Compactifications and consistent truncations in supergravity...GRaB100 Compacti cations and consistent truncations in supergravity James Liu University of Michigan 8 July 2015 1.Introduction

GRaB100

Compactifications and consistent truncations insupergravity

James Liu

University of Michigan

8 July 2015

1. Introduction to supersymmetry and supergravity

2. Compactification on manifolds without fluxes

3. AdS×Sphere compactifications and consistent truncations

JTL

Compactification with fluxes

I For compactifications without fluxes, the criteria for preservingsupersymmetry is the presence of parallel spinors ∇iε = 0

⇒ Xn is a Ricci-flat manifold with special holonomy

dim H2n SU(n) Kahler (Calabi-Yau)4n Sp(2n) Hyperkahler (K3 for n = 1)7 G2

8 Spin(7)

I However there are many interesting situations where fluxes areturned on

I Moduli stabilization with internal fluxesI Landscape of string vacuaI Freund-Rubin compactifications (AdS vacua)

JTL

Compactification with fluxes

I For compactifications without fluxes, the criteria for preservingsupersymmetry is the presence of parallel spinors ∇iε = 0

⇒ Xn is a Ricci-flat manifold with special holonomy

dim H2n SU(n) Kahler (Calabi-Yau)4n Sp(2n) Hyperkahler (K3 for n = 1)7 G2

8 Spin(7)

I However there are many interesting situations where fluxes areturned on

I Moduli stabilization with internal fluxesI Landscape of string vacuaI Freund-Rubin compactifications (AdS vacua)

JTL

Spontaneous compactification of 11D supergravity

I The bosonic Lagrangian of 11D supergravity is given by

e−1L = R ∗ 1− 12F4 ∧ ∗F4 − 1

6F4 ∧ F4 ∧ A4

with corresponding equations of motion

RMN − 12gMNR = 1

2·3! (FMPQRFNPQR − 1

6gMNF2)

dF4 = 0

d ∗ F4 = 12F4 ∧ F4

I The 4-form field strength naturally selects out four dimensions

⇒ Make a 4 + 7 split Freund and Rubin, PLB 97, 233 (1980)

ds211 = gµν(x)dxµdxν + gij(y)dy idy j

F4 = (3/L)vol4

JTL

Spontaneous compactification of 11D supergravity

I The bosonic Lagrangian of 11D supergravity is given by

e−1L = R ∗ 1− 12F4 ∧ ∗F4 − 1

6F4 ∧ F4 ∧ A4

with corresponding equations of motion

RMN − 12gMNR = 1

2·3! (FMPQRFNPQR − 1

6gMNF2)

dF4 = 0

d ∗ F4 = 12F4 ∧ F4

I The 4-form field strength naturally selects out four dimensions

⇒ Make a 4 + 7 split Freund and Rubin, PLB 97, 233 (1980)

ds211 = gµν(x)dxµdxν + gij(y)dy idy j

F4 = (3/L)vol4

JTL

Solving the equations of motion

I The 4-form satisfies dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0

⇒ the 4-form equations are automatically satisfied

I From the Einstein equation, we find

Rµν = − 3

L2gµν Rij =

3

2L2gij

I This leads to the maximally symmetric solution AdS4 × S7

with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S7 but can compactify on other

Einstein manifolds as well]

I Similarly, the AdS7 × S4 solution is obtained by taking F4 tohave internal flux

JTL

Solving the equations of motion

I The 4-form satisfies dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0

⇒ the 4-form equations are automatically satisfied

I From the Einstein equation, we find

Rµν = − 3

L2gµν Rij =

3

2L2gij

I This leads to the maximally symmetric solution AdS4 × S7

with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S7 but can compactify on other

Einstein manifolds as well]

I Similarly, the AdS7 × S4 solution is obtained by taking F4 tohave internal flux

JTL

Solving the equations of motion

I The 4-form satisfies dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0

⇒ the 4-form equations are automatically satisfied

I From the Einstein equation, we find

Rµν = − 3

L2gµν Rij =

3

2L2gij

I This leads to the maximally symmetric solution AdS4 × S7

with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S7 but can compactify on other

Einstein manifolds as well]

I Similarly, the AdS7 × S4 solution is obtained by taking F4 tohave internal flux

JTL

Preserving supersymmetry

I We may examine the Killing spinor equation (arising from thegravitino variation)

δψM = DMε ≡[∇M − 1

288 (ΓMPQRS − 8δPMΓQRS)FPQRS

]ε

Setting Fµνρσ = (3/L)εµνρσ gives

δψµ = [∇µ + 12LΓµΓ0123]ε

δψi = [∇i − 14LΓiΓ

0123]ε

I To proceed, we decompose the D = 11 Dirac matrices

Γµ = γµ ⊗ 1 Γi = γ5 ⊗ γi ⇒ ε = ε4 ⊗ η7

Then

δψµ = [∇µ − i2Lγµγ

5]ε⊗ η Killing spinors on AdS4

δψi = ε⊗ [∇i + i4Lγi ]η Killing spinors on S7

JTL

Preserving supersymmetry

I We may examine the Killing spinor equation (arising from thegravitino variation)

δψM = DMε ≡[∇M − 1

288 (ΓMPQRS − 8δPMΓQRS)FPQRS

]ε

Setting Fµνρσ = (3/L)εµνρσ gives

δψµ = [∇µ + 12LΓµΓ0123]ε

δψi = [∇i − 14LΓiΓ

0123]ε

I To proceed, we decompose the D = 11 Dirac matrices

Γµ = γµ ⊗ 1 Γi = γ5 ⊗ γi ⇒ ε = ε4 ⊗ η7

Then

δψµ = [∇µ − i2Lγµγ

5]ε⊗ η Killing spinors on AdS4

δψi = ε⊗ [∇i + i4Lγi ]η Killing spinors on S7

JTL

Killing spinors on spheres

I Consider the Killing spinor equation

[∇i + im2 γi ]η = 0 (m = const)

I Integrability gives

0 = [∇i + im2 γi ,∇j + im

2 γj ]η

=(

[∇i ,∇j ]− m2

2 γij

)η = 1

4

[Rij

kl −m2(δki δlj − δli δkj )

]γklη

Maximal supersymmetry then requires

Rijkl = m2(gikgjl − gilgjk)

which corresponds to a maximally symmetric space ofpositive curvature ⇒ S7

I A similar calculation with [∇µ − ig2 γµγ

5]η gives

Rµνρσ = −g2(gµρgνσ − gµσgνρ) ⇒ AdS4

JTL

Killing spinors on spheres

I Consider the Killing spinor equation

[∇i + im2 γi ]η = 0 (m = const)

I Integrability gives

0 = [∇i + im2 γi ,∇j + im

2 γj ]η

=(

[∇i ,∇j ]− m2

2 γij

)η = 1

4

[Rij

kl −m2(δki δlj − δli δkj )

]γklη

Maximal supersymmetry then requires

Rijkl = m2(gikgjl − gilgjk)

which corresponds to a maximally symmetric space ofpositive curvature ⇒ S7

I A similar calculation with [∇µ − ig2 γµγ

5]η gives

Rµνρσ = −g2(gµρgνσ − gµσgνρ) ⇒ AdS4

JTL

Gauged D = 4, N = 8 supergravity

I The compactification of D = 11 supergravity on S7 ismaximally supersymmetric

⇒ gives N = 8 in four dimensionsI What symmetries do we expect?

AdS4 × S7

SO(2, 3)× SO(8) + susy ⇒ OSp(4|8)

I The linearized Kaluza-Klein spectrum was obtained by

Casher, Englert, Nicolai and Rooman, NPB 243, 173 (1984);Sezgin, PLB 138, 57 (1984);Biran, Casher, Englert, Rooman and Spindel, PLB 134, 179 (1984)

Decompose the D = 11 fields in terms of sphericalharmonics on S7

I States are classified by their SO(2, 3)× SO(8) quantumnumbers

D(E0, j) (l1, l2, l3, l4)

JTL

Gauged D = 4, N = 8 supergravity

I The compactification of D = 11 supergravity on S7 ismaximally supersymmetric

⇒ gives N = 8 in four dimensionsI What symmetries do we expect?

AdS4 × S7

SO(2, 3)× SO(8) + susy ⇒ OSp(4|8)

I The linearized Kaluza-Klein spectrum was obtained by

Casher, Englert, Nicolai and Rooman, NPB 243, 173 (1984);Sezgin, PLB 138, 57 (1984);Biran, Casher, Englert, Rooman and Spindel, PLB 134, 179 (1984)

Decompose the D = 11 fields in terms of sphericalharmonics on S7

I States are classified by their SO(2, 3)× SO(8) quantumnumbers

D(E0, j) (l1, l2, l3, l4)

JTL

The Kaluza-Klein spectrum on S7

E0 j SO(8) rep KK level n12 (n + 6) 2 (n, 0, 0, 0)12 (n + 5) 3

2 (n, 0, 0, 1)12 (n + 7) 3

2 (n − 1, 0, 1, 0) n ≥ 112 (n + 4) 1− (n, 1, 0, 0)12 (n + 6) 1+ (n − 1, 0, 1, 1) n ≥ 112 (n + 8) 1− (n − 2, 1, 0, 0) n ≥ 212 (n + 3) 1

2 (n + 1, 0, 1, 0)12 (n + 5) 1

2 (n − 1, 1, 1, 0) n ≥ 112 (n + 7) 1

2 (n − 2, 1, 0, 1) n ≥ 212 (n + 9) 1

2 (n − 2, 0, 0, 1) n ≥ 212 (n + 2) 0+ (n + 2, 0, 0, 0)12 (n + 4) 0− (n, 0, 2, 0)12 (n + 6) 0+ (n − 2, 2, 0, 0) n ≥ 212 (n + 8) 0− (n − 2, 0, 0, 2) n ≥ 212 (n + 10) 0+ (n − 2, 0, 0, 0) n ≥ 2

JTL

The massless sector

I The lowest Kaluza-Klein level (n = 0) corresponds to themassless sector

Field D(E0, j) SO(8) repeaµ D(3, 2) (0, 0, 0, 0) 1ψIµ D( 5

2 ,32 ) (0, 0, 0, 1) 8s

AIJµ D(2, 1) (0, 1, 0, 0) 28

χIJK D( 32 ,

12 ) (1, 0, 1, 0) 56s

S [IJKL]+ D(1, 0) (2, 0, 0, 0) 35v

P [IJKL]− D(2, 0) (0, 0, 2, 0) 35c

I Field content of gauged N = 8 supergravity

I Is there a consistent truncation to the supergravity sector?

Here there is no clean separation of scales

AdS radius = 12S

7 radius

I de Wit and Nicolai — full non-linear reduction is expected tobe consistent [Consistency of the AdS7 × S4 reduction was

demonstrated by Nastase, Vaman and van Nieuwenhuizen]

JTL

The massless sector

I The lowest Kaluza-Klein level (n = 0) corresponds to themassless sector

Field D(E0, j) SO(8) repeaµ D(3, 2) (0, 0, 0, 0) 1ψIµ D( 5

2 ,32 ) (0, 0, 0, 1) 8s

AIJµ D(2, 1) (0, 1, 0, 0) 28

χIJK D( 32 ,

12 ) (1, 0, 1, 0) 56s

S [IJKL]+ D(1, 0) (2, 0, 0, 0) 35v

P [IJKL]− D(2, 0) (0, 0, 2, 0) 35c

I Field content of gauged N = 8 supergravityI Is there a consistent truncation to the supergravity sector?

Here there is no clean separation of scales

AdS radius = 12S

7 radius

I de Wit and Nicolai — full non-linear reduction is expected tobe consistent [Consistency of the AdS7 × S4 reduction was

demonstrated by Nastase, Vaman and van Nieuwenhuizen]

JTL

Consistent truncation

I We can always find a consistent truncation by truncating tosinglets on S7 (ie singlets of SO(8))

Field D(E0, j) KK level n

eaµ D(3, 2) 0ϕ D(6, 0) 2 ← Breathing mode

I This is a bosonic truncationBremer, Duff, Lu, Pope and Stelle, NPB 543, 321 (1999)

I Since we restrict to spherical symmetry, the Freund-Rubinansatz is easily generalized to add a breathing mode

ds211 = e2αϕgµνdxµdxν + 4L2e2βϕds2(S7)

Fµνρσ = (3/L)e2γϕεµνρσ

JTL

Consistent truncation

I We can always find a consistent truncation by truncating tosinglets on S7 (ie singlets of SO(8))

Field D(E0, j) KK level n

eaµ D(3, 2) 0ϕ D(6, 0) 2 ← Breathing mode

I This is a bosonic truncationBremer, Duff, Lu, Pope and Stelle, NPB 543, 321 (1999)

I Since we restrict to spherical symmetry, the Freund-Rubinansatz is easily generalized to add a breathing mode

ds211 = e2αϕgµνdxµdxν + 4L2e2βϕds2(S7)

Fµνρσ = (3/L)e2γϕεµνρσ

JTL

Reduction of the equations of motion

I To show that this is a consistent truncation, we examine theequations of motion

I For F4

dF4 = 0 automatic

d ∗ F4 = 12F4 ∧ F4 ⇒ d(e(2γ−4α+7β)ϕ) = 0

⇒ γ = 12 (4α− 7β)

I For the Einstein equation

(11)RMN − 12gMN

(11)R = 12·3! (FMPQRFN

PQR − 16gMNF

2)

we have(11)Rµν = − 3

L2gµνe

(4γ−6α)ϕ

(11)Rij =3

2L2gije

(4γ+2β−8α)ϕ

JTL

Reduction of the equations of motion

I To show that this is a consistent truncation, we examine theequations of motion

I For F4

dF4 = 0 automatic

d ∗ F4 = 12F4 ∧ F4 ⇒ d(e(2γ−4α+7β)ϕ) = 0

⇒ γ = 12 (4α− 7β)

I For the Einstein equation

(11)RMN − 12gMN

(11)R = 12·3! (FMPQRFN

PQR − 16gMNF

2)

we have(11)Rµν = − 3

L2gµνe

(4γ−6α)ϕ

(11)Rij =3

2L2gije

(4γ+2β−8α)ϕ

JTL

Reduction of the Einstein equation

I For the breathing mode metric, we calculate

(11)Rµν = Rµν − αgµνϕ− (2α + 7β)(∇µ∇νϕ+ αgµν∂ϕ2)

+(α(2α + 7β) + 7β(α− β))∂µϕ∂νϕ(11)Rij = Rij − βe2(β−α)ϕgij [ϕ+ (2α + 7β)∂ϕ2]

– Note the simplification when 2α + 7β = 0

This corresponds to the Einstein frame

I We set 2α + 7β = 0 (this also gives γ = 3α) to get

− 3

L2gµνe

6αϕ = (11)Rµν = Rµν − αgµνϕ− 187 α

2∂µϕ∂νϕ

3

2L2gije

247 αϕ = (11)Rij = Rij + 2

7αe− 18

7 αϕgijϕ

JTL

Reduction of the Einstein equation

I For the breathing mode metric, we calculate

(11)Rµν = Rµν − αgµνϕ− (2α + 7β)(∇µ∇νϕ+ αgµν∂ϕ2)

+(α(2α + 7β) + 7β(α− β))∂µϕ∂νϕ(11)Rij = Rij − βe2(β−α)ϕgij [ϕ+ (2α + 7β)∂ϕ2]

– Note the simplification when 2α + 7β = 0

This corresponds to the Einstein frame

I We set 2α + 7β = 0 (this also gives γ = 3α) to get

− 3

L2gµνe

6αϕ = (11)Rµν = Rµν − αgµνϕ− 187 α

2∂µϕ∂νϕ

3

2L2gije

247 αϕ = (11)Rij = Rij + 2

7αe− 18

7 αϕgijϕ

JTL

Reduction of the Einstein equation

I Let

Rij =6

(2L)2gij (S7 with radius 2L)

I Then

Rµν = 187 α

2∂µϕ∂νϕ+54

(2L)2gµν( 1

6e6αϕ − 7

18e187 αϕ)

αϕ =21

(2L)2(e6αϕ − e

187 αϕ)

I These equations can be obtained from the Lagrangian

e−1L4 = R − 18α2

7∂ϕ2 − V (ϕ)

where

V =27

L2

(16e

6αϕ − 718e

187 αϕ

)

JTL

Reduction of the Einstein equation

I Let

Rij =6

(2L)2gij (S7 with radius 2L)

I Then

Rµν = 187 α

2∂µϕ∂νϕ+54

(2L)2gµν( 1

6e6αϕ − 7

18e187 αϕ)

αϕ =21

(2L)2(e6αϕ − e

187 αϕ)

I These equations can be obtained from the Lagrangian

e−1L4 = R − 18α2

7∂ϕ2 − V (ϕ)

where

V =27

L2

(16e

6αϕ − 718e

187 αϕ

)

JTL

A closer look at the breathing mode

I We can obtain a canonical kinetic term by setting α =√

7/6

I The breathing mode potential has a minimum at ϕ = 0

V = − 6

L2+

9

L2ϕ2 + · · ·

Λ m2ϕ = 18

L2

I We can insert this mass into the expression for E0

E0 = 32 +

√( 32 )2 + (mL)2 = 3

2 + 92 = 6

This is the value of E0 obtained from the linearizedKaluza-Klein analysis

JTL

A consistent supersymmetric truncation

I Can we retain the breathing mode and still have susy?

We should not truncate to singlets on S7

I However, for consistency, we still want to truncate to singletsunder a transitively acting subgroup of SO(8)[Duff and Pope, NPB 255, 355 (1985)]

I Motivation from the squashed S7

U(1) −→ S7

↓ ds2(S7) = ds2(CP3) + η2 dη = 2J

CP3

I The isometry group decomposes as

SO(8) ⊃ SU(4)× U(1)

Qα : 8s → 60 + 11 + 1−1

I Truncating to SU(4) singlets preserves two out of eightsupersymmetries ⇒ N = 2 in D = 4

JTL

A consistent supersymmetric truncation

I Can we retain the breathing mode and still have susy?

We should not truncate to singlets on S7

I However, for consistency, we still want to truncate to singletsunder a transitively acting subgroup of SO(8)[Duff and Pope, NPB 255, 355 (1985)]

I Motivation from the squashed S7

U(1) −→ S7

↓ ds2(S7) = ds2(CP3) + η2 dη = 2J

CP3

I The isometry group decomposes as

SO(8) ⊃ SU(4)× U(1)

Qα : 8s → 60 + 11 + 1−1

I Truncating to SU(4) singlets preserves two out of eightsupersymmetries ⇒ N = 2 in D = 4

JTL

Where do SU(4) singlets come from?

I There are only a limited set of SO(8) representations in theKaluza-Klein spectrum that give rise to SU(4) singlets

D(E0, j) SO(8) U(1) charges KK levelD(3, 2) (0, 0, 0, 0) 0 0 gravitonD( 5

2 ,32 ) (0, 0, 0, 1) 1,−1 0 gravitini

D(2, 1) (0, 1, 0, 0) 0 0 graviphotonD(5, 1) (0, 1, 0, 0) 0 2 massive vectorD( 9

2 ,12 ) (0, 1, 0, 1) 1,−1 2

D( 112 ,

12 ) (0, 0, 0, 1) 1,−1 2

D(4, 0) (0, 2, 0, 0) 0 2 squashingD(5, 0) (0, 0, 0, 2) 0, 2,−2 2D(6, 0) (0, 0, 0, 0) 0 2 breathing

I A consistent truncation will yield D = 4, N = 2 supergravitycoupled to a massive vector multiplet (one hyper coupled toone vector)

JTL

Squashed Sasaki-Einstein truncation

I In fact, we can replace S7 by a Sasaki-Einstein manifold SE7

– SE2n+1 is Sasaki-Einstein if the cone dr2 + r2ds2(SE2n+1) isRicci flat and Kahler

This is exactly what we need to preserve supersymmetry

I A Sasaki-Einstein manifold admits a preferred Reeb vectorfield and a fibration

ds2(SE2n+1) = ds2(KE2n) + η2 dη = 2J

I Here the Kahler-Einstein base admits a global (1, 1) and (n, 0)forms J and Ω satisfying

J∧Ω = 0 Ω∧Ω∗ = (−i)n2

(2J)n/n! dJ = 0 dΩ = (n+1)iη∧Ω

I This Sasaki-Einstein structure is key to constructing themassive consistent truncation

JTL

Squashed Sasaki-Einstein truncation

I In fact, we can replace S7 by a Sasaki-Einstein manifold SE7

– SE2n+1 is Sasaki-Einstein if the cone dr2 + r2ds2(SE2n+1) isRicci flat and Kahler

This is exactly what we need to preserve supersymmetry

I A Sasaki-Einstein manifold admits a preferred Reeb vectorfield and a fibration

ds2(SE2n+1) = ds2(KE2n) + η2 dη = 2J

I Here the Kahler-Einstein base admits a global (1, 1) and (n, 0)forms J and Ω satisfying

J∧Ω = 0 Ω∧Ω∗ = (−i)n2

(2J)n/n! dJ = 0 dΩ = (n+1)iη∧Ω

I This Sasaki-Einstein structure is key to constructing themassive consistent truncation

JTL

The reduction ansatz

I Starting from 11-dimensional supergravity, we need to makean ansatz for the metric and three-form potentialGauntlett, Kim, Varela and Waldram, JHEP 0904, 102 (2009)

I For the metric, we take

ds211 = e−73 vgµνdx

µdxν + e23 v(e−2uds2(KE6) + e12u(η + A1)2

)↑—J, Ω

I The four-dimensional fields from the metric are

gµν , Aµ, u, v

I For F4, we expand in a basis of invariant tensors η, J, Ω

F4 = f vol4 + H3 ∧ (η + A1) + H2 ∧ J + H1 ∧ J ∧ (η + A1)

+2hJ ∧ J +√

3[χ1 ∧ Ω + χ(η + A1) ∧ Ω + c .c .]

I The fields in F4 are

H3, H2, H1, χ1, χ∗1 , f , h, χ, χ

∗

JTL

The reduction ansatz

I Starting from 11-dimensional supergravity, we need to makean ansatz for the metric and three-form potentialGauntlett, Kim, Varela and Waldram, JHEP 0904, 102 (2009)

I For the metric, we take

ds211 = e−73 vgµνdx

µdxν + e23 v(e−2uds2(KE6) + e12u(η + A1)2

)↑—J, Ω

I The four-dimensional fields from the metric are

gµν , Aµ, u, v

I For F4, we expand in a basis of invariant tensors η, J, Ω

F4 = f vol4 + H3 ∧ (η + A1) + H2 ∧ J + H1 ∧ J ∧ (η + A1)

+2hJ ∧ J +√

3[χ1 ∧ Ω + χ(η + A1) ∧ Ω + c .c .]

I The fields in F4 are

H3, H2, H1, χ1, χ∗1 , f , h, χ, χ

∗

JTL

Solving the equations of motion

I The ansatz is the most general one compatible withsymmetries

Now we just have to work out the equations of motionI The F4 Bianchi identity is solved by taking

H3 = dB2

H2 = dB1 + 2B2 + hF2

H1 = dh

χ1 = − i4Dχ = − i

4 (dχ− 4iA1χ)

I In addition, the F4 equation of motion leads to a constrainton f

f = 6e−73 v (1 + h2 + |χ|2)

[This corresponds to L = 1/2 normalization where the vacuum

solution is F4 = (3/L)vol4]I The four-dimensional fields from F4 are now

B2, B1, h, χ, χ∗

JTL

Solving the equations of motion

I The ansatz is the most general one compatible withsymmetries

Now we just have to work out the equations of motionI The F4 Bianchi identity is solved by taking

H3 = dB2

H2 = dB1 + 2B2 + hF2

H1 = dh

χ1 = − i4Dχ = − i

4 (dχ− 4iA1χ)

I In addition, the F4 equation of motion leads to a constrainton f

f = 6e−73 v (1 + h2 + |χ|2)

[This corresponds to L = 1/2 normalization where the vacuum

solution is F4 = (3/L)vol4]I The four-dimensional fields from F4 are now

B2, B1, h, χ, χ∗

JTL

The effective four-dimensional Lagrangian

I The equations of motion can be obtained from the Lagrangian

e−1L = R − 42∂u2 − 72∂v

2 − 32e

−8u−2v∂h2 − 32e

6u−2v |Dχ|2

− 14e

12u+3vF 2µν − 1

12e−12u+4vH2

µνρ − 34e

4u+vH2µν − V

+6A1 ∧ H3 + interactions

where

V = −48e2u−3v + 6e16u−3v + 24h2e8u−5v

+18(1 + h2 + |χ|2)2e−7v + 24e−6u−5v |χ|2

I The fields are

metric : gµν

scalars : u, v , h, χ, χ∗

vectors : A1,B1 (B2 dualizes to a Stuckelberg scalar)

JTL

Connection with the linearized Kaluza-Klein analysis

I We may expand the scalar potential about its minimumu = v = h = χ = χ∗ = 0

V = −24 + 16(42u2) + 72( 72v

2) + 40( 32h

2) + 40( 32 |χ|

2) + · · ·

I The minimum of the potential gives AdS4 with radiusL = 1/2, while the scalars have masses

m2u = 16 m2

v = 72 m2h = m2

χ = 40

E0 = 4 6 5

I This matches the bosonic part of the Kaluza-Klein spectrum

D(E0, j) SO(8) U(1) charges KK level FieldD(3, 2) (0, 0, 0, 0) 0 0 gµνD(2, 1) (0, 1, 0, 0) 0 0 A1

D(5, 1) (0, 1, 0, 0) 0 2 B1D(4, 0) (0, 2, 0, 0) 0 2 uD(5, 0) (0, 0, 0, 2) 0, 2,−2 2 h, χ, χ∗

D(6, 0) (0, 0, 0, 0) 0 2 v

JTL

A Kaluza-Klein consistency condition

I Is it possible to find other consistent truncations that retainstates in the Kaluza-Klein tower?

Consistency in the absence of a group-theoretic argument israther delicate

I Consider a general Kaluza-Klein reduction where we gauge theinternal symmetry

ds2D = ds2d + gij(dyi + K I iAI

µdxµ)(dy j + K J jAJ

νdxν)

here K I i (y) are Killing vectors in the internal space

I If this is all we had, then the reduced Einstein equation wouldhave the form

Rµν − 12gµνR + Λgµν = 1

2 (F IµρF

J ρν − 1

4gµνFIρσF

I ρσ)Y IJ(y)

whereY IJ(y) = gijK

I iK J j

I Unless Y IJ is independent of y , this gives an inconsistenttruncation since the LHS is independent of y

JTL

A Kaluza-Klein consistency condition

I Is it possible to find other consistent truncations that retainstates in the Kaluza-Klein tower?

Consistency in the absence of a group-theoretic argument israther delicate

I Consider a general Kaluza-Klein reduction where we gauge theinternal symmetry

ds2D = ds2d + gij(dyi + K I iAI

µdxµ)(dy j + K J jAJ

νdxν)

here K I i (y) are Killing vectors in the internal space

I If this is all we had, then the reduced Einstein equation wouldhave the form

Rµν − 12gµνR + Λgµν = 1

2 (F IµρF

J ρν − 1

4gµνFIρσF

I ρσ)Y IJ(y)

whereY IJ(y) = gijK

I iK J j

I Unless Y IJ is independent of y , this gives an inconsistenttruncation since the LHS is independent of y

JTL

Consistency of sphere reductions

I It is easy to check that Killing vectors on spheres do not giveY IJ independent of y

How can sphere reductions be consistent?

I The reduction ansatz also involves form-fields

For 11-dimensional supergravity

F4 =3

Lvol4 + L ∗4 F I ∧ dK I

This gives an additional term in the lower-dimensional Einsteinequation, so that Y IJ is modified to

Y IJ(y) = K Ii K

J j + 12L

2∇iKIj ∇iK J j −→ δIJ

S7

I A similar argument applies for the S4 reduction of11-dimensional supergravity and the S5 reduction of IIBsupergravity

JTL

Consistency of sphere reductions

I It is easy to check that Killing vectors on spheres do not giveY IJ independent of y

How can sphere reductions be consistent?

I The reduction ansatz also involves form-fields

For 11-dimensional supergravity

F4 =3

Lvol4 + L ∗4 F I ∧ dK I

This gives an additional term in the lower-dimensional Einsteinequation, so that Y IJ is modified to

Y IJ(y) = K Ii K

J j + 12L

2∇iKIj ∇iK J j −→ δIJ

S7

I A similar argument applies for the S4 reduction of11-dimensional supergravity and the S5 reduction of IIBsupergravity

JTL

Consistent truncations with non-Abelian gauge bosons

I Abelian gauge bosons often arise with K I i = const

⇒ easy to make consistent

I However it is more difficult to retain non-Abelian gaugebosons in a consistent truncation

– Sphere reductions with maximal supersymmetry do not allowfor a breathing mode

– IIB theory on AdS5 × T 1,1 has SU(2)× SU(2)× U(1)isometry, but the SU(2)× SU(2) gauge fields cannot be keptin a consistent truncation [The graviphoton gauges the U(1)]

I Are there other general principles for obtaining consistenttruncations?

JTL

Consistent truncations with non-Abelian gauge bosons

I Abelian gauge bosons often arise with K I i = const

⇒ easy to make consistent

I However it is more difficult to retain non-Abelian gaugebosons in a consistent truncation

– Sphere reductions with maximal supersymmetry do not allowfor a breathing mode

– IIB theory on AdS5 × T 1,1 has SU(2)× SU(2)× U(1)isometry, but the SU(2)× SU(2) gauge fields cannot be keptin a consistent truncation [The graviphoton gauges the U(1)]

I Are there other general principles for obtaining consistenttruncations?

JTL

Concluding remarks

I Compactifications and liftings allow us to relate theories invarious dimensions and with different amounts ofsupersymmetries

I Unless we truncate, we end up with an infinite Kaluza-Kleintower of massive states

Linearized analysis allows us to determine the spectrum

I A consistent truncation is one where a solution to thetruncated system is guaranteed to satisfy the full equations ofmotion of the original theory without further constraints

Truncations to the singlet sector of the isometry group (or asubgroup of the isometry group) are automatically consistent

Truncations to the supergravity sector (lowest Kaluza-Kleinlevel) are expected to be consistent

JTL

Additional references

I Freedman and Van Proeyen, Supergravity, CambridgeUniversity Press (2012)

I Duff, Nilsson and Pope, Kaluza-Klein supergravity, Phys.Rept. 130, 1 (1986)

I Font and Theisen, Introduction to String Compactification,http://www.aei.mpg.de/˜theisen/cy.html

I Pope, Kaluza-Klein Theory,http://people.physics.tamu.edu/pope/ihplec.pdf

I Morrison, TASI lectures on compactification and duality,hep-th/0411120