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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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]
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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]
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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
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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
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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
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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
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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
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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
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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
)
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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
)
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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
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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
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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
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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)
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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
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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
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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
∗
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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
∗
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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∗
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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∗
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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)
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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
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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
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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
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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
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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
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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?
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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?
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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
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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