Exact solutions in supergravity

Exact solutions in supergravity

James T. Liu 25 July 2005

Lecture 1: Introduction and overview of supergravity Lecture 2: Conditions for unbroken supersymmetry Lecture 3: BPS black holes and branes

Lecture 4: The LLM bubbling AdS construction

Lecture 1 Outline

• A quick introduction to supersymmetry

The supersymmetry algebra

Local supersymmetry as supergravity

• Eleven dimensional supergravity

Field content, Lagrangian and transformation rules

• Ten dimensional supergravities

The low energy limit of superstring theory A look at IIA, IIB and N = 1 supergravity

Supersymmetry

• Supersymmetry relates bosons with fermions

Fermion ↔ Boson or

δ(Fermion) = ∂(Boson), δ(Boson) = (Fermion) where  = spinor parameter of transformation

• For example, in the Wess-Zumino model (on-shell formulation)

L = −¹₂∂A² − ¹₂∂B² − ¹₂iχγ^µ∂_µχ (two real scalars A, B and one Majorana fermion χ)

δA = ¹₂iχ, δB = ¹₂γ₅χ, δχ = ¹₂γ^µ∂_µ(A − iγ₅B)

Invariance of the action

• For the Wess-Zumino model, we can check

δL = δL_B + δL_F = −∂^µA∂_µδA − ∂^µB∂_µδB − iχγ^µ∂_µδχ

= −¹₂i∂^µ(A + iγ₅B)∂_µχ − ¹₂iχγ^µ∂_µγ^ν∂_ν(A − iγ₅B)

= ¹₂i (A + iγ⁵B)χ − ¹₂iχ (A − iγ⁵B)

= 0 (up to integration by parts)

• Furthermore, the commutator of two transformations yields

[δ₁, δ₂]A = ¹₂ξ^µ∂_µA [δ₁, δ₂]B = ¹₂ξ^µ∂_µB

[δ₁, δ₂]χ = ¹₂ξ^µ∂_µχ − ¹₄γ^µξ_µγ^µ∂_µχ where ξ^µ = i₂γ^µ₁

The supersymmetry algebra

• An extension of the Poincar´e algebra

In D = 4: M_µν = Lorentz generators P_µ = Translations

Qⁱ_α = “supertranslations” where i = 1, . . . , N with

[M_µν, M_ρσ] = i(η_µρM_νσ − η_µσM_νρ + η_νσM_µρ − η_νρM_µσ) [M_µν, P_ρ] = i(η_µρP_ν − η_νρP_µ)

[M_µν, Qⁱ_α] = ¹₂i(γ_µν)_αβQ^{β i}

{Qⁱ_α, Q^j_β} = 2(γ^µC)_αβδ^ijP_µ + C_αβX^[ij] + (γ₅C)_αβY ^[ij]

The central charge Z = X + iY commutes with all generators and plays an important role in extended supersymmetry

Extended supersymmetry

• For D = 4 there is an upper limit on N

Consider massless representations of supersymmetry (given by helicities) N = 1: (0, ¹₂) Wess-Zumino; (¹₂, 1) vector; (³₂, 2) graviton

N = 2: (−¹₂, 0, ¹₂) hypermatter; (0, ¹₂, 1) vector; (1, ³₂, 2) graviton N = 4: (−1, −¹₂, 0, ¹₂, 1) vector; (0, ¹₂, 1, ³₂, 2) graviton

N = 8: (−2, −³₂, −1, −¹₂, 0, ¹₂, 1, ³₂, 2) graviton

• Spins less than or equal to 2 give a restriction

N ≤ 8 in D = 4

and N = 8 must include gravity

Local supersymmetry is supergravity

• In fact, gravity shows up naturally once we impose local supersymmetry

 → (x)

Consider the action of two supersymmetries

[δ₁, δ₂](field) = [₁Q, ₂Q](field) = ξ^µ(x)∂_µ(field) + · · ·

where ξ^µ(x) = i₂γ^µ₁

• This is a general coordinate transformation, and the gravitino ψ_µ(x) acts as a spin-³₂ gauge field

Simple supergravity in D = 4

• Constructed by Freedman, van Nieuwenhuizen and Ferrara and Deser and Zumino in 1976

• To lowest order in fermions

e⁻¹L = R + ¹₂iψ_µγ^µνσ∇_νψ_σ with δψ_µ = ∇_µ δe_µ^α = ¹₄iγ^αψ_µ

• We can check

e⁻¹δL = (R^µν − ¹₂g^µνR)δg_µν + iψ_µγ^µνσ∇νδψ_σ

= ¹₂i(R^µν − ¹₂g^µνR)γ_µψ_ν + iψ_µγ^µνσ∇_ν∇_σ

= ¹₂i(R^µν − ¹₂g^µνR)γ_µψ_ν + ¹₈iψ_µγ^µνσR_νσληγ^λη

= ¹₂i(R^µν − ¹₂g^µνR)γ_µψ_ν + ¹₂i(R^µν − ¹₂g^µνR)ψ_µγ_ν = 0

A general coordinate transformation

• In addition, we verify explicitly that the commutator of two supersymmetries yields a general coordinate transformation

[δ₁, δ₂]e_µ^α = ¹₄i(₂γ^α∇µ₁ − 1γ^a∇µ₂) = ∇_µξ^a

where ξ^a = ¹₄i₂γ^a₁

• This implies that

[δ₁, δ₂]g_µν = 2∇_(µξ_ν) = ∇_µξ_ν + ∇_νξ_µ

which is the expected result

Closure of the algebra on fermions is more involved, as it involves (fermi)² terms in δψ_µ

An upper limit on spacetime dimension

• In four dimensions, we found the restriction N ≤ 8

We can also consider the restriction on spacetime dimension D

• Need to match fermions and bosons in D dimensions

Bosons Fermions

scalar 1 Dirac spinor ¹₂ × 2^bD/2c complex

vector D − 2 Dirac gravitino ¹₂(D − 3) × 2^bD/2c complex n-form ` n

D−2

´

graviton ¹₂(D − 1)(D − 2) − 1

Eventually the fermion degrees of freedom grow too large

For spins less than or equal to 2, the maximum is D = 11, N = 1

Eleven dimensional supergravity

• The maximum dimension for (conventional) supergravity is eleven

A relatively simple theory with field content

g_{M N} 44 metric

A_{M N P} 84 3-form potential ψ_M 128 gravitino

• This can be related to the D = 4, N = 8 theory by dimensional reduction

Kaluza-Klein reduction on T⁷

In fact, the arguments for maximum N and maximum D are closely related

D = 11 supergravity: the setup

• Eleven dimensional supergravity was constructed in 1976 by Cremmer, Julia and Scherk

For fermions coupled to gravity, introduce a vielbein and spin-connection M, N, P, . . . = curved indices, A, B, C, . . . = flat indices

Work in signature (− + · · · +) with

{Γ_A, Γ_B} = 2η_AB, ψ_M = ψ_M^T C⁻¹, C⁻¹Γ_AC = −Γ^T_A

The spin-connection is given by ω_M^AB = ω_M^AB(e) + K_M^AB with contorsion K_M^AB = −₁₆¹ (ψ_NΓ_M^{ABN P}ψ_P + 4ψ_MΓ^[Aψ^B] + 2ψ^AΓ_Mψ^B)

D = 11 supergravity: the Lagrangian

• The supergravity Lagrangian is given by

e⁻¹L₁₁ = R(ω) − _2·4!¹ F_{M N P Q}² + ¹₆(_4!4!3!¹ εM N P QRST U V W X

F_{M N P Q}F_{RST U}A_{V W X})

−¹₂iψ_MΓ^{M N P}∇_N(¹₂(ω + ˆω))ψ_p

−_8·4!¹ i(ψ_MΓ^{M N P QRS}ψ_N + 12ψ^PΓ^QRψ^S)(¹₂(F + ˆF ))_{P QRS}

where the supercovariant quantities are

ˆ

ω_M^AB = ω_M^AB + ₁₆¹ ψ_NΓ_M^{ABN P}ψ_P Fˆ_{M N P Q} = F_{M N P Q} + ³₂iψ_[MΓ_{N P}ψ_Q]

• The bosonic Lagrangian is simple (in a form notation)

L₁₁ = R ∗1 − ¹₂F₍₄₎ ∧ ∗F₍₄₎ + ¹₆F₍₄₎ ∧ F₍₄₎ ∧ A₍₃₎

D = 11 supergravity: equations of motion

• The above Lagrangian gives rise to the equations of motion

R_{M N}( ˆω) − ¹₂g_{M N}R( ˆω) = ₁₂¹ ( ˆF_{M P QR}Fˆ_N^{P QR} − ¹₈g_{M N}Fˆ²) Γ^{M N P}∇ˆ_N( ˆω)ψ_P = 0

∇_M( ˆω) ˆF^{M N P Q} + ¹₂(_4!4!¹ εN P QRST U V W XYFˆ_{RST U}Fˆ_{V W XY} ) = 0 where the supercovariant derivative is

∇ˆM = ∇_M + ₂₈₈¹ (Γ_M^{N P QR} − 8δ_M^NΓ^{P QR})F_{N P QR}

• Often we are only concerned with the bosonic equations

R_{M N} = ₁₂¹ (F_{M P QR}F_N^{P QR} − ₁₂¹ g_{M N}F²) d ∗F₍₄₎ + ¹₂F₍₄₎ ∧ F₍₄₎ = 0

D = 11 supergravity: transformation laws

• The above system has a large set of symmetries

− general covariance

− SO(1, 10) local Lorentz symmetry

− abelian 3-form gauge symmetry δA₍₃₎ = dΛ₍₂₎

− N = 1 local supersymmetry

δe_M^A = ¹₄iΓ^Aψ_M

δA_{M N P} = −³₄iΓ_{[M N}ψ_{P ]} δψ_M = ∇ˆ_M( ˆω)

• Closure of supersymmetry is expressed as

[δ₁, δ₂] = δ_g.c. + δ_l.l. + δ_gauge + δ_susy + e.o.m.

Connection to string theory

• We will return to eleven dimensional supergravity

But note that supergravity by itself cannot be the ultimate theory UV divergences of quantum gravity not really cured by supersymmetry

• May be viewed as the low energy limit of superstring theory (M-theory)

Useful as an effective theory below the Planck scale

• String theory gives rise to a tower of massive states

1

4α⁰M² = 0, 1, 2, . . . (closed string)

by integrating out the massive fields, we are left with a theory of

• massless fields • spin-2 graviton • supersymmetry

This must be supergravity

Ten dimensional supersymmetry

• There are five consistent supersymmetric string theories and three supergravities in D = 10

SUSY String Supergravity

32 Type IIA N = 2A (non-chiral)

Type IIB N = 2B (chiral)

16 E₈ × E₈ Heterotic N = 1 SO(32) Heterotic N = 1 SO(32) Type I N = 1

• In fact, these D = 10 string theories can be related to each other and to D = 11 supergravity through dualities

D=11 HE

HO

I

IIB IIA

Massless string states

• In the RNS formulation, spacetime supersymmetry is obtained form worldsheet supersymmetry and the GSO projection

• The open string tower of states

1

4α⁰M² Bosonic Fermionic

... ... ...

1 ψ_−1/2^µ ψ_−1/2^ν ψ_−1/2^ρ |N Si₊, ψ_−3/2^µ |N Si₊ α^µ₋₁|Ri₊, α^µ₋₁|Ri₋ ψ_−1/2^µ α^ν₋₁|N Si+ ψ₋₁^µ |Ri−, ψ₋₁^µ |Ri+ 1

2 ψ_−1/2^µ ψ_−1/2^ν |N Si₋, α^µ₋₁|N Si₋

0 ψ_−1/2^µ |N Si₊ |Ri₊, |Ri₋

−¹₂ |N Si−

• After GSO, the massless states are ψ_−1/2^µ |N Si₊ and |Ri₊

8_v + 8_s → Physical states of D = 10, N = 1 super-Yang-Mills

Massless string states

• For the closed string, we take two copies (left and right movers) Type IIA (non-chiral combination)

(8_v + 8_s)_L ⊗ (8v + 8_c)_R = (1 + 28 + 35)_{N SN S} + (8_v + 56_t)_RR +(8_c + 56_s + 8_s + 56_c)_{RN S+N SR} Bosonic states: (g_µν, B_µν, φ)_{N SN S} + (F₍₂₎, F₍₄₎)_RR

Type IIB (chiral)

(8_v + 8_s)_L ⊗ (8v + 8 + s)_R = (1 + 28 + 35)_{N SN S} + (1 + 28 + 35_t)_RR +(8_c + 56_s + 8_c + 56_s)_{RN S+N SR} Bosonic states: (g_µν, B_µν, φ)_{N SN S} + (F₍₁₎, F₍₃₎, F₍₅₎⁺ )_RR

• The N SN S sector is identical (also for the Heterotic string)

Type IIA supergravity

• The IIA theory may be obtained by reduction from eleven dimensions

• The Kaluza-Klein idea

Split your spacetime

MD = M_d × T^D−d, x^M = (x^µ, y^m) Assume fields are independent of the internal coordinates

Φ(x^M) = ϕ(x^µ)

• This may be generalized to non-trivial internal manifolds as well

T^D−d → X^D−d and Φ(x^M) = ϕ_i(x^µ)fⁱ(y^m)

where fⁱ(y^m) are harmonics on X

Type IIA supergravity: reduction from D = 11

• Consider the bosonic sector of D = 11 supergravity

g_{M N} → g_µν, g_{µ 11}, g_{11 11} A_{M N P} → A_µνρ, A_{µν 11} These are the fields of IIA supergravity

• For the actual reduction, write

ds²₁₁ = e^{− 23φ}ds²₁₀ + e^43φ(dy + C₍₁₎)² A₍₃₎ = C₍₃₎ + B₍₂₎ ∧ dy

This yields the ten-dimensional fields

(g_µν, B₍₂₎, φ) + (C₍₁₎, C₍₃₎)

Type IIA supergravity: the bosonic Lagrangian

• We insert the above reduction ansatz into the D = 11 Lagrangian

L₁₁ = R ∗1 − ¹₂F₍₄₎ ∧ ∗F₍₄₎ + ¹₆F₍₄₎ ∧ F₍₄₎ ∧ A₍₃₎

to obtain

L_IIA = e^−2φ[R ∗1 + 4dφ ∧ ∗dφ − ¹₂H₍₃₎ ∧ ∗H₍₃₎]

−¹₂F₍₂₎ ∧ ∗F₍₂₎ − ¹₂Fe₍₄₎ ∧ ∗ eF₍₄₎

−¹₂F₍₄₎ ∧ F₍₄₎ ∧ B₍₂₎

where H₍₃₎ = dB₍₂₎, F₍₂₎ = dC₍₁₎, F₍₄₎ = dC₍₃₎ and

Fe₍₄₎ = F₍₄₎ − C₍₁₎ ∧ H₍₃₎

D = 10, N = 1 supergravity

• Type IIA supergravity admits a further truncation to N = 1

We obtain the gravity sector of the Heterotic string by removing the RR sector (C₍₁₎, C₍₃₎) → 0

L_{N =1} = e^−2φ[R ∗1 + 4dφ ∧ ∗dφ − ¹₂H₍₃₎ ∧ ∗H₍₃₎]

• This theory is anomalous, but can be cured by adding an E₈ × E₈ or SO(32) gauge sector

The result is the effective Lagrangian for the Heterotic string

L_het = e^−2φ[R ∗1 + 4dφ ∧ ∗dφ − ¹₂He₍₃₎ ∧ ∗ eH₍₃₎ + α⁰Tr_vF₍₂₎ ∧ ∗F₍₂₎]

where He₍₃₎ = dB₍₂₎ − ¹₄α⁰ω₍₃₎^{Y M} with ω₍₃₎^{Y M} = Tr_v(A₍₁₎dA₍₁₎ + ²₃A³₍₁₎)

Type IIB supergravity

• The final possibility in ten dimensions (N = 2B) cannot be obtained from dimensional reduction

• This theory has a self-dual field strength F₍₅₎ = ∗F₍₅₎, and hence does not admit a covariant action formulation

Self-dual fields also show up in six-dimensional N = (1, 0) and N = (2, 0) supergravity

• However, we can write down an action that gives rise to all equations of motion except the self-dual condition (which must subsequently be imposed by hand)

• Recall that we have the fields

(g_µν, B₍₂₎, φ) + (C₍₀₎, C₍₂₎, C₍₄₎)

Type IIB supergravity: the bosonic Lagrangian

• The IIB Lagrangian has the form

L_IIB = e^−2φ[R ∗1 + 4dφ ∧ ∗dφ − ¹₂H₍₃₎ ∧ ∗H₍₃₎]

−¹₂F₍₁₎ ∧ ∗F₍₁₎ − ¹₂Fe₍₃₎ ∧ ∗ eF₍₃₎ − ¹₄Fe₍₅₎ ∧ ∗ eF₍₅₎

−¹₂C₍₄₎ ∧ H₍₃₎ ∧ F₍₃₎

where H₍₃₎ = dB₍₂₎, F₍₁₎ = dC₍₀₎, F₍₃₎ = dC₍₂₎, F₍₅₎ = dC₍₄₎ and

Fe₍₃₎ = F₍₃₎ − C₍₀₎ ∧ H₍₃₎, Fe₍₅₎ = F₍₅₎ − ¹₂C₍₂₎ ∧ H₍₃₎ + ¹₂B₍₂₎ ∧ F₍₃₎

• In addition, we have to impose F₍₅₎ self-duality by hand on the solution

Supergravities in ten and eleven dimensions

• A summary of the four supergravities we have considered

D = 11, N = 1: g_{M N} A₍₃₎

D = 10, N = 2A: g_µν φ C₍₁₎ B₍₂₎ C₍₃₎

D = 10, N = 2B: g_µν φ, C₍₀₎ B₍₂₎, C₍₂₎ C₍₄₎

D = 10, N = 1: g_µν φ A^a₍₁₎ B₍₂₎

Why study supergravity?

• Backgrounds for string compactification

Calabi-Yau manifolds, flux compactifications

• Low energy dynamics of string theory

String solutions, black holes, D-brane probes of geometry

• AdS/CFT, gravity duals to supersymmetric gauge theories

• Phenomenology and string cosmology

Also braneworlds and large extra dimensions

Supersymmetric backgrounds

• Many of the exact results in string theory and AdS/CFT depend on understanding backgrounds with some fraction of unbroken supersymmetry, ie BPS configurations

use of powerful non-renormalization theorems

{Q, Q} ∼ P + Z → M ≥ |Z|

Single particle representations with M = |Z| have discretely fewer states than those with M > |Z|

Since we do not expect representations to change discontinuously, these states are protected, and are know as BPS states

• Allows us to investigate connections between strong and weak coupling

Next time

• In the next lecture, we will examine the conditions for unbroken supersymmetry, and will look at some examples of supersymmetric vacua preserving various fractions of supersymmetry

• In the third lecture, we will look at the supersymmetry of BPS branes and AdS vacua and will introduce the notion of generalized holonomy as a means of classifying supersymmetric backgrounds with fluxes

• In the final lecture, we will review the recent construction of bubbling 1/2 BPS solutions of IIB theory by Lin, Lunin and Maldacena (LLM)