Compactifications and Consistent Truncations in Supergravity

Compactiﬁcations and consistent truncations in supergravity

James Liu

University of Michigan

7 July 2015

1. Introduction to supersymmetry and supergravity 2. Compactification on manifolds without fluxes 3. AdS×Sphere compactifications and consistent truncations

GRaB100 I The original ﬁelds can be expanded in a complete set of harmonics on Xn µ i X (a) µ i ϕ(x , y ) = φ (x )fa(y ) a

I By expanding in a complete set, the theory is unchanged, but given a lower-dimensional interpretation – A finite number of D-dimensional fields expand into an infinite number of( D − n)-dimensional fields – Isometries of Xn give rise to symmetries of the lower-dimensional theory

The Kaluza-Klein idea

I We may obtain a lower-dimensional theory by compactifying a D-dimensional theory on an n-dimensional manifold Xn

X M = {x µ, y i } where M =0 , 1,..., D − n − 1, D − n,..., D − 1 µ i

JTL The Kaluza-Klein idea

I We may obtain a lower-dimensional theory by compactifying a D-dimensional theory on an n-dimensional manifold Xn

X M = {x µ, y i } where M =0 , 1,..., D − n − 1, D − n,..., D − 1 µ i

I The original ﬁelds can be expanded in a complete set of harmonics on Xn µ i X (a) µ i ϕ(x , y ) = φ (x )fa(y ) a

JTL 1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR

I Now expand in Fourier modes on the circle ∞ µ X µ iny/R ϕ(x , y) = φn(x )e n=−∞

I The scalar equation then reduces to 2 2 X 2 2 iny/R (d + ∂y − mD )ϕ = 0 ⇒ (d − (n/R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R

Example: Reduction of a scalar on S 1

I Consider a complex scalar in D-dimensional Minkowski space

∗ M 2 2 2 LD = −∂M ϕ ∂ ϕ − mD |ϕ| ⇒ (D − mD )ϕ = 0

JTL I The scalar equation then reduces to 2 2 X 2 2 iny/R (d + ∂y − mD )ϕ = 0 ⇒ (d − (n/R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R

Example: Reduction of a scalar on S 1

I Consider a complex scalar in D-dimensional Minkowski space

∗ M 2 2 2 LD = −∂M ϕ ∂ ϕ − mD |ϕ| ⇒ (D − mD )ϕ = 0

1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR

I Now expand in Fourier modes on the circle ∞ µ X µ iny/R ϕ(x , y) = φn(x )e n=−∞

JTL Example: Reduction of a scalar on S 1

I Consider a complex scalar in D-dimensional Minkowski space

∗ M 2 2 2 LD = −∂M ϕ ∂ ϕ − mD |ϕ| ⇒ (D − mD )ϕ = 0

1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR

I Now expand in Fourier modes on the circle ∞ µ X µ iny/R ϕ(x , y) = φn(x )e n=−∞

I The scalar equation then reduces to 2 2 X 2 2 iny/R (d + ∂y − mD )ϕ = 0 ⇒ (d − (n/R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R JTL 2 I The equations of motion( d − mn)φn = 0 can be obtained from the d-dimensional Lagrangian

X ∗ µ 2 2 Ld = [−∂µφn∂ φn − mn|φn| ] n

[Obtained by inspection, or by substituting the Fourier expansion

of ϕ into LD and integrating over the extra dimension]

The Kaluza-Klein tower

I A single complex scalar decomposes into an inﬁnite Kaluza-Klein tower {φn} with corresponding masses {mn}

I Furthermore, internal momentum on the circle pn = n/R can be identiﬁed with U(1) charge D-dimensional momentum conservation ⇔ d-dimensional charge conservation

JTL The Kaluza-Klein tower

I A single complex scalar decomposes into an inﬁnite Kaluza-Klein tower {φn} with corresponding masses {mn}

I Furthermore, internal momentum on the circle pn = n/R can be identiﬁed with U(1) charge D-dimensional momentum conservation ⇔ d-dimensional charge conservation

2 I The equations of motion( d − mn)φn = 0 can be obtained from the d-dimensional Lagrangian

X ∗ µ 2 2 Ld = [−∂µφn∂ φn − mn|φn| ] n

[Obtained by inspection, or by substituting the Fourier expansion

of ϕ into LD and integrating over the extra dimension]

JTL I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0

I We thus end up with two possibilities ϕ(xµ, y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ {φn(x )} φ0(x )

The zero-mode sector

I In general, for an interacting theory, if we had a field with charge n, we would also have to have fields of charge2 n,3 n, etc by charge conservation Once we include at least one field with n 6= 0, we must keep the entire Kaluza-Klein tower

JTL I We thus end up with two possibilities ϕ(xµ, y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ {φn(x )} φ0(x )

The zero-mode sector

I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0

JTL The zero-mode sector

I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0

I We thus end up with two possibilities ϕ(xµ, y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ {φn(x )} φ0(x )

JTL I The truncation is consistent if the d-dimensional equations of motion imply the D-dimensional equations of motion For the complex scalar on S 1, the zero-mode reduction ansatz µ µ ϕ(x , y) = φ0(x ) is a consistent truncation since

2 2 (d − mD )φ0 = 0 ⇒ (D − mD )ϕ = 0

I The zero-mode truncation is straightforward for torus reductions, but can be less obvious for more general Xn We can always obtain a consistent truncation by restricting to singlets under the isometry group of Xn

Consistent truncations

JTL I The zero-mode truncation is straightforward for torus reductions, but can be less obvious for more general Xn We can always obtain a consistent truncation by restricting to singlets under the isometry group of Xn

Consistent truncations

I If we expand in a complete set of modes on Xn, then we have simply rewritten the original D-dimensional theory in a d-dimensional manner I However, it is often useful to consider a consistent truncation Truncate to a ﬁnite set of d-dimensional ﬁelds (often just the zero modes) I The truncation is consistent if the d-dimensional equations of motion imply the D-dimensional equations of motion For the complex scalar on S 1, the zero-mode reduction ansatz µ µ ϕ(x , y) = φ0(x ) is a consistent truncation since

2 2 (d − mD )φ0 = 0 ⇒ (D − mD )ϕ = 0

JTL Consistent truncations

2 2 (d − mD )φ0 = 0 ⇒ (D − mD )ϕ = 0

JTL I We can work out the linearized Kaluza-Klein spectrum from the linearized Einstein equation

¯ M ¯ D hMN = 0, ∂ hMN = 0

Kaluza-Klein gravity

1 I Now consider the reduction of pure gravity on S

−1 1 e LD = R ⇒ RMN − 2 gMN R = 0 ⇒ RMN = 0

I There are several complications

– Tensor decomposition gMN −→ gµν , gµy , gyy – Gauge degrees of freedom from general coordinate transformations – The Einstein equation is non-linear

JTL Kaluza-Klein gravity

1 I Now consider the reduction of pure gravity on S

−1 1 e LD = R ⇒ RMN − 2 gMN R = 0 ⇒ RMN = 0

I There are several complications

– Tensor decomposition gMN −→ gµν , gµy , gyy – Gauge degrees of freedom from general coordinate transformations – The Einstein equation is non-linear

I We can work out the linearized Kaluza-Klein spectrum from the linearized Einstein equation

¯ M ¯ D hMN = 0, ∂ hMN = 0

JTL I The zero mode (n = 0) is massless and needs to be treated separately

¯0 µ¯0 µ¯0 d hMN = 0 ∂ hµν = 0 ∂ hµy = 0

Kaluza-Klein gravity

I Use the standard Kaluza-Klein decomposition

¯ µ X ¯n µ iny/R hMN (x , y) = hMN (x )e n

I The linearized Einstein equation reduces to n ( − m2)h¯n = 0 m = d n MN n R along with the gauge conditions

µ¯n ¯n µ¯n ¯n ∂ hµν + imnhyν = 0 ∂ hµy + imnhyy = 0

JTL Kaluza-Klein gravity

I Use the standard Kaluza-Klein decomposition

¯ µ X ¯n µ iny/R hMN (x , y) = hMN (x )e n

I The linearized Einstein equation reduces to n ( − m2)h¯n = 0 m = d n MN n R along with the gauge conditions

µ¯n ¯n µ¯n ¯n ∂ hµν + imnhyν = 0 ∂ hµy + imnhyy = 0

I The zero mode (n = 0) is massless and needs to be treated separately

¯0 µ¯0 µ¯0 d hMN = 0 ∂ hµν = 0 ∂ hµy = 0

JTL We can transform to the Einstein frame by shifting 2 h → hˆ − η φ µν µν d − 2 µν so that ˆ µ ˆ 1 ˆρ d hµν = 0 ∂ (hµν − 2 hρ) = 0

The zero-mode sector

I We decompose the zero-mode metric as 1 ρ 0 hµν Aµ ¯0 hµν − ηµν ( 2 hρ + φ) Aµ hMN = ⇒ hMN = 1 ρ Aν 2φ Aν φ − 2 hρ

I The zero-mode sector gives massless spins0,1 and2

d φ = 0 µ d Aµ = 0 ∂ Aµ = 0 µ 1 ρ d hµν = 0 ∂ (hµν − 2 hρ) = ∂ν φ

– Note that the dilaton mixes with the metric

JTL The zero-mode sector

I We decompose the zero-mode metric as 1 ρ 0 hµν Aµ ¯0 hµν − ηµν ( 2 hρ + φ) Aµ hMN = ⇒ hMN = 1 ρ Aν 2φ Aν φ − 2 hρ

I The zero-mode sector gives massless spins0,1 and2

d φ = 0 µ d Aµ = 0 ∂ Aµ = 0 µ 1 ρ d hµν = 0 ∂ (hµν − 2 hρ) = ∂ν φ

– Note that the dilaton mixes with the metric We can transform to the Einstein frame by shifting 2 h → hˆ − η φ µν µν d − 2 µν so that ˆ µ ˆ 1 ˆρ d hµν = 0 ∂ (hµν − 2 hρ) = 0

JTL Pure gravity reduced on a circle gives ‘Einstein-Maxwell-Dilaton’ gravity coupled to a tower of massive spin-2 ﬁelds

The Kaluza-Klein tower

I For massive (n 6= 0) Kaluza-Klein modes, the gauge condition M ∂ h¯MN = 0 can be used to solve for h¯µy and h¯yy

¯n i µ¯n ¯n 1 µ ν ¯n hyν = − ∂ hµν hyy = − 2 ∂ ∂ hµν mn mn

¯n I Thus only hµν is independent (there is still residual gauge freedom to ﬁx)

⇒ massive spin-2 with mass mn = n/R

JTL The Kaluza-Klein tower

I For massive (n 6= 0) Kaluza-Klein modes, the gauge condition M ∂ h¯MN = 0 can be used to solve for h¯µy and h¯yy

¯n i µ¯n ¯n 1 µ ν ¯n hyν = − ∂ hµν hyy = − 2 ∂ ∂ hµν mn mn

¯n I Thus only hµν is independent (there is still residual gauge freedom to ﬁx)

⇒ massive spin-2 with mass mn = n/R

Pure gravity reduced on a circle gives ‘Einstein-Maxwell-Dilaton’ gravity coupled to a tower of massive spin-2 ﬁelds

JTL I The full non-linear reduction ansatz can be obtained by gauging the U(1) isometry 2 µ ν 2φ µ 2 dsD = gµν dx dx + e (dy + Aµdx ) or 2φ 2φ gµν + e AµAν e Aµ gMN = 2φ 2φ e Aν e

Natural vielbein basis

a a µ 9 φ µ E = eµdx E = e (dy + Aµdx )

The zero-mode reduction

I In the zero-mode sector, we found that the D-dimensional graviton reduces to a combination of a d-dimensional graviton, gauge ﬁeld and real scalar hµν Aµ hMN → Aν 2φ

JTL Natural vielbein basis

a a µ 9 φ µ E = eµdx E = e (dy + Aµdx )

The zero-mode reduction

I In the zero-mode sector, we found that the D-dimensional graviton reduces to a combination of a d-dimensional graviton, gauge ﬁeld and real scalar hµν Aµ hMN → Aν 2φ

I The full non-linear reduction ansatz can be obtained by gauging the U(1) isometry 2 µ ν 2φ µ 2 dsD = gµν dx dx + e (dy + Aµdx ) or 2φ 2φ gµν + e AµAν e Aµ gMN = 2φ 2φ e Aν e

JTL The zero-mode reduction

I In the zero-mode sector, we found that the D-dimensional graviton reduces to a combination of a d-dimensional graviton, gauge ﬁeld and real scalar hµν Aµ hMN → Aν 2φ

I The full non-linear reduction ansatz can be obtained by gauging the U(1) isometry 2 µ ν 2φ µ 2 dsD = gµν dx dx + e (dy + Aµdx ) or 2φ 2φ gµν + e AµAν e Aµ gMN = 2φ 2φ e Aν e

Natural vielbein basis

a a µ 9 φ µ E = eµdx E = e (dy + Aµdx )

JTL Reduction of the Einstein equation

I The D-dimensional Einstein equation is simply RMN = 0 I Using the above reduction ansatz gives (in vielbein components)

(d) 1 2φ c 0 = Rab = Rab − ∂aφ∂bφ − ∇a∇bφ − 2 e Fac Fb 1 −2φ b 3φ 0 = Ra9 = − 2 e ∇ (e Fba) 2 1 2φ ab 0 = R99 = −∂φ − φ + 4 e FabF

I These equations of motion can be derived from a d-dimensional Lagrangian

−1 φ 1 3φ 2 e Ld = e R − 4 e Fµν

I Note that the Einstein term is non-standard Neither the string frame e−1L = e2ϕ[R + ··· ] nor the Einstein frame e−1L = R − · · ·

JTL I Choosing ρ = φ gives the string frame Lagrangian

−1 str (d−1)φ 2 1 2 e Ld = e [R + d(d − 1)∂φ − 4 Fµν ]

I Choosing ρ = −φ/(d − 2) gives the Einstein frame Lagrangian

−1 E d − 1 2 1 2 d−1 φ 2 e L = R − ∂φ − e d−2 F d d − 2 4 µν

This is the same shift we found in the linearized analysis

Weyl scaling

I We can perform a Weyl scaling

2ρ −2ρ 2 gµν → e gµν ⇒ R → e [R −(d −1)(d −2)∂ρ −2(d −2)ρ]

JTL I Choosing ρ = −φ/(d − 2) gives the Einstein frame Lagrangian

−1 E d − 1 2 1 2 d−1 φ 2 e L = R − ∂φ − e d−2 F d d − 2 4 µν

This is the same shift we found in the linearized analysis

Weyl scaling

I We can perform a Weyl scaling

2ρ −2ρ 2 gµν → e gµν ⇒ R → e [R −(d −1)(d −2)∂ρ −2(d −2)ρ]

I Choosing ρ = φ gives the string frame Lagrangian

−1 str (d−1)φ 2 1 2 e Ld = e [R + d(d − 1)∂φ − 4 Fµν ]

JTL Weyl scaling

I We can perform a Weyl scaling

2ρ −2ρ 2 gµν → e gµν ⇒ R → e [R −(d −1)(d −2)∂ρ −2(d −2)ρ]

I Choosing ρ = φ gives the string frame Lagrangian

−1 str (d−1)φ 2 1 2 e Ld = e [R + d(d − 1)∂φ − 4 Fµν ]

I Choosing ρ = −φ/(d − 2) gives the Einstein frame Lagrangian

−1 E d − 1 2 1 2 d−1 φ 2 e L = R − ∂φ − e d−2 F d d − 2 4 µν

This is the same shift we found in the linearized analysis

JTL I Note that

dA(3) = dC(3) + dB(2) ∧ dy = (dC(3) − dB(2) ∧ A1) + dB(2) ∧ (dy + A)

Fe(4) H(3)

Interplay between the form-ﬁeld reduction and the Kaluza-Klein (metric) gauge ﬁeld

Kaluza-Klein supergravity

I Reduction of supergravity proceeds in a similar manner, except we have to consider additional ﬁelds I For the case of 11-dimensional supergravity, we have the 3-form potential

AMNP → Aµνρ, Aµν 11 or A(3) = C(3) + B(2) ∧ dy

JTL Kaluza-Klein supergravity

I Reduction of supergravity proceeds in a similar manner, except we have to consider additional ﬁelds I For the case of 11-dimensional supergravity, we have the 3-form potential

AMNP → Aµνρ, Aµν 11 or A(3) = C(3) + B(2) ∧ dy

I Note that

dA(3) = dC(3) + dB(2) ∧ dy = (dC(3) − dB(2) ∧ A1) + dB(2) ∧ (dy + A)

Fe(4) H(3)

Interplay between the form-ﬁeld reduction and the Kaluza-Klein (metric) gauge ﬁeld

JTL Look for manifolds Xn that preserve partial supersymmetry

geometry ↔ symmetry ↔ supersymmetry

Reducing the number of supercharges

I Torus reductions lower the dimension of the theory, but preserve the supersymmetries

To reduce the number of supersymmetries, we must replace n T by some other Xn

I Although we can completely eliminate supersymmetry (a bosonic truncation), usually we still want to preserve some fraction of the original supersymmetries

JTL Reducing the number of supercharges

I Torus reductions lower the dimension of the theory, but preserve the supersymmetries

To reduce the number of supersymmetries, we must replace n T by some other Xn

I Although we can completely eliminate supersymmetry (a bosonic truncation), usually we still want to preserve some fraction of the original supersymmetries

Look for manifolds Xn that preserve partial supersymmetry

geometry ↔ symmetry ↔ supersymmetry

JTL I Since a classical background is purely bosonic, we are left with

δ(Fermion) = 0 for unbroken [or partially broken] supersymmetry

Conditions for unbroken supersymmetry

I Suppose the vacuum is invariant under a supersymmetry generated by Q Q|vaci = 0

I Then hvac|[Q, ϕ]|vaci = 0, which indicates that δϕ = 0 This must hold for all ﬁelds ϕ in the theory I In supergravity, we require

0 = δ(Fermion) = ∂(Boson) and 0 = δ(Boson) = ¯(Fermion)

JTL Conditions for unbroken supersymmetry

I Suppose the vacuum is invariant under a supersymmetry generated by Q Q|vaci = 0

I Then hvac|[Q, ϕ]|vaci = 0, which indicates that δϕ = 0 This must hold for all ﬁelds ϕ in the theory I In supergravity, we require

0 = δ(Fermion) = ∂(Boson) and 0 = δ(Boson) = ¯(Fermion)

I Since a classical background is purely bosonic, we are left with

δ(Fermion) = 0 for unbroken [or partially broken] supersymmetry

JTL Isometries of the background geometry ↔ Killing vectors Unbroken supersymmetry ↔ Killing spinors

Killing spinors

I In supergravity, we have at a minimum a graviton and one or more gravitinos

a i a δeµ = 4 γ¯ ψµ, δψµ = Dµ

I The condition for unbroken supersymmetry is then δψµ = 0 or Dµ = 0

This is known as the Killing spinor equation [There may be additional fermion variations that need to be satisﬁed as well]

JTL Killing spinors

I In supergravity, we have at a minimum a graviton and one or more gravitinos

a i a δeµ = 4 γ¯ ψµ, δψµ = Dµ

I The condition for unbroken supersymmetry is then δψµ = 0 or Dµ = 0

This is known as the Killing spinor equation [There may be additional fermion variations that need to be satisﬁed as well]

Isometries of the background geometry ↔ Killing vectors Unbroken supersymmetry ↔ Killing spinors

JTL I The Killing spinor equation then reduces to ∇i (y) = 0 on an n-dimensional Riemannian manifold ⇒ covariantly constant spinor

I The number of d-dimensional supersymmetries is given by the number of independent Killing spinors on Xn

Reduction of Poincar´esupergravity

I Consider the reduction of D-dimensional supergravity to d dimensions

1,D−1 1,d−1 M = M × Xn {x M } → {x µ} + {y i }

We split all D-dimensional fields into background plus fluctuations The simplest case has all background fields vanishing except the internal metric gij (y) on Xn

JTL Reduction of Poincar´esupergravity

I Consider the reduction of D-dimensional supergravity to d dimensions

1,D−1 1,d−1 M = M × Xn {x M } → {x µ} + {y i }

We split all D-dimensional fields into background plus fluctuations The simplest case has all background fields vanishing except the internal metric gij (y) on Xn

I The Killing spinor equation then reduces to ∇i (y) = 0 on an n-dimensional Riemannian manifold ⇒ covariantly constant spinor

I The number of d-dimensional supersymmetries is given by the number of independent Killing spinors on Xn

JTL – Make an ansatz and solve the ﬁrst order equations Coupled PDEs can still be a challenge

– Study the integrability condition

∇i = 0 ⇒ [∇i , ∇j ] = 0

– Investigate the holonomy group The group formed by parallel transport around closed loops

Solving the Killing spinor equation

I How do we go about looking for Killing spinors?

JTL – Study the integrability condition

∇i = 0 ⇒ [∇i , ∇j ] = 0

– Investigate the holonomy group The group formed by parallel transport around closed loops

Solving the Killing spinor equation

I How do we go about looking for Killing spinors?

– Make an ansatz and solve the ﬁrst order equations Coupled PDEs can still be a challenge

JTL – Investigate the holonomy group The group formed by parallel transport around closed loops

Solving the Killing spinor equation

I How do we go about looking for Killing spinors?

– Make an ansatz and solve the ﬁrst order equations Coupled PDEs can still be a challenge

– Study the integrability condition

∇i = 0 ⇒ [∇i , ∇j ] = 0

JTL Solving the Killing spinor equation

I How do we go about looking for Killing spinors?

– Make an ansatz and solve the ﬁrst order equations Coupled PDEs can still be a challenge

– Study the integrability condition

∇i = 0 ⇒ [∇i , ∇j ] = 0

– Investigate the holonomy group The group formed by parallel transport around closed loops

JTL Integrability of the Killing spinor equation

I Consider the integrability condition

1 kl ∇i = 0 ⇒ [∇i , ∇j ] = 0 → 4 Rijkl γ = 0

This is a condition on curvature for the existence of a Killing spinor I We can perform an additional contraction

kl j kl Rijkl γ = 0 ⇒ Rijkl γ γ = 0 This can be reexpressed using Dirac matrix identities as

j kl jkl jk l l 0 = Rijkl γ γ = Rijkl (γ + 2g γ ) = −2Ril γ

which is consistent with the vacuum Einstein equation Rij = 0

JTL Supersymmetry and the Einstein equation

I In Euclidean signature, we can demonstrate that the existence of at least one unbroken supersymmetry implies the Einstein equation For a ﬁxed iˆ

j k j Rijˆ γ = 0 ⇒ (Rikˆ γ )(Rijˆ γ ) = 0 kj kj jk ⇒ Rikˆ Rijˆ (γ + g ) = 0 ⇒ Rijˆ Rikˆ g = 0

2 jk I Since 6= 0, we must have |Rijˆ | ≡ Rijˆ Rikˆ g = 0

Assuming a positive definite metric, then all components of Rijˆ must vanish Ricci-flatness is a necessary condition for (partially) unbroken supersymmetry [The situation is more complicated in Lorentzian signature or with background fluxes turned on]

JTL Holonomy and supersymmetry

I Integrability can be given a geometrical interpretation

1 ab [∇i , ∇j ] = 2 Rij Σab whereΣ ab is a SO(n) rotation generator This represents the eﬀect of parallel transport of a spinor around an inﬁnitesimal loop

I The group of rotations acting on forms the holonomy group

H ⊆ SO(n) = spinor of SO(n)

I The number of preserved supersymmetries = the number of singlets in the decomposition of under SO(n) ⊃ H

JTL I In Lorentzian signature, the classiﬁcation is based on either a timelike or a null Killing vector ξµ =γ ¯ µ R.L. Bryant, math.DG/0004073

Holonomy classiﬁcation

I For an irreducible non-symmetric Riemannian metric, the possible holonomy groups are dim H Ricci flat? nSO (n) generic metric 2nU (n) complex Kähler 2nSU (n) yes Calabi-Yau 4n Sp(2n) yes hyperkähler 4n Sp(2n) × Sp(2) quaternionic 7 G2 yes exceptional holonomy 8 Spin(7) yes exceptional holonomy

M. Berger, Bull. Soc. Math. France 83, 225 (1955)

JTL Holonomy classiﬁcation

M. Berger, Bull. Soc. Math. France 83, 225 (1955)

I In Lorentzian signature, the classiﬁcation is based on either a timelike or a null Killing vector ξµ =γ ¯ µ R.L. Bryant, math.DG/0004073

JTL I Another look at the table of supersymmetric theories D 32 168 11 M 10 IIAI 9 N = 2 N = 1 8 N = 2 N = 1 7 N = 4 N = 2 6 (2, 2) (1, 1) (1, 0) 5 N = 8 N = 4 N = 2 4 N = 8 N = 4 N = 2

Torus compactiﬁcation revisited

n I For T , we have trivial holonomy (the space is ﬂat) Maximally symmetric and maximally supersymmetric

∇i = 0 ⇒ ∂i = 0 ⇒ (y) = 0

JTL Torus compactiﬁcation revisited

n I For T , we have trivial holonomy (the space is ﬂat) Maximally symmetric and maximally supersymmetric

∇i = 0 ⇒ ∂i = 0 ⇒ (y) = 0

I Another look at the table of supersymmetric theories D 32 168 11 M 10 IIAI 9 N = 2 N = 1 8 N = 2 N = 1 7 N = 4 N = 2 6 (2, 2) (1, 1) (1, 0) 5 N = 8 N = 4 N = 2 4 N = 8 N = 4 N = 2

JTL The case of SU(2) holonomy

I In four Euclidean dimensions, the tangent space group is

SO(4) = SU(2)+ × SU(2)−

I The spinor of SO(4) transforms as 4 = (2, 1) + (1, 2), corresponding to the decomposition

5 = + + − ± = ±γ ±

I This allows us to rewrite the integrability condition as

ab +ab −ab Rij γab = 0 ⇒ Rij γab− = 0 and Rij γab+ = 0

±ab 1 c d 1 cd where Rij = 2 (δa δb ± 2 ab )Rijcd

JTL I A compact manifold of SU(2) holonomy is known as a K3 surface See eg Aspinwall, K3 Surfaces and string duality, hep-th/9611137

The case of SU(2) holonomy

−ab I Killing spinors exist for a self-dual connection, Rij = 0

+ab Rij γab− = 0 ⇒ − = 0 with + allowed

[+ must still solve the ﬁrst order equation ∇i + = 0] I In terms of holonomy

4 = (2, 1) + (1, 2) → 2 + 1 + 1 under SO(4) ⊃ SU(2)−

I Thus X4 with SU(2) holonomy preserves half of the 1 supersymmetries ⇒ 2 -BPS

JTL The case of SU(2) holonomy

−ab I Killing spinors exist for a self-dual connection, Rij = 0

+ab Rij γab− = 0 ⇒ − = 0 with + allowed

[+ must still solve the ﬁrst order equation ∇i + = 0] I In terms of holonomy

4 = (2, 1) + (1, 2) → 2 + 1 + 1 under SO(4) ⊃ SU(2)−

I Thus X4 with SU(2) holonomy preserves half of the 1 supersymmetries ⇒ 2 -BPS

I A compact manifold of SU(2) holonomy is known as a K3 surface See eg Aspinwall, K3 Surfaces and string duality, hep-th/9611137

JTL Calabi-Yau (complex) 3-fold

I M on CY3 → D = 5, N = 2 IIA or IIB on CY3 → D = 4, N = 2 Heterotic on CY3 → D = 4, N = 1

The case of SU(3) holonomy

I In six Euclidean dimensions, the tangent space group is

SO(6) = SU(4)

I For SU(3) holonomy, the complex spinor decomposes as

SO(6) ⊃ U(3) ⊃ SU(3)

4 → 3−1 + 13 → 3 + 1

I The resulting manifold is Ricci-ﬂat and K¨aherand preserves a quarter of the supersymmetries

JTL The case of SU(3) holonomy

I In six Euclidean dimensions, the tangent space group is

SO(6) = SU(4)

I For SU(3) holonomy, the complex spinor decomposes as

SO(6) ⊃ U(3) ⊃ SU(3)

4 → 3−1 + 13 → 3 + 1

I The resulting manifold is Ricci-ﬂat and K¨aherand preserves a quarter of the supersymmetries

Calabi-Yau (complex) 3-fold

I M on CY3 → D = 5, N = 2 IIA or IIB on CY3 → D = 4, N = 2 Heterotic on CY3 → D = 4, N = 1

JTL I Look at scalars, p-forms and the metric [Fermions can also be analyzed]

Linearized Kaluza-Klein reduction

I The linearized Kaluza-Klein spectrum is obtained by expanding the fluctuations of the D-dimensional fields in harmonics on Xn (0) I Background plus fluctuations:Φ D = ΦD + ϕ µ i µ i I Expand on Xn: ϕ(x , y ) = φ(x )f (y ) i I Solve for the spectrum of f (y ) on Xn and rewrite the D-dimensional eom in terms of this spectrum I For compactifications without fluxes, there is a separation between massless modes and the massive Kaluza-Klein tower We can remove the massive tower by (formally) taking the size of the compactification manifold to zero The massless d-dimensional modes are obtained by restricting to zero modes on Xn

JTL Linearized Kaluza-Klein reduction

I Look at scalars, p-forms and the metric [Fermions can also be analyzed]

JTL I Zero modes: For compact Xn, there is a single zero mode

f (y) = constant

Reduction of scalars

I The reduction of a D-dimensional scalar is straightforward

∆0 ≡ −D = −d − n

Writing ϕ(x µ, y i ) = φ(x µ)f (y i ) gives

∆0 = −d + λ

where nf (y) = −λf (y)

JTL Reduction of scalars

I The reduction of a D-dimensional scalar is straightforward

∆0 ≡ −D = −d − n

Writing ϕ(x µ, y i ) = φ(x µ)f (y i ) gives

∆0 = −d + λ

where nf (y) = −λf (y)

I Zero modes: For compact Xn, there is a single zero mode

f (y) = constant

JTL I A p-form can be decomposed on Md × Xn µ i µ i Ap(x , y ) = αq(x ) ∧ βp−q(y )

I ∆p then reduces as

∆p = ∆q(Md ) + ∆p−q(Xn) with the result ∆p = ∆q(Md ) + λ

where∆ p−qβ = λβ

p-form gauge ﬁelds

I Consider a p-form potential with ﬁeld strength Fp+1 = dAp I The Bianchi identity and (source-free) equation of motion are dF = 0 and d ∗ F = 0 I In Lorenz gauge d ∗ A = 0, the equation of motion can be expressed in terms of the Hodge-de Rham operator

∗ ∗ ∆p = d d + dd where d∗ = (−1)n(p+1)+1 ∗ d∗ is the codiﬀerential on p-forms with n the dimension of the manifold

JTL p-form gauge ﬁelds

∗ ∗ ∆p = d d + dd where d∗ = (−1)n(p+1)+1 ∗ d∗ is the codiﬀerential on p-forms with n the dimension of the manifold I A p-form can be decomposed on Md × Xn µ i µ i Ap(x , y ) = αq(x ) ∧ βp−q(y )

I ∆p then reduces as

∆p = ∆q(Md ) + ∆p−q(Xn) with the result ∆p = ∆q(Md ) + λ

where∆ p−qβ = λβ JTL I Consider the2-form BMN → Bµν, Bµi , Bij

Bµν = bµν (x) b0 = 1

Bµi = bµ(x)fi (y) b1

Bij = b(x)fij (y) b2

p-form gauge ﬁelds

I Zero modes: Zero modes are given by( p − q)-forms β on Xn satisfying dβ = 0 But we do not want β to be pure gauge, so we take β 6= dΛ ⇒ zero modes are counted by cohomology

number of zero modes of βp−q = bp−q

(bp−q = the( p − q)-th Betti number)

JTL p-form gauge ﬁelds

I Zero modes: Zero modes are given by( p − q)-forms β on Xn satisfying dβ = 0 But we do not want β to be pure gauge, so we take β 6= dΛ ⇒ zero modes are counted by cohomology

number of zero modes of βp−q = bp−q

(bp−q = the( p − q)-th Betti number)

I Consider the2-form BMN → Bµν, Bµi , Bij

Bµν = bµν (x) b0 = 1

Bµi = bµ(x)fi (y) b1

Bij = b(x)fij (y) b2

JTL I We decompose hMN → hµν, hµi , hij

hµν = hµν (x) d-dimensional graviton

hµi = Aµ(x)fi (y) gauged isometries of Xn

hij = φ(x)hij (y) moduli

In the Ricci-ﬂat case, the moduli are determined by demanding (0) Rij (g + h) = 0, ie the Einstein equation on Xn

Metric ﬂuctuations

I We decompose the metric into background plus ﬂuctuations (0) gMN = gMN + hMN This is just linearized gravity I The modes are governed by the Lichnerowicz operator

PQ P ∆LhMN = −hMN − 2RMPNQ h + 2R(M hN)Q (acting on transverse traceless ﬂuctuations)

JTL Metric ﬂuctuations

I We decompose the metric into background plus ﬂuctuations (0) gMN = gMN + hMN This is just linearized gravity I The modes are governed by the Lichnerowicz operator

PQ P ∆LhMN = −hMN − 2RMPNQ h + 2R(M hN)Q (acting on transverse traceless ﬂuctuations)

I We decompose hMN → hµν, hµi , hij

hµν = hµν (x) d-dimensional graviton

hµi = Aµ(x)fi (y) gauged isometries of Xn

hij = φ(x)hij (y) moduli

In the Ricci-ﬂat case, the moduli are determined by demanding (0) Rij (g + h) = 0, ie the Einstein equation on Xn

JTL I Metric ﬂuctuations are given by

1,1 K¨ahler: hij¯ h j 2,1 complex structure: hi¯ Ωjkl h

I The other ﬁelds depend on the model

Calabi-Yau compactiﬁcation

I For Calabi-Yau compactiﬁcations, we make further use of K¨ahlergeometry The Hodge diamond is given by

h0,0 1 h1,0 h0,1 0 0 h2,0 h1,1 h0,2 0 h1,1 0 h3,0 h2,1 h1,2 h0,3 = 1 h2,1 h2,1 1 3,1 2,2 1,3 1,1 h h h ¡ m0 h 0 h3,2 h2,3 ¡ 0 0 h3,3 Ωijk 1

JTL Calabi-Yau compactiﬁcation

I For Calabi-Yau compactiﬁcations, we make further use of K¨ahlergeometry The Hodge diamond is given by

h0,0 1 h1,0 h0,1 0 0 h2,0 h1,1 h0,2 0 h1,1 0 h3,0 h2,1 h1,2 h0,3 = 1 h2,1 h2,1 1 3,1 2,2 1,3 1,1 h h h ¡ m0 h 0 h3,2 h2,3 ¡ 0 0 h3,3 Ωijk 1

I Metric ﬂuctuations are given by

1,1 K¨ahler: hij¯ h j 2,1 complex structure: hi¯ Ωjkl h

I The other ﬁelds depend on the model

JTL h1,1 vectors and h2,1 + 1 hypers

IIA on CY3

I IIA supergravity contains the bosonic ﬁelds

gMN , BMN , φ, AM , AMNP NSNSRR

I The zero mode reduction gives N = 2 in four dimensions

IIA ﬁeld1 h1,1 h2,1 1

hMN hµν hij¯ hij + hi¯j¯ BMN bij¯ bµν φφ AM Aµ AMNP Aµij¯ Aij¯k¯ + Aijk¯ Aijk + Ai¯j¯k¯ gravity vector hyper hyper

JTL IIA on CY3

I IIA supergravity contains the bosonic ﬁelds

gMN , BMN , φ, AM , AMNP NSNSRR

I The zero mode reduction gives N = 2 in four dimensions

IIA ﬁeld1 h1,1 h2,1 1

hMN hµν hij¯ hij + hi¯j¯ BMN bij¯ bµν φφ AM Aµ AMNP Aµij¯ Aij¯k¯ + Aijk¯ Aijk + Ai¯j¯k¯ gravity vector hyper hyper

h1,1 vectors and h2,1 + 1 hypers

JTL h2,1 vectors and h1,1 + 1 hypers

IIB on CY3

I IIB supergravity diﬀers in the RR sector

+ gMN , BMN , φ, χ, AMN , AMNPQ NSNSRR

I The zero mode reduction gives N = 2 in four dimensions

IIA ﬁeld1 h1,1 h2,1 1

hMN hµν hij¯ hij + hi¯j¯ BMN bij¯ bµν φφ χχ

AMN Aij¯ Aµν + AMNPQ Aµijk Aµνij¯ Aµijk¯ gravity hyper vector hyper

JTL IIB on CY3

I IIB supergravity diﬀers in the RR sector

+ gMN , BMN , φ, χ, AMN , AMNPQ NSNSRR

I The zero mode reduction gives N = 2 in four dimensions

IIA ﬁeld1 h1,1 h2,1 1

hMN hµν hij¯ hij + hi¯j¯ BMN bij¯ bµν φφ χχ

AMN Aij¯ Aµν + AMNPQ Aµijk Aµνij¯ Aµijk¯ gravity hyper vector hyper

h2,1 vectors and h1,1 + 1 hypers

JTL Mirror symmetry

1,1 2,1 I The IIA and IIB reductions interchange h and h – Mirror symmetry I In the language of D = 4, N = 2 scalars in the vector multiplets live on a special K¨ahler manifold M scalars in the hypermultiplets live on a quaternionic manifold Q

I The moduli spaces are thus

vectors hypers

IIA : Mh1,1 × Qh2,1+1

IIB : Mh2,1 × Qh1,1+1 6 universal hypermultiplet

JTL Next time

I In the next (final) lecture, we examine the Freund-Rubin (AdS×Sphere) compactification Turn on volume filling fluxes I We also revisit consistent truncations – When is it possible to retain massive states in a consistent truncation?

JTL