Compactifications and Consistent Truncations in Supergravity

Compactiﬁcations and consistent truncations in supergravity

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

GRaB100 I However there are many interesting situations where ﬂuxes are turned on

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

Compactiﬁcation with ﬂuxes

I For compactiﬁcations without ﬂuxes, the criteria for preserving supersymmetry is the presence of parallel spinors ∇i = 0

⇒ Xn is a Ricci-flat manifold with special holonomy dim H 2nSU (n) Kähler(Calabi-Yau) 4n Sp(2n) Hyperkähler(K3 for n = 1) 7 G2 8 Spin(7)

JTL Compactiﬁcation with ﬂuxes

I For compactiﬁcations without ﬂuxes, the criteria for preserving supersymmetry is the presence of parallel spinors ∇i = 0

⇒ Xn is a Ricci-flat manifold with special holonomy dim H 2nSU (n) Kähler(Calabi-Yau) 4n Sp(2n) Hyperkähler(K3 for n = 1) 7 G2 8 Spin(7)

I However there are many interesting situations where ﬂuxes are turned on

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

JTL I The4-form ﬁeld strength naturally selects out four dimensions ⇒ Make a4+7 split Freund and Rubin, PLB 97, 233 (1980)

2 µ ν i j ds11 = gµν (x)dx dx + gij (y)dy dy

F4 = (3/L)vol4

Spontaneous compactiﬁcation of 11D supergravity

I The bosonic Lagrangian of 11D supergravity is given by

−1 1 1 e L = R ∗ 1 − 2 F4 ∧ ∗F4 − 6 F4 ∧ F4 ∧ A4 with corresponding equations of motion

1 1 PQR 1 2 RMN − 2 gMN R = 2·3! (FMPQR FN − 6 gMN F ) dF4 = 0 1 d ∗ F4 = 2 F4 ∧ F4

JTL Spontaneous compactiﬁcation of 11D supergravity

I The bosonic Lagrangian of 11D supergravity is given by

−1 1 1 e L = R ∗ 1 − 2 F4 ∧ ∗F4 − 6 F4 ∧ F4 ∧ A4 with corresponding equations of motion

1 1 PQR 1 2 RMN − 2 gMN R = 2·3! (FMPQR FN − 6 gMN F ) dF4 = 0 1 d ∗ F4 = 2 F4 ∧ F4

I The4-form ﬁeld strength naturally selects out four dimensions ⇒ Make a4+7 split Freund and Rubin, PLB 97, 233 (1980)

2 µ ν i j ds11 = gµν (x)dx dx + gij (y)dy dy

F4 = (3/L)vol4

JTL I From the Einstein equation, we ﬁnd 3 3 R = − g R = g µν L2 µν ij 2L2 ij

7 I This leads to the maximally symmetric solution AdS4 × S with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S 7 but can compactify on other Einstein manifolds as well] 4 I Similarly, the AdS7 × S solution is obtained by taking F4 to have internal ﬂux

Solving the equations of motion

I The4-form satisﬁes dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0 ⇒ the4-form equations are automatically satisﬁed

JTL 4 I Similarly, the AdS7 × S solution is obtained by taking F4 to have internal ﬂux

Solving the equations of motion

I The4-form satisfies dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0 ⇒ the4-form equations are automatically satisfied I From the Einstein equation, we find 3 3 R = − g R = g µν L2 µν ij 2L2 ij

7 I This leads to the maximally symmetric solution AdS4 × S with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S 7 but can compactify on other Einstein manifolds as well]

JTL Solving the equations of motion

I The4-form satisfies dF4 = 0, d ∗ F4 = 0 and F4 ∧ F4 = 0 ⇒ the4-form equations are automatically satisfied I From the Einstein equation, we find 3 3 R = − g R = g µν L2 µν ij 2L2 ij

7 I This leads to the maximally symmetric solution AdS4 × S with radii of curvature

LAdS = L and LS7 = 2L

[We are not restricted to the round S 7 but can compactify on other Einstein manifolds as well] 4 I Similarly, the AdS7 × S solution is obtained by taking F4 to have internal ﬂux

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

5 Γµ = γµ ⊗ 1 Γi = γ ⊗ γi ⇒ = ε4 ⊗ η7 Then

i 5 δψµ = [∇µ − 2L γµγ ]ε ⊗ η Killing spinors on AdS4 i 7 δψi = ε ⊗ [∇i + 4L γi ]η Killing spinors on S

Preserving supersymmetry

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

1 PQRS P QRS δψM = DM ≡ ∇M − 288 (ΓM − 8δM Γ )FPQRS

Setting Fµνρσ = (3/L)µνρσ gives 1 0123 δψµ = [∇µ + 2L ΓµΓ ] 1 0123 δψi = [∇i − 4L Γi Γ ]

JTL Preserving supersymmetry

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

1 PQRS P QRS δψM = DM ≡ ∇M − 288 (ΓM − 8δM Γ )FPQRS

Setting Fµνρσ = (3/L)µνρσ gives 1 0123 δψµ = [∇µ + 2L ΓµΓ ] 1 0123 δψi = [∇i − 4L Γi Γ ]

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

5 Γµ = γµ ⊗ 1 Γi = γ ⊗ γi ⇒ = ε4 ⊗ η7 Then

i 5 δψµ = [∇µ − 2L γµγ ]ε ⊗ η Killing spinors on AdS4 i 7 δψi = ε ⊗ [∇i + 4L γi ]η Killing spinors on S

JTL ig 5 I A similar calculation with[ ∇µ − 2 γµγ ]η gives 2 Rµνρσ = −g (gµρgνσ − gµσgνρ) ⇒ AdS4

Killing spinors on spheres

I Consider the Killing spinor equation

im [∇i + 2 γi ]η = 0 (m = const)

I Integrability gives

im im 0 = [∇i + 2 γi , ∇j + 2 γj ]η m2 1 kl 2 k l l k = [∇i , ∇j ] − 2 γij η = 4 Rij − m (δi δj − δi δj ) γkl η Maximal supersymmetry then requires

2 Rijkl = m (gik gjl − gil gjk ) which corresponds to a maximally symmetric space of positive curvature ⇒ S7

JTL Killing spinors on spheres

I Consider the Killing spinor equation

im [∇i + 2 γi ]η = 0 (m = const)

I Integrability gives

im im 0 = [∇i + 2 γi , ∇j + 2 γj ]η m2 1 kl 2 k l l k = [∇i , ∇j ] − 2 γij η = 4 Rij − m (δi δj − δi δj ) γkl η Maximal supersymmetry then requires

2 Rijkl = m (gik gjl − gil gjk ) which corresponds to a maximally symmetric space of positive curvature ⇒ S7 ig 5 I A similar calculation with[ ∇µ − 2 γµγ ]η gives 2 Rµνρσ = −g (gµρgνσ − gµσgνρ) ⇒ AdS4

JTL 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 spherical harmonics on S 7 I States are classiﬁed by their SO(2, 3) × SO(8) quantum numbers D(E0, j)(l1, l2, l3, l4)

Gauged D = 4, N = 8 supergravity

7 I The compactiﬁcation of D = 11 supergravity on S is maximally supersymmetric ⇒ gives N = 8 in four dimensions I What symmetries do we expect? 7 AdS4 × S SO(2, 3) × SO(8) + susy ⇒ OSp(4|8)

JTL Gauged D = 4, N = 8 supergravity

7 I The compactiﬁcation of D = 11 supergravity on S is maximally supersymmetric ⇒ gives N = 8 in four dimensions I What symmetries do we expect? 7 AdS4 × S 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 spherical harmonics on S 7 I States are classiﬁed by their SO(2, 3) × SO(8) quantum numbers D(E0, j)(l1, l2, l3, l4)

JTL The Kaluza-Klein spectrum on S 7

E0 j SO(8) rep KK level n 1 2 (n + 6) 2 (n, 0, 0, 0) 1 3 2 (n + 5) 2 (n, 0, 0, 1) 1 3 2 (n + 7) 2 (n − 1, 0, 1, 0) n ≥ 1 1 − 2 (n + 4) 1 (n, 1, 0, 0) 1 + 2 (n + 6) 1 (n − 1, 0, 1, 1) n ≥ 1 1 − 2 (n + 8) 1 (n − 2, 1, 0, 0) n ≥ 2 1 1 2 (n + 3) 2 (n + 1, 0, 1, 0) 1 1 2 (n + 5) 2 (n − 1, 1, 1, 0) n ≥ 1 1 1 2 (n + 7) 2 (n − 2, 1, 0, 1) n ≥ 2 1 1 2 (n + 9) 2 (n − 2, 0, 0, 1) n ≥ 2 1 + 2 (n + 2) 0 (n + 2, 0, 0, 0) 1 − 2 (n + 4) 0 (n, 0, 2, 0) 1 + 2 (n + 6) 0 (n − 2, 2, 0, 0) n ≥ 2 1 − 2 (n + 8) 0 (n − 2, 0, 0, 2) n ≥ 2 1 + 2 (n + 10) 0 (n − 2, 0, 0, 0) n ≥ 2

JTL I Is there a consistent truncation to the supergravity sector? Here there is no clean separation of scales

1 7 AdS radius = 2 S radius

I de Wit and Nicolai — full non-linear reduction is expected to 4 be consistent [Consistency of the AdS7 × S reduction was demonstrated by Nastase, Vaman and van Nieuwenhuizen]

The massless sector

I The lowest Kaluza-Klein level (n = 0) corresponds to the massless sector Field D(E0, j) SO(8) rep a eµ D(3, 2) (0, 0, 0, 0) 1 I 5 3 ψµ D( 2 , 2 ) (0, 0, 0, 1) 8s IJ Aµ D(2, 1) (0, 1, 0, 0) 28 IJK 3 1 χ D( 2 , 2 ) (1, 0, 1, 0) 56s [IJKL]+ S D(1, 0) (2, 0, 0, 0) 35v [IJKL]− P D(2, 0) (0, 0, 2, 0) 35c I Field content of gauged N = 8 supergravity

JTL The massless sector

1 7 AdS radius = 2 S radius

I de Wit and Nicolai — full non-linear reduction is expected to 4 be consistent [Consistency of the AdS7 × S reduction was demonstrated by Nastase, Vaman and van Nieuwenhuizen] JTL I Since we restrict to spherical symmetry, the Freund-Rubin ansatz is easily generalized to add a breathing mode

2 2αϕ µ ν 2 2βϕ 2 7 ds11 = e gµν dx dx + 4L e ds (S ) 2γϕ Fµνρσ = (3/L)e µνρσ

Consistent truncation

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

Field D(E0, j) KK level n a eµ D(3, 2)0 ϕ D(6, 0)2 ← Breathing mode

I This is a bosonic truncation Bremer, Duﬀ, L¨u,Pope and Stelle, NPB 543, 321 (1999)

JTL Consistent truncation

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

Field D(E0, j) KK level n a eµ D(3, 2)0 ϕ D(6, 0)2 ← Breathing mode

I This is a bosonic truncation Bremer, Duﬀ, L¨u,Pope and Stelle, NPB 543, 321 (1999) I Since we restrict to spherical symmetry, the Freund-Rubin ansatz is easily generalized to add a breathing mode

2 2αϕ µ ν 2 2βϕ 2 7 ds11 = e gµν dx dx + 4L e ds (S ) 2γϕ Fµνρσ = (3/L)e µνρσ

JTL I For the Einstein equation

(11) 1 (11) 1 PQR 1 2 RMN − 2 gMN R = 2·3! (FMPQR FN − 6 gMN F ) we have 3 (11)R = − g e(4γ−6α)ϕ µν L2 µν 3 (11)R = g e(4γ+2β−8α)ϕ ij 2L2 ij

Reduction of the equations of motion

I To show that this is a consistent truncation, we examine the equations of motion I For F4

dF4 = 0 automatic

1 (2γ−4α+7β)ϕ d ∗ F4 = 2 F4 ∧ F4 ⇒ d(e ) = 0 1 ⇒ γ = 2 (4α − 7β)

JTL Reduction of the equations of motion

I To show that this is a consistent truncation, we examine the equations of motion I For F4

dF4 = 0 automatic

1 (2γ−4α+7β)ϕ d ∗ F4 = 2 F4 ∧ F4 ⇒ d(e ) = 0 1 ⇒ γ = 2 (4α − 7β)

I For the Einstein equation

(11) 1 (11) 1 PQR 1 2 RMN − 2 gMN R = 2·3! (FMPQR FN − 6 gMN F ) we have 3 (11)R = − g e(4γ−6α)ϕ µν L2 µν 3 (11)R = g e(4γ+2β−8α)ϕ ij 2L2 ij

JTL I We set2 α + 7β = 0 (this also gives γ = 3α) to get 3 − g e6αϕ = (11)R = R − αg ϕ − 18 α2∂ ϕ∂ ϕ L2 µν µν µν µν 7 µ ν 24 18 3 αϕ (11) 2 − αϕ g e 7 = R = R + αe 7 g ϕ 2L2 ij ij ij 7 ij

Reduction of the Einstein equation

I For the breathing mode metric, we calculate

(11) 2 Rµν = Rµν − αgµν ϕ − (2α + 7β)(∇µ∇ν ϕ + αgµν ∂ϕ ) +(α(2α + 7β) + 7β(α − β))∂µϕ∂ν ϕ (11) 2(β−α)ϕ 2 Rij = Rij − βe gij [ϕ + (2α + 7β)∂ϕ ]

– Note the simpliﬁcation when2 α + 7β = 0 This corresponds to the Einstein frame

JTL Reduction of the Einstein equation

I For the breathing mode metric, we calculate

(11) 2 Rµν = Rµν − αgµν ϕ − (2α + 7β)(∇µ∇ν ϕ + αgµν ∂ϕ ) +(α(2α + 7β) + 7β(α − β))∂µϕ∂ν ϕ (11) 2(β−α)ϕ 2 Rij = Rij − βe gij [ϕ + (2α + 7β)∂ϕ ]

– Note the simpliﬁcation when2 α + 7β = 0 This corresponds to the Einstein frame

I We set2 α + 7β = 0 (this also gives γ = 3α) to get 3 − g e6αϕ = (11)R = R − αg ϕ − 18 α2∂ ϕ∂ ϕ L2 µν µν µν µν 7 µ ν 24 18 3 αϕ (11) 2 − αϕ g e 7 = R = R + αe 7 g ϕ 2L2 ij ij ij 7 ij

JTL I These equations can be obtained from the Lagrangian

18α2 e−1L = R − ∂ϕ2 − V (ϕ) 4 7 where 27 1 6αϕ 7 18 αϕ V = e − e 7 L2 6 18

Reduction of the Einstein equation

I Let 6 R = g (S 7 with radius 2L) ij (2L)2 ij

I Then

18 2 54 1 6αϕ 7 18 αϕ R = α ∂ ϕ∂ ϕ + g ( e − e 7 ) µν 7 µ ν (2L)2 µν 6 18

21 6αϕ 18 αϕ α ϕ = (e − e 7 ) (2L)2

JTL Reduction of the Einstein equation

I Let 6 R = g (S 7 with radius 2L) ij (2L)2 ij

I Then

18 2 54 1 6αϕ 7 18 αϕ R = α ∂ ϕ∂ ϕ + g ( e − e 7 ) µν 7 µ ν (2L)2 µν 6 18

21 6αϕ 18 αϕ α ϕ = (e − e 7 ) (2L)2

I These equations can be obtained from the Lagrangian

18α2 e−1L = R − ∂ϕ2 − V (ϕ) 4 7 where 27 1 6αϕ 7 18 αϕ V = e − e 7 L2 6 18

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 6 9 V = − + ϕ2 + ··· L2 L2 2 18 Λ mϕ = L2

I We can insert this mass into the expression for E0 q 3 3 2 2 3 9 E0 = 2 + ( 2 ) + (mL) = 2 + 2 = 6

This is the value of E0 obtained from the linearized Kaluza-Klein analysis

JTL 7 I Motivation from the squashed S U(1) −→ S 7 2 7 2 2 ↓ ds (S ) = ds (CP3) + η dη = 2J

CP3

I The isometry group decomposes as

SO(8) ⊃ SU(4) × U(1) α Q : 8s → 60 + 11 + 1−1

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

A consistent supersymmetric truncation

I Can we retain the breathing mode and still have susy? We should not truncate to singlets on S 7

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

JTL A consistent supersymmetric truncation

I Can we retain the breathing mode and still have susy? We should not truncate to singlets on S 7

I However, for consistency, we still want to truncate to singlets under a transitively acting subgroup of SO(8) [Duﬀ and Pope, NPB 255, 355 (1985)] 7 I Motivation from the squashed S U(1) −→ S 7 2 7 2 2 ↓ ds (S ) = ds (CP3) + η dη = 2J

CP3

I The isometry group decomposes as

SO(8) ⊃ SU(4) × U(1) α Q : 8s → 60 + 11 + 1−1

I Truncating to SU(4) singlets preserves two out of eight

supersymmetries ⇒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 the Kaluza-Klein spectrum that give rise to SU(4) singlets

D(E0, j) SO(8) U(1) charges KK level D(3, 2) (0, 0, 0, 0)00 graviton 5 3 D( 2 , 2 ) (0, 0, 0, 1)1 , −10 gravitini D(2, 1) (0, 1, 0, 0)00 graviphoton D(5, 1) (0, 1, 0, 0)02 massive vector 9 1 D( 2 , 2 ) (0, 1, 0, 1)1 , −12 11 1 D( 2 , 2 ) (0, 0, 0, 1)1 , −12 D(4, 0) (0, 2, 0, 0)02 squashing D(5, 0) (0, 0, 0, 2)0 , 2, −22 D(6, 0) (0, 0, 0, 0)02 breathing

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

JTL I A Sasaki-Einstein manifold admits a preferred Reeb vector ﬁeld and a ﬁbration

2 2 2 ds (SE2n+1) = ds (KE2n) + η dη = 2J

I Here the K¨ahler-Einsteinbase admits a global (1, 1) and( n, 0) forms J andΩ satisfying

2 J∧Ω = 0 Ω∧Ω∗ = (−i)n (2J)n/n! dJ = 0 dΩ = (n+1)iη∧Ω

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

Squashed Sasaki-Einstein truncation

7 I In fact, we can replace S by a Sasaki-Einstein manifold SE7 2 2 2 – SE2n+1 is Sasaki-Einstein if the cone dr + r ds (SE2n+1) is Ricci ﬂat and K¨ahler This is exactly what we need to preserve supersymmetry

JTL Squashed Sasaki-Einstein truncation

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

2 2 2 ds (SE2n+1) = ds (KE2n) + η dη = 2J

I Here the K¨ahler-Einsteinbase admits a global (1, 1) and( n, 0) forms J andΩ satisfying

2 J∧Ω = 0 Ω∧Ω∗ = (−i)n (2J)n/n! dJ = 0 dΩ = (n+1)iη∧Ω

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

JTL I For F4, we expand in a basis of invariant tensors η, J, Ω F = f vol + H ∧ (η + A ) + H ∧ J + H ∧ J ∧ (η + A ) 4 4 3 √ 1 2 1 1 +2hJ ∧ J + 3[χ1 ∧ Ω + χ(η + A1) ∧ Ω + c.c.]

I The ﬁelds in F4 are ∗ ∗ {H3, H2, H1, χ1, χ1, f , h, χ, χ }

The reduction ansatz

I Starting from 11-dimensional supergravity, we need to make an ansatz for the metric and three-form potential Gauntlett, Kim, Varela and Waldram, JHEP 0904, 102 (2009) I For the metric, we take 7 2 2 − 3 v µ ν 3 v −2u 2 12u 2 ds11 = e gµν dx dx + e e ds (KE6) + e (η + A1) ↑—J, Ω

I The four-dimensional ﬁelds from the metric are

{gµν , Aµ, u, v}

JTL The reduction ansatz

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 F = f vol + H ∧ (η + A ) + H ∧ J + H ∧ J ∧ (η + A ) 4 4 3 √ 1 2 1 1 +2hJ ∧ J + 3[χ1 ∧ Ω + χ(η + A1) ∧ Ω + c.c.]

I The ﬁelds in F4 are ∗ ∗ {H3, H2, H1, χ1, χ1, f , h, χ, χ }

JTL I In addition, the F4 equation of motion leads to a constraint on f − 7 v 2 2 f = 6e 3 (1 + h + |χ| ) [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 }

Solving the equations of motion

I The ansatz is the most general one compatible with symmetries Now we just have to work out the equations of motion I The F4 Bianchi identity is solved by taking

H3 = dB2

H2 = dB1 + 2B2 + hF2

H1 = dh i i χ1 = − 4 Dχ = − 4 (dχ − 4iA1χ)

JTL Solving the equations of motion

I The ansatz is the most general one compatible with symmetries Now we just have to work out the equations of motion I The F4 Bianchi identity is solved by taking

H3 = dB2

H2 = dB1 + 2B2 + hF2

H1 = dh i i χ1 = − 4 Dχ = − 4 (dχ − 4iA1χ)

I In addition, the F4 equation of motion leads to a constraint on f − 7 v 2 2 f = 6e 3 (1 + h + |χ| ) [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

−1 2 7 2 3 −8u−2v 2 3 6u−2v 2 e L = R − 42∂u − 2 ∂v − 2 e ∂h − 2 e |Dχ| 1 12u+3v 2 1 −12u+4v 2 3 4u+v 2 − 4 e Fµν − 12 e Hµνρ − 4 e Hµν − 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 St¨uckelberg scalar)

JTL Connection with the linearized Kaluza-Klein analysis

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

2 7 2 3 2 3 2 V = −24 + 16(42u ) + 72( 2 v ) + 40( 2 h ) + 40( 2 |χ| ) + ···

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

2 2 2 2 mu = 16 mv = 72 mh = mχ = 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 Field D(3, 2) (0, 0, 0, 0)00 gµν D(2, 1) (0, 1, 0, 0)00 A1 D(5, 1) (0, 1, 0, 0)02 B1 D(4, 0) (0, 2, 0, 0)02 u D(5, 0) (0, 0, 0, 2)0 , 2, −22 h, χ, χ∗ D(6, 0) (0, 0, 0, 0)02 v JTL I If this is all we had, then the reduced Einstein equation would have the form

1 1 I J ρ 1 I I ρσ IJ Rµν − 2 gµν R + Λgµν = 2 (FµρFν − 4 gµν FρσF )Y (y) where IJ I i J j Y (y) = gij K K IJ I Unless Y is independent of y, this gives an inconsistent truncation since the LHS is independent of y

A Kaluza-Klein consistency condition

I Is it possible to ﬁnd other consistent truncations that retain states in the Kaluza-Klein tower? Consistency in the absence of a group-theoretic argument is rather delicate I Consider a general Kaluza-Klein reduction where we gauge the internal symmetry

2 2 i I i I µ j J j J ν dsD = dsd + gij (dy + K Aµdx )(dy + K Aν dx )

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

JTL A Kaluza-Klein consistency condition

2 2 i I i I µ j J j J ν dsD = dsd + gij (dy + K Aµdx )(dy + K Aν dx )

here K I i (y) are Killing vectors in the internal space I If this is all we had, then the reduced Einstein equation would have the form

1 1 I J ρ 1 I I ρσ IJ Rµν − 2 gµν R + Λgµν = 2 (FµρFν − 4 gµν FρσF )Y (y) where IJ I i J j Y (y) = gij K K IJ I Unless Y is independent of y, this gives an inconsistent

truncation since the LHS is independent of y JTL I The reduction ansatz also involves form-ﬁelds For 11-dimensional supergravity 3 F = vol + L ∗ F I ∧ dK I 4 L 4 4 This gives an additional term in the lower-dimensional Einstein equation, so that Y IJ is modiﬁed to

IJ I J j 1 2 I i J j IJ Y (y) = Ki K + 2 L ∇i Kj ∇ K −→ δ S 7

4 I A similar argument applies for the S reduction of 11-dimensional supergravity and the S5 reduction of IIB supergravity

Consistency of sphere reductions

I It is easy to check that Killing vectors on spheres do not give Y IJ independent of y How can sphere reductions be consistent?

JTL Consistency of sphere reductions

I It is easy to check that Killing vectors on spheres do not give Y IJ independent of y How can sphere reductions be consistent? I The reduction ansatz also involves form-ﬁelds For 11-dimensional supergravity 3 F = vol + L ∗ F I ∧ dK I 4 L 4 4 This gives an additional term in the lower-dimensional Einstein equation, so that Y IJ is modiﬁed to

IJ I J j 1 2 I i J j IJ Y (y) = Ki K + 2 L ∇i Kj ∇ K −→ δ S 7

4 I A similar argument applies for the S reduction of 11-dimensional supergravity and the S5 reduction of IIB supergravity

JTL I Are there other general principles for obtaining consistent truncations?

Consistent truncations with non-Abelian gauge bosons

I i I Abelian gauge bosons often arise with K = const ⇒ easy to make consistent I However it is more diﬃcult to retain non-Abelian gauge bosons in a consistent truncation – Sphere reductions with maximal supersymmetry do not allow for a breathing mode

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

JTL Consistent truncations with non-Abelian gauge bosons

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

I Are there other general principles for obtaining consistent truncations?

JTL Concluding remarks

I Compactifications and liftings allow us to relate theories in various dimensions and with different amounts of supersymmetries I Unless we truncate, we end up with an infinite Kaluza-Klein tower of massive states Linearized analysis allows us to determine the spectrum I A consistent truncation is one where a solution to the truncated system is guaranteed to satisfy the full equations of motion of the original theory without further constraints Truncations to the singlet sector of the isometry group (or a subgroup of the isometry group) are automatically consistent Truncations to the supergravity sector (lowest Kaluza-Klein level) are expected to be consistent

JTL Additional references

I Freedman and Van Proeyen, Supergravity, Cambridge University 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

JTL