bracket

[[EA,EB]] =12(LEAEB − LEBEA)

We demand

[LEA, LEB]EC − L[[EA,EB]]EC = 4[AB] = 0

On the other hand gaugings expressed in SU(8) indices should transform asscalars

δξF NRM = ξPDPF NR

M − ξQ4 NP(QM) ⇒ 4 NP

(QM) = 0

δξϑM = ξPDPϑM − ξQ4QM ⇒ 4QM = 0

Remarkably, in the particular case when the gaugins are constant we find exactlythe conditions found in [3]

[FM, FN] = −F RMN FR ; F P

(MN) F QPR = F P

(MN) ϑP = 0

Space-time extensionWe now want to couple the gaugings with a four-dimensional metric gµν andvectors AM

µ and perform the SS reduction.Generalized diffeomorphism

E MA (x ,Y ) = E B

A (x)U MB (Y )

LξVM = ξP∂PVM−AM PN Q∂Pξ

QVNThe mixed components of AM P

N Q can onlytake the form

Aµ Mν N = a1δ

νµδ

NM ; AM µ

N ν = a2δνµδ

NM

AM µν N = a3δ

νµδ

NM ; Aµ M

N ν = a4δνµδ

NM

Master flux(LEAEB)M = F C

AB EMC

BeinE MA (x) =

eµα eναAM

ν

0 ΦMA

U MB (Y ) =

δµβ 00 UM

B

Gauge symmetry and four dimensional diffeomorphisms

ξM = (λµ,ΛM)

δeµα = Lλeµα +12ϑNΛNeµα

δAMµ = LλAM

µ − F MNP ΛNAP

µ − ∂µΛM − 12ϑNΛNAM

µ

δΦMA = LλΦM

A − F MNP ΛNΦP

A

FMNP = FMN

P (gaugings)

FµνM = 2∂[µAν]M + FNP

MAµNAν

P

F[µM ]P = (Φ−1)M

N∂µΦNP + F[RM ]

PAµR

F(µM)P = F(RM)

PAµR

Fµνγ = 2 Γ[µν]

γ = −2(e−1)[νβ∂µ]eβγ

ConclusionsWe have shown that the idea of DFT can be extended to a more generalcontext to include the whole U-duality group. We treated the diffeomorphismsand gauge transformations in a unified way. We provide a higher dimensionalorigin for the gaugings of maximal supergravity in terms of an extendedgeometry over a generalized manifold. The generalized Lie derivative providesgaugings that are compatible with supersymmetry, therefore satisfyingautomatically the linear constraint. We also compute the closure conditions ofthe algebra and extract the quadratic constraints, which generalize those foundin maximal supergravity to the case of non-constant fluxes. Based on theseresults, we give a first step towards an action written in terms of thegeneralized fluxes which we hope can be interpreted as generalized Ricci scalar.

References[1] C. Hull and B. Zwiebach, “Double Field Theory”, JHEP 0909 (2009) 099 [arXiv:0904.4664 [hep-th]].[2] G. Aldazabal, W. Baron, D. Marques and C. Nunez, “The effective action of Double Field Theory”, JHEP 1111

(2011) 052 [arXiv:1109.0290 [hep-th]].[3] Arnaud Le Diffon, Henning Samtleben, Mario Trigiante, “N = 8 Supergravity with Local Scaling Symmetry”,

JHEP 04 (2011) 079 [arXiv:1103.2785 [hep-th]].

rosabalj@ib.cnea.gov.ar http://fisica.cab.cnea.gov.ar/particulas