gauged double field theoryactive.pd.infn.it/g4/seminars/2012/files/marques.pdf · gauged double...

Gauged Double Field Theory

Diego Marques

IPhT CEA Saclay

Padova, 22 February 2012

Aldazabal, Baron, DM and Nunez [1109.0290]

Grana and DM [1201.2924]

Diego Marques IPhT CEA


Motivation



Outline

Introduction to Double Field Theory (DFT)(Field content, action, symmetries and constraints)

Generalities of dimensional reductions of DFT(Reduction ansatz, symmetries, effective actions and constraints)

Four dimensional effective action: gauged supergravity,generalized backgrounds and fluxes

Ten dimensional effective action: non-Abelian Heteroticstrings



Double Field TheoryThe idea

T-duality explicit in field theory

Defined on a double space XM =(x i , xi

)For tori:

Restricted DFT: can always rotate to a frame in which fieldsdepend only on x i



Double Field TheoryField content

Field content

HMN(x i , xi ) , d(x i , xi )

Generalized 2D × 2D metric (constrained)

H =

(g−1 −g−1bbg−1 g − bg−1b

)∈ O(D,D) , η =

(0 11 0

)

Invariant dilaton d

e−2d =√ge−2φ



Double Field TheoryAction

SDFT =

∫dxdx e−2d R(H, d)

R(H, d) = 4HMN∂M∂Nd − 4HMN∂Md∂Nd + 4∂MHMN∂Nd − ∂M∂NHMN

+1

8HMN∂MHKL∂NHKL −

1

2HMN∂MHKL∂KHNL

Particular frame ∂M =(∂ i , ∂i

)= (0, ∂i )

SDFT → SNSNS =

∫dx√ge−2φ

(R + 4(∂φ)2 − 1

12H2

), H = db



Double Field TheoryGlobal symmetries

d ′ = d , X ′ = UX , H′ = UTH U , U ∈ O(D,D)

Lead to Buscher rules

g ′aa = 1/gaa , g ′ai = −bai/gaa , g ′ij = gij − (gaigaj − baibaj )/gaa

b′ai = −gai/gaa , b′ij = bij − (gaibaj − baigaj )/gaa

Double Field Theory is invariant under generalized T-dualities.



Double Field TheoryLocal symmetries

Also invariant under gauge transformations with parameter ξM

δξe−2d = ∂M

(ξMe−2d

)δξHMN = ξP∂PHMN +

(∂MξP − ∂Pξ

M)HPN +

(∂NξP − ∂Pξ

N)HMP

Parameterizing ξM =(εi , ε

i)

in the frame ∂M = (0, ∂i )

δξgij = Lεg ij

δξbij = Lεbij + ∂i εj − ∂j εi



Double Field TheoryGauge invariance constraint

For the action of DFT

SDFT =

∫dxdx e−2d R

we have

δξR = ξM∂MR+ G (ξ,H, d) , δξe−2d = ∂M

(ξMe−2d

)So the action is invariant if

δξSDFT =

∫dxdx

[∂M

(ξMe−2dR

)+ e−2dG (ξ,H, d)

]= 0



Double Field TheoryGauge invariance constraint

Gauge invariance requires

G ≡ −∂P∂NξM∂PHMN − 2∂PξM∂P∂NHMN

+4∂Pd∂M∂PξNHMN

+4∂Pd∂PξN∂MHMN + 4∂Nd∂

PξM∂PHMN

+1

4HMN∂PξM∂PHKL∂NHKL

+8HMN∂PξM∂P∂Nd − 8HMN∂Md∂PξN∂Pd

−HMN∂PξM∂PHKL∂KHNL

−2∂M

(∂P∂PξNHMN

)+ 4∂P∂PξM∂NdHMN = 0



Double Field TheoryAlgebra and closure

Demanding that two successive transformations behave as onetransformation

[δξ1 , δξ2 ]VM = δξ12VM − FM(ξ1, ξ2,V )

requires

FM ≡ ξQ[1∂

Pξ2]Q∂PVM + 2∂Pξ[1Q∂

PξM2] V

Q = 0

and defines the C-bracket

ξ12 = [ξ1, ξ2]MC = 2ξN[1∂Nξ

M2] − ξ

P[1∂

Mξ2]P



Double Field TheoryClosure of gauge transformations

The C-bracket does not satisfy the Jacobi identity

JMC (ξ1, ξ2, ξ3) = 3[ξ[1, [ξ2, ξ3]]C]MC =

3

2∂M(ξP[1ξ

Q2 ∂Pξ3]Q

)This can be a problem because it must generate trivialtransformations

3[δξ[1 , [δξ2 , δξ3] ]] = δJC(ξ1,ξ2,ξ3)

so gauge transformations close provided

HMN ≡ δJC(ξ1,ξ2,ξ3)V

MN =

3

2∂P(ξR[1ξ

S2 ∂Rξ3]S

)∂PV

MN = 0



Double Field TheoryRestricted DFT

Sufficient conditions to solve the constraints

Weak constraint (Level matching condition on tori)

∂P∂PA = 0

Strong constraint

∂PA ∂PB = 0

Where A and B generically denote fields and gauge parameters.

Restricted DFT ⇔ Weak + Strong constraints



Double Field TheoryRelated works and extensions

Heterotic formulation: Andriot; Hohm, Kwak

Type II unification: Hohm, Kwak, Zwiebach; Coimbra,Strickland-Constable, Waldram

Massive Type II: Hohm, Kwak

U-duality, M-theory: Berman, Copland, Godazgar, Perry,Thompson; West; Coimbra, Strickland-Constable, Waldram




Frame doubled geometry: Hohm, Kwak; Jeon, Lee, Park

Generalized geometry: Coimbra, Strickland-Constable,Waldram; many others...

Non-geometry: Andriot, Hohm, Larfors, Lust, Patalong

Double geometry: Hull, Reid-Edwards; Dall’Agata, Prezas,Samtleben, Trigiante

Riemann Tensor: Hohm, Zwiebach

Supersymmetric DFT: Hohm, Kwak; Jeon, Lee, Park




Branes and solitons: Bergshoeff, Riccioni; Albertsson, Dai,Kao, Lin; Jensen

Dimensional reductions: Aldazabal, Baron, DM, Nunez;Geissbhuler; Grana, DM

Double Sigma Models: Hull; Berman, Copland, Thompson

Noncommutativity and nonassociativity: Lust, Blumenhagen,Deser, Plauschinn, Rennecke



Gauged Double Field TheoryBeyond restricted DFT

The strong and weak constraints

∂M∂M = 0

are sufficient to satisfy

G (∂M∂M) = F (∂M∂M) = H(∂M∂M) = 0

but in principle not necessary.

Is there some other solution?



Gauged Double Field TheoryScherk-Schwarz procedure

Start with a theory defined over (X,Y) coordinates:

1 Choose reduction ansatz: give explicit dependence ψ(X,Y)

2 Verify that the Y dependence factorizes out of the gaugetransformations

3 Plug the ansatz in the action and integrate the Y dependence

Effective theory defined over X coordinates.



Gauged Double Field TheoryReduction ansatz

We choose the ansatz (not the most general)

H(X,Y) = U(Y)T H(X) U(Y) , d(X,Y) = d(X) + λ(Y)

ξ(X,Y) = U(Y) ξ(X) , U(Y) ∈ O(D,D)

Constraints

External coordinates X remain untwisted

Dual external (internal) coordinates are external (internal)



Gauged Double Field TheoryTwisted invariance constraint

Inserting the ansatz in G ...

G = G (ξ, H, d) +1

2HAB∂D ξA∂DHEF HGF f

GBE

+1

2ξC HABHEF HGH fBFH fP[A

C f PGE ]

−3

2ξC HABHEF HGF fD[AC f

DE ]

G

−3

2ξG HDH f B

AD f AC[B fGH]C = 0

where the gaugings are defined by

fABC ≡ 3 ηD[A (U−1)MB(U−1)N

C ] ∂MUDN



Gauged Double Field TheoryTwisted closure constraints

Inserting the ansatz in F and H...

FA = FA(ξ1, ξ2, V )− 3fF [CD fF

E ]A ξC

[1 ξD2] V

E = 0

HA = HA(ξ1, ξ2, ξ3, V ) +1

2fBCE ∂D

(ξB1 ξ

C2 ξ

E3

)∂DV

A = 0

So gauge invariance and closure can be obtained if

fE [AB fC ]DE = 0

and∂E∂

E V AB = 0 , ∂E V

AB∂

EW CD = 0



Gauged Double Field TheoryNew solutions do not necessarily obey weak or strong constraints

The fluxes can be rewritten as

fABC = 3Ω[ABC ] , ΩABC ≡ ηCD(U−1)MA(U−1)N

B∂MUDN

The strong constraint for the duality twists implies

ΩEABΩECD = 0

while Jacobi identities read

fE [AB fE

C ]D = ΩE [ABΩEC ]D = 0

Therefore, a subset of the new solutions is not annihilated by thestrong constraint (similar argument for weak).



Gauged Double Field TheoryThe effective action

DFT evaluated on these new configurations is effectively describedby a lower-dimensional Gauged DFT...

DFT Gauged DFT

φ = Uφ φ , fABC (U)

δU ξ

δξ(fABC )

[ , ]C [ , ]f

SDFT SGDFT (fABC )



Gauged Double Field TheoryTwisted gauge transformations

Effective gauge transformations can be defined via

δξVA

B ≡ (U−1)AC UD

B δξV C

D

and give

δξV A

B = δξV A

B − f ACD ξ

C VDB + f D

CB ξC V A

D

Remember:

fABC ≡ 3 ηD[A (U−1)MB(U−1)N

C ] ∂MUDN

Must be Y-independent.



Gauged Double Field TheoryTwisted bracket

An effective bracket can be defined via

[ξ1, ξ2]AC ≡ (U−1)AB [ξ1, ξ2]Bf

taking the form

[ξ1, ξ2]Af = [ξ1, ξ2]AC − f ABC ξB

1 ξC2



Gauged Double Field TheoryThe effective action

Plugging the reduction ansatz in the action of Double Field Theorygives

SGDFT = v

∫dX e−2d

(R(H, d) +Rf (H)

)

Rf = −1

2f A

CDHCF HDE∂F HAE −1

12f A

CD fF

EBHAF HCE HDB

−1

4f A

CD fC

AF HDF



Gauged Double Field TheoryConsistency



Relation to gauged supergravitiesThe setup



Relation to gauged supergravitiesEffective fields



Relation to gauged supergravitiesGauge transformations

ξ =(εµ, εµ, λ

A)

= (diffeos, B transfs, gauge transfs)

δξgµν = Lεgµν

δξBµν = LεBµν + (∂µεν − ∂ν εµ)

δξAA

µ = LεAAµ − ∂µλA + f A

BC λBACµ

δξHAB = fAC

D λCHDB + fBCD λCHAD



Relation to gauged supergravitiesEffective action

Seff = v

∫dnx√ge−2φ

R + 4 ∂µΦ∂µΦ− 1

4HABFAµνFB

µν

− 1

12GµνρGµνρ +

1

8DµHABD

µHAB − V

with scalar potential

V =1

4f C

DA f DCBHAB +

1

12f E

AC f FBDHAB HCD HEF



Relation to gauged supergravitiesField Strengths and covariant derivatives

Field strengths

FAµν = ∂µA

Aν − ∂νAA

µ + f ABCA

Bµ AC

ν

Gµρλ = 3∂[µBρλ] + fABCAAµA

BρA

Cλ + 3∂[µA

AρAλ]A

Covariant derivative

DµHAB = ∂µHAB + f CADA

DµHCB + f C

BDADµHAC

The global symmetry group is O(d , d) if gaugings transform asspurions.



Backgrounds and fluxesO(d , d) democracy

The duality twist can be thought of as an internal generalized2d-bein [Dall’agata, Prezas, Samtleben, Trigiante]

H(Y) = UT (Y)U(Y) =

(g−1 −g−1bbg−1 g − bg−1b

)

O(d , d) democracy: the gaugings are defined in a completelycovariant way, so geometric and non-geometric fluxes aretreated on an equal footing

fABC (U(Y))



Relation to gauged N = 4 supergravityDifferences

Global symmetry SL(2)× O(6, 6 + N)

SL(2) mixes electric and magnetic sector... Restrict to electricsector.

Take N = 0.

All possible deformations are parameterized by gaugings

fABC , fA

Take fA = 0



Relation to gauged N = 4 supergravityComparison

When N = 0 and fA = 0 (almost) perfect agreement withidentifications

GDFT Gauged N = 4 sugra

fABC fABC

HAB MAB

e−2φ 2Im(τ)AAµ AA

µ

Gµνρ 2e4φεσµνρ∂σRe(τ)

f[ABE fC ]DE = 0 f[AB

E fC ]DE = 0



Relation to gauged N = 4 supergravityRemarks

N 6= 0: Complete DFT with additional gauge vectorsH ∈ O(10, 10 + N) [Maharana, Schwarz; Hohm, Kwak]

The inclusion of gaugings fA requires a more general ansatzinvolving warp factors for some fields[Derendinger, Petropoulos, Prezas; Gueissbuhler].

Missing term in the scalar potential

fABC fABC

Restricts to trunctations of maximal supergravity [Dibitteto,Guarino, Roest].



Relation to gauged N = 4 supergravityThe missing term

The missing part of the action, proportional to fABC fABC can be

obtained from an extra term in DFT [Geissbuhler]

4SDFT = −1

6

∫dxdxe−2dFabcFabc

Fabc ≡ 3Sd [a(E−1)Mb(E−1)N

c]∂MEdN

where EaM is a generalized vielbein for HMN . After compactifying

4SGDFT = −1

6v

∫dXe−2d

(fABC f

ABC + FabcFabc)

Recover the full electric sector of N = 4 supergravity.



Relation to Heterotic stringThe setup

Global symmetry group O(10, 10 + 496) with newparametrization incorporating vectors

H(g , b,A)

Ungauged DFT defined on a 10 + 10 + 496-dimensional space

XM = (x i , xi , yα)

Metric of the global symmetry group

η =

0 1 01 0 00 0 1

So yα are their own duals. Strong constraint would remove alltheir dependence.



Relation to Heterotic stringThe effective action

Twisting the coordinates yα leads to gaugings fαβγ and theheterotic supergravity action is reproduced

SHet = v

∫d10x√ge−2φ

[R + 4∂iφ∂

iφ− 1

12HijkH

ijk − 1

4δαβF

αij F

ijβ

]Hijk = 3

(∂[ibjk] − δαβA[i

α∂jAk]β)

+ δασfσβγA[i

αAjβAk]

γ

Fαij = 2∂[iAj]α + f αβγA[i

βAj]γ

together with the gauge transformations.



Summary

Double field theory promotes a string duality to a symmetry.Gives an action for objects in generalized geometry defined ona doubled space. Scherk-Schwarz flux compactifications ofDFT lead to lower-dimensional gauged DFTs, which includeheterotic and gauged supergravities. They feature gaugingsassociated to non-geometric backgrounds in string theory,which from the perspective of the doubled geometry arehowever geometric.



Summary

The weak and strong constraints are sufficient to solve thegauge consistency conditions of DFT, but not necesarry.

The consistency conditions formally admit solutions with atruly doubled internal space.

For these configurations, DFT is effectively described by alower dimensional GDFT, which include gauged and heteroticsupergravities.



Open questions

What orbits of gaugings can be turned on?

U ∈ O(D,D)Constant fABC

Jacobi identities

Are there genuinely non-geometric orbits?

Worldsheet (weak-like) constraints?

Supersymmetry? RR fields? Branes? Exceptional extensions?Non-commutativity? Non-associativity? Otherbackgrounds?...

Relation between DFT and string theory beyond tori?



gauged double field theoryactive.pd.infn.it/g4/seminars/2012/files/marques.pdf · gauged double...

Documents