Exact solutions in supergravity theory

Exact solutions in supergravity theory

James T. Liu 26 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 3 Outline

• D = 11 supergravity with F₍₄₎ 6= 0

The AdS₄ × S⁷ (Freund-Rubin) background The supergravity M2-brane

• Generalized holonomy when F₍₄₎ 6= 0

Definition of generalized holonomy A look at the M2-brane

Turning on fluxes

• We now consider the general case with charges or fluxes turned on

For example, we can consider D = 11 supergravity

∇ˆ_M = 0 ∇ˆ_M = ∇_M + ₂₈₈¹ (Γ_M^{N P QR} − 8δ_M^NΓ^{P QR})F_{N P QR}

• Different approaches to solving this system

− Ansatz based on symmetry

− Generalized holonomy

− G-structure analysis

− Combination of methods, etc.

Use of symmetry: the Freund-Rubin ansatz

• The presence of F₍₄₎ hints at a natural compactification to four dimensions, M¹¹ = M⁴ × X⁷

dˆs²₁₁ = ds²₄ + ds²₇(X⁷) F₍₄₎ = m₍₄₎

• Solve the equations of motion

F₍₄₎ eom: dF₍₄₎ = 0 and d ∗F₍₄₎ + ¹₂F₍₄₎ ∧ F₍₄₎ = 0

⇒ m = constant

Einstein: R_{M N} = ₁₂¹ (F_{M N}² − ₁₂¹ g_{M N}F²) Decompose D = 11 indices:

µ, ν, . . . = 0, 1, 2, 3 m, n, . . . = 4 . . . 10

Freund-Rubin: the bosonic solution

• Use F_µν² = −6m²g_µν and F² = −24m² to obtain

R_µν = −3(m/3)²g_µν AdS₄

R_mn = 6(m/6)²g_mn Einstein (e.g. S⁷)

• Maximum symmetry for the 4 + 7 split is

M¹¹ = AdS₄ × S⁷

• There are three other interesting cases

− A 7 + 4 split gives AdS₇ × S⁴

− IIB theory with self-dual F₍₅₎ gives AdS₅ × S⁵

− The D1-D5 system gives AdS₃ × S³ × T⁴

Freund-Rubin: Killing spinors

• To explore supersymmetry, we reduce

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

along the µ and m directions to obtain

∇ˆ_µ = ∇_µ − ₃₆¹ m_µνρσΓ^νρσ

∇ˆ_m = ∇_m + ₂₈₈¹ m_µνρσΓ_mΓ^µνρσ Introduce Γ⁵ = iΓ⁰¹²³

∇ˆ_µ = ∇_µ + ₂ⁱ(m/3)Γ_µΓ⁵ ∇ˆ_m = ∇_m − ₂ⁱ(m/6)Γ_mΓ⁵

Killing spinor in AdS Killing spinor on spheres

Freund-Rubin: Killing spinors

• The basic solution, AdS₄ × S⁷, has maximal supersymmetry

• Of course, we could replace S⁷ by a manifold with reduced supersymmetry

7-dimensional Einstein manifold with ‘weak’ G₂ holonomy yields N = 1 inD = 4

• Conversely, we can ask how many backgrounds there are preserving maximal supersymmetry

Answered by J. Figueroa-O’Farrill and G. Papadopoulos, JHEP 0303, 048 (2003) 1. M^1,10 Flat Minkowski space

2. AdS₄ × S⁷ Freund-Rubin 3. AdS₇ × S⁴

4. Hpp pp-wave geometry F₍₄₎ = µ dx⁻ ∧ dx¹ ∧ dx² ∧ dx³ ds²₁₁ = 2dx⁺dx⁻ − µ²(~x² + ¹₄~y ²)(dx⁻)² + d~x₃² + d~y₆²

Maximal supersymmetry in D = 11

• Classification performed through integrability

Define M_{M N} ≡ [ ˆ∇_M, ˆ∇_N] ≡ ¹₄R

For maximal supersymmetry, R vanishes for all 

This implies the matrix R = c_MΓ^M + c_{M N}Γ^{M N} + · · · + c_{M N P QR}Γ^{M N P QR} must vanish

→ c_M, c_{M N}, . . . , c_{M N P QR} all vanish

• Working out the consequences of these conditions is sufficient to prove the result

Use of symmetry: the M2-brane

• The 3-form potential A₍₃₎ of D = 11 supergravity couples naturally to the worldsheet of the eleven-dimensional supermembrane (M2-brane)

• The M2-brane carries mass (G_{M N}) and charge (electric F₍₄₎)

BPS → mass = charge

• We obtain the M2 geometry by considering a longitudinal + transverse split, M¹¹ = M³ × M⁸

Take the ansatz: ds²₁₁ = e^2A(y)dx²_µ + e^2B(y)dy_i²

F₍₄₎ = dx⁰ ∧ dx¹ ∧ dx² ∧ de^2C(y)

Goal: obtain a supersymmetric solution for A(y), B(y) and C(y)

The M2-brane: supersymmetry

• We may work out the Killing spinor equations

In the longitudinal direction

∇ˆ_µ = ∂_µ + ¹₂ω_µνiΓ^νi − ₁₂¹ F_µνρiΓ^νρi

= ∂_µ + ¹₆Γ_µΓⁱe^−3A∂_i(e^3A + e^2CΓ⁰¹²) In the transverse direction

∇ˆ_i = ∂_i + ¹₄ω_ijkΓ^jk + ₇₂¹ F_µνρjΓ_i^µνρj − ₃₆¹ F_iµνρΓ^µνρ

= ∂_i + ¹₆e^−3A∂_ie^2CΓ⁰¹² + ¹₂Γ_i^j(∂_jB − ¹₆e^−3A∂_je^2CΓ⁰¹²)

• We may build a projection by taking e^3A = e^2C and B = −¹₃C

⇒ e^3A = e^−6B = e^2C

The M2-brane: supersymmetry

• The resulting expressions for the supercovariant derivative are

∇ˆ_µ = ∂_µ + ²₃Γ_µΓⁱ∂_iC P₊

∇ˆi = ∂_i − ¹₃∂_iC − ¹₃(Γ_i^j − 2δ_i^j)∂_jC P₊ where P_± = ¹₂(1 ± Γ⁰¹²)

• Killing spinors are easily obtained

 = e^13CP₋₀

Note that partial supersymmetry does not imply the equations of motion . . . only that the metric and form-field are given in terms of a single function C(y)

The M2-brane: supergravity solution

• To obtain a solution, we impose (one) equation of motion

d ∗F₍₄₎ + ¹₂F₍₄₎ ∧ F₍₄₎ = 0 → d ∗F₍₄₎ = 0 → d ∗₈ de^−2C = 0

Hence e^−2C is harmonic in transverse space

e^−2C = H = 1 + X

i

q_i

|~y − ~y_i|⁶

• The complete solution

ds²₁₁ = H^−2/3dx²_µ + H^1/3dy_i² F₍₄₎ = dx⁰ ∧ dx¹ ∧ dx² ∧ d„ 1

H

«

 = H^−1/6P₋₀ Harmonic superposition for BPS objects

The M2-brane: BPS and time independence

• We may construct a vector from spinor bilinears

V ^µ = Γ^µ = H^−1/3₀Γ^µ₀ = ₀Γ^µ₀

A constant vector pointing in the time direction, V = ∂

∂t

Γ_M = real Γ^T_µ = −C⁻¹Γ_µC C = Γ⁰ ψ = ψ^TC⁻¹

• We can prove that V ^µ is a Killing vector

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

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

→ ∇_MV_N = ¹₆F_{M N P Q}Γ^{P Q} + _6!¹ ∗ F_{M N P QRST}Γ^{P QRST}

The supergravity p-brane ansatz

• Generalize the M2-brane solution to other objects and other dimensions

• Effective description in terms of the metric (mass M), a dilatonic scalar and an electrically charged (p + 2)-form field strength (charge Q)

e⁻¹L = R − ¹₂dφ ∧ ∗dφ − ¹₂e^αφF_(p+2) ∧ ∗F_(p+2)

F_(p+2) → A_(p+1) → (p + 1)-dimensional world volume → p-brane

• Ansatz

ds²₁₁ = e^2A(y)dx²_µ + e^2B(y)dy²_i

φ = φ(x) F_(p+2) = dx⁰ ∧ . . . ∧ dx^p ∧ de^2C(y)

The p-brane solution

• The solution depends on the parameter ∆

a² = ∆ − 2d ˜d D − 2

d = p + 1 (worldvolume dim) d = D − d − 2 (dual brane dim)˜

and is given by

ds² = H^{−4 ˜d/∆(D−2)}dx²_µ + H^{4d/∆(D−2)}dy_i²

e^φ = H^2a/∆ F_(p+2) = 2∆^−1/2dx⁰ ∧ . . . ∧ dx^p ∧ d(1/H)

• The function H is harmonic in the transverse space

H(y) = 1 + X

i

q_i

|~y − ~y_i|^d˜

Generalized holonomy classification

• The presence of F₍₄₎ modifies the covariant derivative

∇ˆM = ∇_M + ₂₈₈¹ (Γ_M^{N P QR} − 8δ_M^NΓ^{P QR})F_{N P QR} = ∂_M + ¹₄Ω_M

where

Ω_M = ω_M^ABΓ_AB + ₇₂¹ (Γ_M^{N P QR} − 8δ_M^NΓ^{P QR})F_{N P QR} is a generalized connection taking values in SL(32; R)

• Introduce Generalized holonomy H ⊆ SL(32; R) for the generalized connection Ω_M

[D_M(Ω), D_N(Ω)] = ¹₄R_{M N}(Ω)

The number of preserved supersymmetries = the number of singlets in the decomposition of the 32 of SL(32; R) under H

The number of singlets

• For  a 32-component spinor, we may choose a basis of Killing spinors

 = 2 6 6 4

1 0 0...

3 7 7 5

2 6 6 4

0 1 0...

3 7 7 5

etc.

To preserve n supersymmetries, the generalized holonomy must act as

R ∈ 0

@ SL(32; R)

1

A →

0

@

1 R^n,32−n

0 SL(32 − n; R) 1 A

Hence H ⊆ SL(32 − n; R) n ⊕ⁿR³²⁻ⁿ

However H does not have to be so large, so long as the singlet counting is correct

Generalized holonomy for the Freund-Rubin background

• For the Fruend-Rubin ansatz, ∇ˆ_M split into

∇ˆµ = ∇_µ + ₂ⁱ(m/3)Γ_µΓ⁵ ∇ˆm = ∇_m − ₂ⁱ(m/6)Γ_mΓ⁵

Killing spinor in AdS Killing spinor on spheres

• Focus on the 7-sphere part Integrability yields

[ ˆ∇m, ˆ∇n] = [∇_m, ∇_n] − ¹₄(m/6)²[Γ_mΓ⁵, Γ_nΓ⁵]

= ¹₄[R_mnpq − (m/6)²(g_mpg_nq − gmqg_np)]Γ^pq

• Substituting in the equation of motion R_mn = 5(m/6)²g_mn gives

[ ˆ∇m, ˆ∇n] = ¹₄C_mnpqΓ^pq

Higher order integrability

• First order integrability: [ ˆ∇m, ˆ∇n] = ¹₄C_mnpqΓ^pq For the round S⁷, the Weyl tensor vanishes

⇒ trivial generalized holonomy

Puzzle for the squashed S⁷:

left-squashing gives N = 1 while right-squashing gives N = 0 but first order integrability cannot distinguish these cases

• Second order integrability:

[ ˆ∇l, [ ˆ∇m, ˆ∇n]] = ¹₄(∇_lC_mn^abΓ_ab − 2i(m/6)Cmnla

Γ_aΓ⁵)

left- and right-squashing give opposite signs for m

Generalized holonomy for the squashed S

• The generalized structure group is SO(8)

SO(7) is generated by ¹₂iΓ_ab;

SO(8) is generated by appending Γ_aΓ⁵ to the above set

• At first order, we only see SO(7)

SO(7) ⊃ G₂ : 8 → 7 + 1

• At second order, we see SO(8)

SO(8) ⊃ Spin(7) : 8_s → 7 + 1 N = 1 for left-squashing SO(8) ⊃ Spin(7) : 8_s → 8 + 1 N = 0 for left-squashing

Generalized holonomy for the M2-brane

• Given ∇ˆ _M = ∂_M + ¹₄Ω_M, the M2 background yields the generalized connection

Ω_µ = ⁸₃∂_iC(Γ_µΓⁱP₊) Ω_i = ⁴₃∂_iCΓ⁰¹² − ⁴₃∂_jC(Γ_i^jP₊)

• We may view these as Lie algebra generators

T_ij = Γ_ijP₊ K_{µ i} = Γ_µiP₊ K_{µ ijk} = Γ_µijkP₊

where K_{µ ijk} arises from second order integrability

• The result is

Hol_M2 = SO(8)₊ n 12R^2(8s) ⊂ SL(16; R) n 16R¹⁶

Generalized holonomies of other objects

• A similar analysis may be performed for the basic objects of M-theory

n Background Generalized holonomy

32 E^1,10, AdS₇ × S⁴, AdS₄ × S⁷, Hpp {1}

18,. . . ,26 plane waves R⁹

16 M5 SO(5) n 6R⁴⁽⁴⁾

16 M2 SO(8) n 12R^2(8s)

16 MW R⁹

16 MK SU (2)

The M-wave and KK monopole are pure geometry solutions

• Is there a generalized holonomy classification generalizing Berger’s classification of Riemannian holonomy?

Next time

• Direct use of the Killing spinor equations in constructing new supersymmetric vacua

G-structure (Killing tensor) analysis

• The LLM construction of all 1/2 BPS bubbling AdS₅ × S⁵ backgrounds with SO(4) × SO(4) isometry