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Gauged SUGRAs in light of double field theory Jose Juan Fern andez-Melgarejo KIAS February 24, 2014 Outline 1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions Outline 1 Introduction 2 Gauged supergravities 3 Double

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Gauged SUGRAs in light

  • f double field theory

Jose Juan Fern´ andez-Melgarejo KIAS February 24, 2014

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Outline

1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

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Outline

1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

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Supergravity essentials

Quantum field theory Field content Action / eom’s

Supersymmetry transformations Global symmetry group

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Examples: N = 2A D = 10 supergravity

Field content

  • gµν , φ , Bµν , C (3)

µνρ , C (1) µ

, ψ±

µ , χ±

. Lagrangian of the bosonic sector e−1L2A = R − 1

2(∂φ)2

− 1

2e−φ|H|2 − 1 2

  • d=1,3

e(4−d)φ/2|G (d+1)|2 − 1

2 ⋆

  • dC (3) ∧ dC (3) ∧ B
  • Global symmetry group

R+ × R+

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Examples: N = 2B D = 10 supergravity

Field content

  • gµν , Bµν , φ , C (0) , C (2) , C (4)

µνρσSD , ψI µα , λI α

  • Lagrangian of the bosonic sector

e−1L2B = R − 1

2(∂φ)2

− 1

2e−φ|H|2 − 1 2

  • d=0,2,4

|G (d+1)|2 − 1

2 ⋆

  • C (4) ∧ dC (2) ∧ B
  • Global symmetry group

SL(2, R) × R+ Self-duality relation G (5) = ⋆G (5) . (1)

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Examples: D = 11 supergravity

Field content {eµa , Cµνρ , ψµ} Lagrangian S = 1 2κ2

  • d11xe[eaµebνRµνab − ¯

ψµγµνρDνψρ − 1 24F µνρσFµνρσ − √ 2 192 ¯ ψν

  • γαβγδνρ + 12γαβgγνgδρ

ψρ(Fαβγδ + ˜ Falphaβγδ) − 2 √ 2 (144)2 e−1ǫα′β′γ′δ′αβγδµνρFα′β′γ′δ′FαβγδCµνρ] Global symmetry group: R+ (trombone symmetry) N = 8 D = 4 → perturbatively UV finite up to 3 loops

Bern, Carrasco, Dixon, Johansson, Kosower, Roiban’07

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Massive supergravities

Massive SUGRAs

Deform a theory such that SUSY is preserved L = L0 + Ldef(m)

Gauged SUGRA: deformed theory in which vector fields gauge a Yang-Mills subgroup of the global symmetry group Relation between

RR (p + 1)-form potentials in d = 10 N = 2A/2B D-branes in string theory Polchinski’95

How to find higher-rank fields? (extended field content)

Democratic formulations Bergshoeff et al.’01 E11 + U-duality arguments Bergshoeff,Riccioni’10’11’12 Gauged supergravities

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Massive supergravity

There is not a general procedure Dimensional reduction

  • n tori T n: Abelian gaugings
  • n twisted tori: non-Abelian gaugings

Scherk-Schwarz dimensional reductions: non-Abelian gaugings

Embedding tensor formalism: systematic procedure to scan all the possible gaugings Tensor hierarchy + embedding tensor: including all possible deformations

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Outline

1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

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Scherk-Schwarz dimensional reduction

Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action {gij, bij, φ} to d = (D − n) dimensions S =

  • dDx
  • |g|e−2φ
  • R − 4(∂φ)2 +

1 2 · 3!HµνρHµνρ

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Scherk-Schwarz dimensional reduction

Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action {gij, bij, φ} to d = (D − n) dimensions

gij =

  • gµν + gpqApµAqν

Apµgpn gmpApν gmn

  • bij =
  • bµν − 1

2(ApµVpν − ApνVpµ) + ApµAqνbpq

Vnµ − bnpApµ −Vmν + bmpApν bmn

  • gµν =

gµν(x) , bµν = bµν(x) Am

µ = ua m(y)

Aa

µ(x) ,

Vmµ = ua

m(y)

Vaµ(x) , gmn = ua

m(y)ub n(y)

gab(x) , bmn = ua

m(y)ub n(y)

bab(x) + vmn(y) .

  • AA

µ = (

Vaµ, Aa

µ)

  • MAB =
  • g ab

− g ac bcb

  • bac

g cb

  • gab −

bac g cd bdb

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Scherk-Schwarz dimensional reduction

Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action {gij, bij, φ} to d = (D − n) dimensions

S =

  • ddx
  • ge−2

φ

  • R + 4∂µ

φ∂µ φ − 1 4

  • MABFAµνFB

µν

− 1 12GµνρGµνρ + 1 8Dµ MABDµ MAB + V

  • FA

µν = ∂µ

AA

ν − ∂ν

AA

µ − fBC A

AB

µ

AC

ν

Gµρλ = 3∂[µ bρλ] − fABC AA

µ

AB

ρ

AC

λ + 3∂[µ

AA

ρ

Aλ]A Dµ MAB = ∂µ MAB − fAD

C

AD

µ

MCB − fBD

C

AD

µ

MAC V = −1 4fDA

C fCB D

MAB − 1 12fAC

E fBD F

MAB MCD MEF − 1 6fABCf ABC

fabc = 3(∂[avbc] + f[ab

dvc]d)

fabc = 0 fabc = uam ∂mubn ucn − ubm ∂muan ucn f abc = 0

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Embedding tensor formalism ϑI A

Systematic study of the most general gaugings

Cordaro et al.’98 / Nicolai,Samtleben’01 / deWit et al.’02 / deWit et al.’03

Promote subgroup G0 ⊂ G to be local (nv × dim G) matrix ϑI A ⇒ XI⋄⋄ = ϑI A(tA)⋄⋄ Constraints

quadratic (gauge) ϑK AXIJ K − ϑJ BXIB A = 0 linear (SUSY) g ⊗ V = θ1 ⊕ θ2 ⊕ . . . ⊕ θn

St¨ uckelberg couplings ⇒ tensor hierarchy Deformation consequences      field strengths scalar potential fermionic SUSY transf . . .

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Embedding tensor formalism ϑI A

Systematic study of the most general gaugings

Cordaro et al.’98 / Nicolai,Samtleben’01 / deWit et al.’02 / deWit et al.’03

Promote subgroup G0 ⊂ G to be local (nv × dim G) matrix ϑI A ⇒ XI⋄⋄ = ϑI A(tA)⋄⋄ Constraints

quadratic (gauge) ϑK AϑI B(tB)J K − ϑJ BϑI CfCB A = 0 linear (SUSY) g ⊗ V = θ1 ⊕ θ2 ⊕ . . . ⊕ θn

St¨ uckelberg couplings ⇒ tensor hierarchy Deformation consequences      field strengths scalar potential fermionic SUSY transf . . .

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N = 2 d = 9 supergravity

One undeformed maximal theory Gates’86 Global symmetry: SL(2, R) × (R+)2 Field content

Vielbein: eµa Scalar fields: ϕ, τ ≡ χ + ie−φ p-form potentials: Aµ0, Aµ1, Aµ2, Bµν1, Bµν2, Cµνρ fermions: ψµ, ˜ λ, λ

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Magnetic duals

p-forms A(1)I → F (2)I ⇒ ˜ F (7)

I

∼ ⋆F I ⇒ ˜ A(6)

I

B(2)i → H(3)i ⇒ ˜ H(6)

i

∼ ⋆Hi ⇒ ˜ B(5)

i

C (3) → G (4) ⇒ ˜ G (5) ∼ ⋆G ⇒ ˜ C (4) eom’s magnetic electric

  • fields ≡ Bianchi’s
  • electric

magnetic

  • fields

7-, 8- and 9-forms? ˜ AA

(d−2)

⇒ JA ≡ d ˜ AA

(d−2) = G AB ⋆ jB

Noether currents ˜ A♯

(d−1)

⇒

  • ♯ dm♯ ∧ ˜

A♯

(d−1)

Deformation parameters ˜ A♭

(d)

⇒

  • ♭ Q♭˜

A♭

(d)

Quadratic constraints

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

1 Deform supersymmetric transformations

  • f fermionic fields

δǫψµ = δ0ψµ + f γµǫ + kγµǫ∗ δǫ˜ λ = δ0˜ λ + ˜ gǫ + ˜ hǫ∗ δǫλ = δ0λ + gǫ + hǫ∗ Dµǫ ≡

  • ∇µ + i

2

  • 1

2eφD5 µχ + AI µϑI mPm

  • + 9

14γµ AIϑI 4

ǫ

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

2 Covariant derivatives and gauge

transformations of the 0-forms Dϕ = dϕ + AIϑI AkAϕ Dτ = dτ + AIϑI AkAτ δΛAI = −DΛI + Z I iΛi

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

3 Closure of the algebra of the 0-forms

[δǫ1, δǫ2] ϕ = ξµDµϕ + ℜe(˜ h)b − ℑm(˜ g)c + ℜe(˜ g)d [δǫ1, δǫ2] τ = ξµDµτ + e−φ [g(c − id) − ihb] [δǫ1, δǫ2] ϕ = Lξϕ + ΛIϑI

AkA ϕ

[δǫ1, δǫ2] τ = Lξτ + ΛIϑI

AkA τ

ℜe(˜ h)b − ℑm(˜ g)c + ℜe(˜ g)d = (ΛI − aI)ϑI

AkA ϕ

g(c − id) − ihb = eφ(ΛI − aI)ϑI

AkA τ

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

4 Deformed field strength of the 1-forms

F I = dAI + 1

2XJK IAJ ∧ AK + Z I iBi

δΛBi = −DΛi − 2hIJ

i

ΛIF J + 1

2AI ∧ δΛAJ

+ Z iΛ X(JK)

I + Z I ihJK i = 0

Z I

iZ i = 0

XI j

ihJK j − 2XI(J LhK)L i = 0

XIZ i − XI j

iZ j = 0

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

5 Closure of the algebra of the 1-forms

[δǫ1, δǫ2] AI µ = LξAI µ − DµΛI + Z I iΛi µ [δǫ1, δǫ2] AI µ = f (ǫ1, ǫ∗

1, ǫ2, ǫ∗ 2)

ΛI = ΛI(ǫ1, ǫ∗

1, ǫ2, ǫ∗ 2)

˜ h, ˜ g, h, g = ˜ h(ϑ), ˜ g(ϑ), h(ϑ), g(ϑ) (ϑ1Aτ + ϑ2A)kAτ = 0

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

6 Deformed field strength of the 2-forms

Hi = DBi − hIJ

iAI ∧ dAJ

− 1

3X[IJ LhK]L iAIJK + Z iC

δΛC = −DΛ + gIi

  • −ΛIHi − F I ∧ Λi + δΛAI ∧ Bi

− 1

3hJK iAIJ ∧ δΛAK

+ Z ˜ Λ 2hIJ

iZ J j + XI j i + Z igIj = 0

Z iZ = 0 XIJ

LgLi + XI i jgJj − XIgJi = 0

(XI − ˜ XI)Z = 0

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Deformation recipe

Deformation ingredients 1 embedding tensor ϑI A (3 × 5) St¨ uckelberg couplings Z •⋄ Fermion shifts f , k, ˜ g, ˜ h, g, h Other ingredients 1 undeformed theory Gauge generators XI⋄• = ϑI AtA⋄• Covariant derivatives Dµ = ∂µ + XIAI µ Gauge parameters Λ(p)

7 Closure of the algebra of the 2-forms

[δǫ1, δǫ2] Bi µν = LξBi µν + δΛBi µν [δǫ1, δǫ2] Bi µν = f (ǫ1, ǫ∗

1, ǫ2, ǫ∗ 2)

Λi µ = Λi µ(ǫ1, ǫ∗

1, ǫ2, ǫ∗ 2)

f , k = f (ϑ), k(ϑ) ϑ21 = 0 ϑ12 = 3

4ϑ25

ϑ13 = 3

4ϑ25

ϑ11 = 3

2ϑ15

ϑ22 = 3

4ϑ15

ϑ23 = − 3

4ϑ15

ϑ04 = − 1

6ϑ05

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Gauged quantities

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Gauged quantities: field strengths

Ungauged F I = dAI Hi = dBi + 1

2δi i(A0 ∧ F i + Ai ∧ F 0)

G = d[C − 1

6εijA0ij] − εijF i ∧

  • Bj + 1

2δj jA0j

Gauged F I = dAI + 1

2XJK IAJ ∧ AK + Z I iBi

Hi = DBi + 1

2δi i(A0 ∧ F i + Ai ∧ F 0) + (XA012) + Z iC

G = D[C − 1

6εijA0ij] − εijF i ∧

  • Bj + 1

2δj jA0j

+ ZijBij + Z ˜ C ˜ G = D˜ C + (OLD) + Z 0jBj ∧ C + (XJijAJ ∧ Bij) + Z i ˜ Hi

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Gauged quantities: gauge transformations

Ungauged δΛAI = −dΛI δΛBi = −dΛi + δi i

  • ΛiF 0 + Λ0F i + 1

2

  • A0δΛAi + AiδΛA0

δΛ[C − 1

6εijA0ij] = −dΛ − εij

  • F iΛj + ΛiHj − δΛAiBj + 1

2δj jA0iδΛAj

Gauged δΛAI = −DΛI + Z I iΛi δΛBi = −DΛi + ΛiF 0 + Λ0F i + 1

2

  • A0δΛAi + AiδΛA0

+ Z iΛ δΛ[C − 1

6εijA0ij] = −DΛ − εij

  • F iΛj + ΛiHj − δΛAi ∧ Bj + 1

2A0iδΛAj

+ Z ˜ Λ δΛ ˜ C = −D˜ Λ + (OLD) + Z i ˜ Λi

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Gauged quantities: Bianchi identities

Ungauged Gauged dF I = 3 DF I = Z I iHi dHi + F 0F i = 4 DHi + F 0F i = Z iG dG − F iHi = 5 DG − F iHi = Z ˜ G d ˜ G + F 0G + 1

2ǫijHiHj

= 6 D˜ G + F 0G + 1

2HiHi

= Z iHi d ˜ Hi + Fi ˜ G − HiG = 7 D˜ Hi + Fi ˜ G − HiG = Zi I ˜ FI d ˜ F0 + F j ˜ Hj − 1

2GG

= 8 D˜ F0 + F i ˜ Hi − 1

2GG

= ϑ0AJA d ˜ Fi + F 0 ˜ Hi − Hi ˜ G = D˜ Fi + F 0 ˜ Hi − Hi ˜ G = ϑi AJA . . . . . .

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Constraints summary

1 Quadratic constraints

Gauge invariance of the deformation parameters ϑX + Xϑ = 0 ZX + XZ = 0 Orthogonality constraints (tensor hierarchy) ϑZ = 0 ZZ = 0

2 Linear constraints

Leibniz rule condition (group representation consistency) X + Z + Z = 0 Closure of the algebra (SUSY)

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Reduction of parameters

30 free parameters   15 from ϑI A 9 from Zi I, Z i, Z 6 from f , k, ˜ g, ˜ h, g, h 8 independent deformations

Z = Z(ϑ) f (ϑ), k(ϑ), ˜ g(ϑ), ˜ h(ϑ), g(ϑ), h(ϑ)

ϑ0m = mm (m = 1, 2, 3) ϑ14 = −m11 ϑ15 = ˜ m4 ϑ05 = − 16

3 mIIB

ϑ24 = mIIA ϑ25 = m4 13 constraints ϑ0m 12ϑi4 + 5ϑi5 ≡ Qmi = 6 ϑi4ϑ05 ≡ Q4i = 2 ϑi5ϑ05 ≡ Q5i = 2 ϑj4 (ϑm

0 Tm)i j

≡ Qi = 2 εijϑi4ϑj5 ≡ Q = 1

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Results

1 Deformed theory

Covariant derivatives Non-Abelian field strengths Gauge transformations Fermionic SUSY transformations Fermion mass terms Scalar potential Bianchi identities

2 N = 2 d = 9

Tensor hierarchy All deformations and their possible combinations Field content

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Outline

1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

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Higher-dimensional origin of gaugings

Embedding tensor maximal lower- dimensional SUGRAs Gauged SUGRAs New gauged SUGRAs D = 11, N = 2A/2B SUGRAs T-duality constructions T n SS and others

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T-duality motivations

SS dim. reductions do not reproduce some ET configurations Habc

Tc

← → ωabc

Tb

← → Qabc

Ta

← → Rabc

Shelton,Taylor,Wecht’05 / Dabholkar,Hull’05

Constructions

Doubled geometry

Hull’04’06 / Dall’Agata et al.’07 / Hull,Reid-Edwards’09

Generalized geometry

Gra˜ na et al.’08 / Berman,Perry’10 / Berman,Godazgar,Perry’11’11 / Coimbra et al.’11 / Hull,Reid-Edwards’09

Double field theory

Aldaz´ abal, Berman, Geissb¨ uhler, Gra˜ na, Hohm, Hull, Jeon, Kwak, Lee, Marqu´ es, Park, Siegel, Suh, Zwiebach, . . .

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Double field theory

T-duality invariant formulation of bosonic NSNS sector Dimensional reduction to 2d = 2D + 2n O(D, D) ⊃ O(n, n) × O(1, d − 1) × O(d − 1, 1) X = (x, ˜ x, y, ˜ y) y, ˜ y x ˜ x Field content gµν Bµν

  • →

H = g−1 −g−1B Bg−1 g − Bg−1B

  • ∈ O(D, D)

e−2φ → e−d =

  • |g|e−2φ

The action S =

  • d2Dx e−2dR(H, d)

R(H, d) = 1 8HMN∂MHPQ∂NHPQ − 1 2HMN∂NHPQ∂QHMP − 2∂Md∂NHMN + 4HMN∂Md∂Nd

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Double field theory

T-duality invariant formulation of bosonic NSNS sector Dimensional reduction to 2d = 2D + 2n O(D, D) ⊃ O(n, n) × O(1, d − 1) × O(d − 1, 1) X = (x, ˜ x, y, ˜ y) y, ˜ y x ˜ x Field content gµν Bµν

  • →

H = g−1 −g−1B Bg−1 g − Bg−1B

  • ∈ O(D, D)

e−2φ → e−d =

  • |g|e−2φ

The action S =

  • d2Dx e−2dR(H, d)

(˜ x truncation) S =

  • dDx
  • |g|e−2φ
  • R − 4(∂φ)2 +

1 2 · 3!HµνρHµνρ

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SLIDE 37

Higher-dimensional origin of gaugings

Embedding tensor maximal lower- dimensional SUGRAs Gauged SUGRAs New gauged SUGRAs D = 11, N = 2A/2B SUGRAs T-duality constructions T n SS and others

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SLIDE 38

Higher-dimensional origin of gaugings

Embedding tensor maximal lower- dimensional SUGRAs Gauged SUGRAs New gauged SUGRAs D = 11, N = 2A/2B SUGRAs T-duality constructions T n SS and others

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SLIDE 39

Maximal d = 9

R+ × SL(2) Embedding tensor

  • θi , κjk

2(+3) ⊕ 4(−1) Quadratic constraints ǫij θi κjk = 0 , 2(−1) θ(i κjk) = 0 . 4(−1) ID θi κij gauging 1 (0, 0) diag(1, 1) SO(2) 2 diag(1, −1) SO(1, 1) 3 diag(1, 0) R+

γ

4 (1, 0) diag(0, 0) R+

β

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SLIDE 40

Maximal d = 9

R+ × SL(2) Embedding tensor

  • θi , κjk

2(+3) ⊕ 4(−1) Quadratic constraints ǫij θi κjk = 0 , 2(−1) θ(i κjk) = 0 . 4(−1) ID θi κij gauging 1 (0, 0) diag(1, 1) SO(2) 2 diag(1, −1) SO(1, 1) 3 diag(1, 0) R+

γ

4 (1, 0) diag(0, 0) R+

β

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SLIDE 41

Maximal d = 8

SL(2) × SL(3) Embedding tensor

  • fα(mn) , ξαm
  • (2, 6′) ⊕ (2, 3)

Quadratic constraints ǫαβ ξαpξβq = 0 ,

  • 1, 3′

f(α

npξβ)p = 0 ,

  • 3, 3′

ǫαβ (ǫmqrfαqnfβrp + fαnpξβm) = 0 .

  • 1, 3′

⊕ (1, 15)

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SLIDE 42

Maximal d = 8

ID f+

mn

f−

mn

ξ+m ξ−m gauging 1 diag(1, 1, 1) diag(0, 0, 0) (0, 0, 0) (0, 0, 0) SO(3) 2 diag(1, 1, −1) SO(2, 1) 3 diag(1, 1, 0) ISO(2) 4 diag(1, −1, 0) ISO(1, 1) 5 diag(1, 0, 0) CSO(1, 0, 2) 6 diag(0, 0, 0) diag(0, 0, 0) (1, 0, 0) (0, 0, 0) Solv2 × Solv3 7 diag(1, 1, 0) diag(0, 0, 0) (0, 0, 1) (0, 0, 0) Solv2 × Solv3 8 diag(1, −1, 0) 9 diag(1, 0, 0) 10 diag(1, −1, 0)

  1 1 1 1   2 9(0, 0, 1)

(0, 0, 0) Solv2 × SO(2) ⋉ Nil3(2)

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SLIDE 43

Maximal d = 8

ID f+

mn

f−

mn

ξ+m ξ−m gauging 1 diag(1, 1, 1) diag(0, 0, 0) (0, 0, 0) (0, 0, 0) SO(3) 2 diag(1, 1, −1) SO(2, 1) 3 diag(1, 1, 0) ISO(2) 4 diag(1, −1, 0) ISO(1, 1) 5 diag(1, 0, 0) CSO(1, 0, 2) 6 diag(0, 0, 0) diag(0, 0, 0) (1, 0, 0) (0, 0, 0) Solv2 × Solv3 7 diag(1, 1, 0) diag(0, 0, 0) (0, 0, 1) (0, 0, 0) Solv2 × Solv3 8 diag(1, −1, 0) 9 diag(1, 0, 0) 10 diag(1, −1, 0)

  1 1 1 1   2 9(0, 0, 1)

(0, 0, 0) Solv2 × SO(2) ⋉ Nil3(2)

slide-44
SLIDE 44

Maximal d = 7

SL(5) Embedding tensor

  • Y(MN)

, Z MN,P

  • 15

⊕ 40′ Quadratic constraints YMQ Z QN,P + 2 ǫMRSTU Z RS,N Z TU,P = 0

QC restricted by SUSY: 5′ ⊕ 45′ ⊕ 70′

slide-45
SLIDE 45

Half-maximal d = 10 and d = 9

d = 10

No vector multiplets No freedom to deform

d = 9

R+ × SO(1, 1) Abelian gauging Fully geometric

slide-46
SLIDE 46

Half-maximal d = 8

R+ × SL(2) × SL(2)

  • O(2,2)

3 − → 2 ⊕ 1 m − → (i, •)

Embedding tensor

fαi• = ǫijaαj ξαi = 4bαi

Quadratic constraints

ǫαβ ǫij (aαi aβj − bαi bβj) = 0 ǫαβ ǫij (aαi bβj + bαi bβj) = 0 ǫij a(αi bβ)j = 0 ǫαβ aα(i bβj) = 0

ID aαi bαi gauging 1 diag( cos α, 0) diag( sin α, 0) Solv2 × SO(1, 1) 2 diag(1, 1) diag(−1, −1) SL(2) × SO(1, 1) 3 diag(1, −1) diag(−1, 1)

slide-47
SLIDE 47

Half-maximal d = 8

R+ × SL(2) × SL(2)

  • O(2,2)

3 − → 2 ⊕ 1 m − → (i, •)

Embedding tensor

fαi• = ǫijaαj ξαi = 4bαi

Quadratic constraints

ǫαβ ǫij (aαi aβj − bαi bβj) = 0 ǫαβ ǫij (aαi bβj + bαi bβj) = 0 ǫij a(αi bβ)j = 0 ǫαβ aα(i bβj) = 0

ID aαi bαi gauging 1 diag( cos α, 0) diag( sin α, 0) Solv2 × SO(1, 1) 2 diag(1, 1) diag(−1, −1) SL(2) × SO(1, 1) 3 diag(1, −1) diag(−1, 1)

slide-48
SLIDE 48

Half-maximal d = 8

R+ × SL(2) × SL(2)

  • O(2,2)

3 − → 2 ⊕ 1 m − → (i, •)

Embedding tensor

fαi• = ǫijaαj ξαi = 4bαi

Quadratic constraints

ǫαβ ǫij (aαi aβj − bαi bβj) = 0 ǫαβ ǫij (aαi bβj + bαi bβj) = 0 ǫij a(αi bβ)j = 0 ǫαβ aα(i bβj) = 0

ID aαi bαi gauging 1 diag( cos α, 0) diag( sin α, 0) Solv2 × SO(1, 1) 2 diag(1, 1) diag(−1, −1) SL(2) × SO(1, 1) 3 diag(1, −1) diag(−1, 1)

slide-49
SLIDE 49

Half-maximal d = 8

QC fABCf ABC = 0 ⇒ non upliftable! Twist matrix U =     1 cosh(m y1 + n ˜ y1) sinh(m y1 + n ˜ y1) 1 sinh(m y1 + n ˜ y1) cosh(m y1 + n ˜ y1)     Gaugings aαi = −bαi = diag

  • −m + n

2 √ 2 , m − n 2 √ 2

  • Orbit 2: m = 0, n = −2

√ 2 Orbit 3: m = −2 √ 2, n = 0

slide-50
SLIDE 50

Half-maximal d = 7

R+ × SL(4)

SO(3,3)

5 − → 1(+4) ⊕ 4(−1) M − → (⋄, m) Embedding tensor Z MN,P Y(MN) − → θ , ξ(mn), M(mn), ˜ M(mn) 1 ⊕ 6 ⊕ 10 ⊕ 10′ Quadratic constraints θξmn = 0 (6)

  • ˜

Mmp + ξmp Mpq − 1 4

  • ˜

MnpMnp

  • δm

q = 0

(15) Mmpξpn + ξmp ˜ Mpn = 0 (15) ǫmnpqξmnξpq = 0 (1)

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SLIDE 51

Half-maximal d = 7

Truncation

✁ ❆

θ , ✟✟

✟ ❍❍ ❍

ξ[mn] , M(mn) , ˜ M(mn) 10 ⊕ 10′

  • fabc, fabc, fabc, f abc

≡

  • Habc, ωabc, Qabc, Rabc

10 ⊕ 10′

  • f SL(4)

← → 10SD ⊕ 10ASD

  • f SO(3, 3)

Dictionary M = diag

  • H123, Q123, Q231, Q312

˜ M = diag

  • R123, ω231, ω312, ω123

Surviving QC ˜ Mmp Mpn − 1 4

  • ˜

Mpq Mpq

  • δm

n = 0

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SLIDE 52

Half-maximal d = 7

ID Mmn/ cos α ˜ Mmn/ sin α range of α gauging 1 diag(1, 1, 1, 1) diag(1, 1, 1, 1) − π

4 < α ≤ π 4

SO(4) , α =

π 4

SO(3) , α =

π 4

2 diag(1, 1, 1, −1) diag(1, 1, 1, −1) − π

4 < α ≤ π 4

SO(3, 1) 3 diag(1, 1, −1, −1) diag(1, 1, −1, −1) − π

4 < α ≤ π 4

SO(2,2) , α =

π 4

SO(2, 1) , α =

π 4

4 diag(1, 1, 1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(3) 5 diag(1, 1, −1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(2, 1) 6 diag(1, 1, 0, 0) diag(0, 0, 1, 1) − π

4 < α ≤ π 4

CSO(2, 0, 2) , α =

π 4

f1 (Solv6) , α =

π 4

7 diag(1, 1, 0, 0) diag(0, 0, 1, −1) − π

2 < α < π 2

   CSO(2, 0, 2) , |α| <

π 4

CSO(1, 1, 2) , |α| >

π 4

g0 (Solv6) , |α| =

π 4

8 diag(1, 1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h1 (Solv6) 9 diag(1, −1, 0, 0) diag(0, 0, 1, −1) − π

4 < α ≤ π 4

CSO(1, 1, 2) , α =

π 4

f2 (Solv6) , α =

π 4

10 diag(1, −1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h2 (Solv6) 11 diag(1, 0, 0, 0) diag(0, 0, 0, 1) − π

4 < α ≤ π 4

l (Nil6(3) ) , α = 0 CSO(1, 0, 3) , α = 0

slide-53
SLIDE 53

Half-maximal d = 7

ID Mmn/ cos α ˜ Mmn/ sin α range of α gauging 1 diag(1, 1, 1, 1) diag(1, 1, 1, 1) − π

4 < α ≤ π 4

SO(4) , α =

π 4

SO(3) , α =

π 4

2 diag(1, 1, 1, −1) diag(1, 1, 1, −1) − π

4 < α ≤ π 4

SO(3, 1) 3 diag(1, 1, −1, −1) diag(1, 1, −1, −1) − π

4 < α ≤ π 4

SO(2,2) , α =

π 4

SO(2, 1) , α =

π 4

4 diag(1, 1, 1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(3) 5 diag(1, 1, −1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(2, 1) 6 diag(1, 1, 0, 0) diag(0, 0, 1, 1) − π

4 < α ≤ π 4

CSO(2, 0, 2) , α =

π 4

f1 (Solv6) , α =

π 4

7 diag(1, 1, 0, 0) diag(0, 0, 1, −1) − π

2 < α < π 2

   CSO(2, 0, 2) , |α| <

π 4

CSO(1, 1, 2) , |α| >

π 4

g0 (Solv6) , |α| =

π 4

8 diag(1, 1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h1 (Solv6) 9 diag(1, −1, 0, 0) diag(0, 0, 1, −1) − π

4 < α ≤ π 4

CSO(1, 1, 2) , α =

π 4

f2 (Solv6) , α =

π 4

10 diag(1, −1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h2 (Solv6) 11 diag(1, 0, 0, 0) diag(0, 0, 0, 1) − π

4 < α ≤ π 4

l (Nil6(3) ) , α = 0 CSO(1, 0, 3) , α = 0

slide-54
SLIDE 54

Half-maximal d = 7

ID Mmn/ cos α ˜ Mmn/ sin α range of α gauging 1 diag(1, 1, 1, 1) diag(1, 1, 1, 1) − π

4 < α ≤ π 4

SO(4) , α =

π 4

SO(3) , α =

π 4

2 diag(1, 1, 1, −1) diag(1, 1, 1, −1) − π

4 < α ≤ π 4

SO(3, 1) 3 diag(1, 1, −1, −1) diag(1, 1, −1, −1) − π

4 < α ≤ π 4

SO(2,2) , α =

π 4

SO(2, 1) , α =

π 4

4 diag(1, 1, 1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(3) 5 diag(1, 1, −1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(2, 1) 6 diag(1, 1, 0, 0) diag(0, 0, 1, 1) − π

4 < α ≤ π 4

CSO(2, 0, 2) , α =

π 4

f1 (Solv6) , α =

π 4

7 diag(1, 1, 0, 0) diag(0, 0, 1, −1) − π

2 < α < π 2

   CSO(2, 0, 2) , |α| <

π 4

CSO(1, 1, 2) , |α| >

π 4

g0 (Solv6) , |α| =

π 4

8 diag(1, 1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h1 (Solv6) 9 diag(1, −1, 0, 0) diag(0, 0, 1, −1) − π

4 < α ≤ π 4

CSO(1, 1, 2) , α =

π 4

f2 (Solv6) , α =

π 4

10 diag(1, −1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h2 (Solv6) 11 diag(1, 0, 0, 0) diag(0, 0, 0, 1) − π

4 < α ≤ π 4

l (Nil6(3) ) , α = 0 CSO(1, 0, 3) , α = 0

slide-55
SLIDE 55

Half-maximal d = 7

ID Mmn/ cos α ˜ Mmn/ sin α range of α gauging 1 diag(1, 1, 1, 1) diag(1, 1, 1, 1) − π

4 < α ≤ π 4

SO(4) , α =

π 4

SO(3) , α =

π 4

2 diag(1, 1, 1, −1) diag(1, 1, 1, −1) − π

4 < α ≤ π 4

SO(3, 1) 3 diag(1, 1, −1, −1) diag(1, 1, −1, −1) − π

4 < α ≤ π 4

SO(2,2) , α =

π 4

SO(2, 1) , α =

π 4

4 diag(1, 1, 1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(3) 5 diag(1, 1, −1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(2, 1) 6 diag(1, 1, 0, 0) diag(0, 0, 1, 1) − π

4 < α ≤ π 4

CSO(2, 0, 2) , α =

π 4

f1 (Solv6) , α =

π 4

7 diag(1, 1, 0, 0) diag(0, 0, 1, −1) − π

2 < α < π 2

   CSO(2, 0, 2) , |α| <

π 4

CSO(1, 1, 2) , |α| >

π 4

g0 (Solv6) , |α| =

π 4

8 diag(1, 1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h1 (Solv6) 9 diag(1, −1, 0, 0) diag(0, 0, 1, −1) − π

4 < α ≤ π 4

CSO(1, 1, 2) , α =

π 4

f2 (Solv6) , α =

π 4

10 diag(1, −1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h2 (Solv6) 11 diag(1, 0, 0, 0) diag(0, 0, 0, 1) − π

4 < α ≤ π 4

l (Nil6(3) ) , α = 0 CSO(1, 0, 3) , α = 0

slide-56
SLIDE 56

Half-maximal d = 7

ID Mmn/ cos α ˜ Mmn/ sin α range of α gauging 1 diag(1, 1, 1, 1) diag(1, 1, 1, 1) − π

4 < α ≤ π 4

SO(4) , α =

π 4

SO(3) , α =

π 4

2 diag(1, 1, 1, −1) diag(1, 1, 1, −1) − π

4 < α ≤ π 4

SO(3, 1) 3 diag(1, 1, −1, −1) diag(1, 1, −1, −1) − π

4 < α ≤ π 4

SO(2,2) , α =

π 4

SO(2, 1) , α =

π 4

4 diag(1, 1, 1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(3) 5 diag(1, 1, −1, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

ISO(2, 1) 6 diag(1, 1, 0, 0) diag(0, 0, 1, 1) − π

4 < α ≤ π 4

CSO(2, 0, 2) , α =

π 4

f1 (Solv6) , α =

π 4

7 diag(1, 1, 0, 0) diag(0, 0, 1, −1) − π

2 < α < π 2

   CSO(2, 0, 2) , |α| <

π 4

CSO(1, 1, 2) , |α| >

π 4

g0 (Solv6) , |α| =

π 4

8 diag(1, 1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h1 (Solv6) 9 diag(1, −1, 0, 0) diag(0, 0, 1, −1) − π

4 < α ≤ π 4

CSO(1, 1, 2) , α =

π 4

f2 (Solv6) , α =

π 4

10 diag(1, −1, 0, 0) diag(0, 0, 0, 1) − π

2 < α < π 2

h2 (Solv6) 11 diag(1, 0, 0, 0) diag(0, 0, 0, 1) − π

4 < α ≤ π 4

l (Nil6(3) ) , α = 0 CSO(1, 0, 3) , α = 0

slide-57
SLIDE 57

Half-maximal d = 7

Orbits 4, 5, 8, 10, 11: geometric U(y)

Aldaz´ abal et al.’11 / Alonso-Alberca et al.’03

Orbits 1, 2, 3, 6, 7, 9: U(y, ˜ y) U =     1 A B 1 C D     , U4 = A B C D

  • = exp
  • tIJφIJ

[tIJ]K L = δL

[IηJ]K

φIJ = αIJy1 + βIJ˜ y1

slide-58
SLIDE 58

Conclusions

U-duality orbits (maximal SUGRA) d 9 8 ♯ orbits 4 10 T-duality orbits (half-maximal SUGRA) d 10 9 8 7 ♯ orbits 1 3 (2) 11∗ (3∗) Twist degeneracy in half-maximal d = 7 (different U’s for the same gauging

Orbit 6 twist 1: WC/SC twist 2: WC/SC ×

slide-59
SLIDE 59

Outline

1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

slide-60
SLIDE 60

Conclusions

Massive supergravities

Structure Embedding tensor formalism All the possible deformations All their possible combinations Extended field content ⇔ extended objects Test for M/stringy constructions

Double field theory

T-duality invariant formulation of NSNS sector of SUGRA Generalized SS reduction Maximal and half-maximal d = 9, 8, 7 SUGRA gaugings ∀ XIJ

K = ϑI A(tA)J K ∃ U(Y) | fABC(U) ∼

= XMN

P

(non-)maximal ⇔ (non-)geometric non-upliftable ⇔ non-geometric description

slide-61
SLIDE 61

Prospects

Full classification of maximal and half-maximal gauged SUGRAs and generation from DFT U-duality formulation Non-linear σ-model in terms of {HAB(g, b), d} Extension of DFT to include D = 11 SUGRA Massive N = 2A SUGRA in terms of DFT Ramond-Ramond and supersymmetric DFT SL(2) × O(6, 6) gaugings for N = 4 D = 4 SUGRA

slide-62
SLIDE 62

Thanks