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 4 Outline

• Brief overview of the G-structure (Killing tensor) analysis

• Worked example: The LLM bubbling AdS construction

Reduction of IIB on S³ × S³ Obtaining the solution

The vacuum AdS₅ × S⁵ background

G-structure (Killing tensor) analysis

• A powerful means of classifying and constructing new supersymmetric backgrounds was pioneered by Gauntlett, Gutowski, Martelli, Pakis, Sparks, Tod, Waldram. . .

Given a Killing spinor , construct all possible tensors T_(n) = Γ_(n)

Killing spinor → background isometries → specialized coordinates

• Outline of the procedure

0) Choose initial isometries of the background geometry 1) Construct all possible spinor bilinears

2) Derive the algebraic identities (Fierz relations) between bilinears 3) Obtain the differential identities and identify additional symmetries

4) Specialize the choice of coordinates and solve the appropriate equations

G-structures

• The resulting supersymmetric backgrounds may be classified according to G-structures

a principle sub-bundle of the frame bundle with fiber in GL(n, R) GL(6, R) → O(6) → SO(6) → SU (3)

metric orientable Calabi-Yau (J₍₂₎, Ω₍₃₎) The failure of ∇ˆ to be compatible with the G-structure is measured by the intrinsic torsion (in this case dJ₍₂₎ and dΩ₍₃₎)

See e.g. Gauntlett et al., hep-th/0411194

• We will not focus on the classification, but will instead illustrate the construction with an example

Bubbling AdS₅×S⁵ geometry [Lin, Lunin and Maldacena, hep-th/0409174] (LLM)

The LLM construction

• By focusing on the near-horizon geometry of a stack of N D3-branes

Strings on AdS₅ × S⁵ ←→ N = 4 super-Yang Mills

• Both sides have the identical isometry group

SU (2, 2|4) ⊃ SO(2, 4) × SO(6)

• The AdS/CFT conjecture then relates

− States in N = 4 super-Yang Mills

− String configurations/giant gravitons in AdS₅ × S⁵

− Exact supergravity backgrounds

• Investigate the 1/2 BPS sector of the theory

. . . given by operators, states or configurations with ∆ = J₁

1/2 BPS states in N = 4 SYM

• Consider the super-Yang-Mills fields A_µ, 4χ^r, 6φⁱ

1/2 BPS chiral primaries are built from operators with conformal dimensional equal to R-charge, ∆ = J₁

X = φ¹ + iφ², Y = φ³ + iφ⁴, Z = φ⁵ + iφ⁶

Tr(X^J), Tr(Xⁿ)Tr(X^{J −n}), etc.

• Reduce the system to matrix quantum mechanics X(x^µ) → X(t)

Free fermion phase space

pλ

λ

1/2 BPS states in supergravity

• A ∆ = J₁ state still preserves SO(4) × SO(4) symmetry

SO(2, 4) × SO(6) ⊃ SO(2)_∆ × SO(4) × SO(2)_J1 × SO(4)

Consider writing AdS₅ × S⁵ as

ds²₁₀ = [− cosh²ρ dt² + dρ² + sinh²ρ dΩ²₃] + [cos²θ dφ² + dθ² + sin²θ deΩ²₃]

= [− cosh²ρ dt² + cos²θ dφ² + dρ² + dθ²] + [sinh²ρ dΩ²₃ + sin²θ deΩ²₃] Giant gravitons rotate on S⁵ (t,φ) but may expand either in AdS₅ (Ω₃) or S⁵ (Ωe₃)

• We thus seek a supergravity solution preserving SO(4) × SO(4) isometry

IIB sugra in D = 10 −→ Effective D = 4 model (breathing mode reduction on S³ × S³)

IIB supergravity on S

× S

• Start with IIB theory in ten dimensions

(g_µν, B₍₂₎, φ, C₍₀₎, C₍₂₎, C₍₄₎)

• Focus on D3-branes dissolving into fluxes

ds²₁₀ = g_µνdx^µdx^ν + e^H(e^GdΩ²₃ + e^−GdeΩ²₃) F₍₅₎ = F₍₂₎ ∧ dΩ3 − ∗4e^−3GF₍₂₎ ∧ deΩ₃

• The reduction on S³ × S³ yields an effective D = 4 Lagrangian

e⁻¹L₄ = e^3H[R + ¹⁵₂ ∂H² − ³₂∂G² − ¹₄e^−3(H+G)F_µν² + 12e^−H cosh G]

• Use of symmetry reduces the system to a simpler one

g_µν, H, G, F₍₂₎ in D = 4

Effective D = 4 supersymmetry

• We reduce the IIB gravitino variation (only non-trivial variation)

δψ_M = [∇_M + _16·5!¹ iF_{N P QRS}Γ^{N P QRS}Γ_M]ε

With the reduction ansatz, this turns into

δψ_µ = [∇_µ − ₁₆¹ e^{− 32(H+G)}F_νλΓ^νλΓ⁽³⁾Γ_µ]ε

δψ_a = [ ˆ∇_a + ¹₄Γ_aΓ^µ∂_µ(H + G) − ₁₆¹ e^{− 32(H+G)}F_µνΓ^µνΓ⁽³⁾Γ_a]ε δψ_˜a = [ ˆ∇_a˜ + ¹₄Γ_˜aΓ^µ∂_µ(H − G) − ₁₆¹ e^{− 32(H+G)}F_µνΓ^µνΓ⁽³⁾Γ_˜a]ε

where Γ⁽³⁾ = −iΓ⁴⁵⁶ lives on the first S³ and we have used the fact that IIB spinors have definite Γ¹¹ chirality

Effective D = 4 supersymmetry

• There is one final step in obtaining effective four-dimensional Killing spinor equations

Choose a Dirac matrix decomposition

Γ_µ = γ_µ × 1 × 1 × σ₁ Γ_a = 1 × σ_a × 1 × σ₂ Γ_˜a = γ₅ × 1 × σ_a˜ × σ₁

along with ε =  × χ × χ ×e » 1

0 –

• We also demand that η and ηe are Killing spinors on S³ × S³

[ ˆ∇_a + ¹₂iη ˆσ_a]χ = 0 [ ˆ∇_a˜ + ¹₂iη ˆeσ_a˜]η = 0e (η,η = ±1)e

The Killing spinor equations

• The result may be written as

δψ_µ = [∇_µ + ₁₆¹ ie^{− 32(H+G)}F_νλγ^νλγ_µ]

δχ_H = [γ^µ∂_µH + e^{− 12H}(ηe^{− 12G} − iηγe ₅e^12G)]

δχ_G = [γ^µ∂_µG − ¹₄ie^{− 32(H+G)}F_µνγ^µν + e⁻

1 2H

(ηe^{− 12G} + iηγe ₅e^12G)]

This looks vaguely like D = 4 gauged supergravity coupled to matter (but this is not a consistent truncation)

• Our goal is to solve these Killing spinor equations in order to obtain all 1/2 BPS solutions

Supersymmetry analysis

• We now apply the G-structure (Killing tensor) analysis

Gauntlett et al. CMP 247, 421 (2004); CQG 20, 4587 (2003);

CQG 20, 5049 (2003); CQG 21, 4335 (2004)

• Start with the spinor bilinears

For  a four-dimensional Dirac spinor, we define the real quantities

f₁ = γ₅ f₂ = i K^µ = γ^µ L^µ = γ^µγ₅ Y ^µν = iγ^µνγ₅ In addition, we may also consider expressions involving (^c · · · ) where ^c = ^TC

• The algebraic identities give relations between these tensors

L² = −K² = f₁² + f₂² etc.

⇒ K^µ is timelike and L^µ is spacelike

The differential identities

• Start with the gravitino variation

∇_µ = −₁₆¹ ie^{− 32(H+G)}F_νλγ^νλγ_µ

∇_µ = ₁₆¹ ie^{− 32(H+G)}F_νλγ_µγ^νλ

• For K_ν = γ_ν we find

∇_µK_ν = ¹₄e^{− 32(H+G)}(f₂F_µν − f₁ ∗ F_µν) → ∇_(µK_ν) = 0 so K^µ is a (timelike) Killing vector (take K = ∂/∂t)

• For L_ν = γ_νγ₅ we find

∇_µL_ν = ¹₄e^{− 32(H+G)}(¹₂g_µνF_λρY ^λρ − 2F_(µ^λY_ν)λ) → dL₍₁₎ = 0 so L₍₁₎ is a (spacelike) closed 1-form (take L₍₁₎ = dy)

Pinning down f

and f

• The gravitino variation also yields

∂_µf₁ = ¹₄e^{− 32(H+G)} ∗ F_µνK^ν ∂_µf₂ = −¹₄e^{− 32(H+G)}F_µνK^ν

magnetic electric

• This may be combined with additional ‘differential’ identities obtained from δχ_H and δχ_G

f₁∂_µ(H − G) = ¹₂e^{− 32(H+G)} ∗ F_µνK^ν f₂∂_µ(H + G) = −¹₂e^{− 32(H+G)}F_µνK^ν

→ d[e^{− 12(H−G)}f₁] = 0 d[e^{− 12(H+G)}f₂] = 0

→ f₁ = be^12(H−G) f₂ = ae^12(H+G) where a and b are constants

Specializing the metric

• We consider one more constraint from the δχ_H equation

 : γ^µ∂_µH = −e^{− 12H}(ηe^{− 12G} − iηγe ₅e

1 2G

)

→ K^µ∂_µH = i “

ηe^{− 12(H+G)}f₂ + ηee ^{− 12(H−G)}f₁”

→ K^µ∂_µH = 0 and aη + bη = 0e

Choose, e.g., a = b = η = −η = 1e or a = b = η = −η = −1e This gives a 1/4 + 1/4 = 1/2 BPS configuration

• This normalization yields

L² = −K² = f₁² + f₂² = e^H(e^G + e^−G) = 2e^H cosh G

Specializing the metric

• We now have enough information to write down the metric in a convenient coordinate system (where ^{K = ∂/∂t} and ^L₍₁₎ ^{= dy})

ds²₄ = −h⁻²(dt + V_idxⁱ)² + h²(e^2γ(dxⁱ)² + dy²)

Combining L₍₁₎ = dy with L₍₁₎ = de^H obtained from δχ_H, we obtain e^H = y h⁻² = 2y cosh G

• We may also perform a similar analysis on the 1-form ω_µ = ^cγ_µ to find

dω = 0 and that it has normalized components along x¹ and x²

→ we are allowed to choose coordinates such that γ = 0

The form field F

• What remains is a determination of F₍₂₎ and dV Recall the differential identities

∂_µf₁ = ¹₄e^{− 32(H+G)} ∗ F_µνK^ν ∂_µf₂ = −¹₄e^{− 32(H+G)}F_µνK^ν

magnetic electric

• With f₁ = e^12(H−G) and f₂ = e^12(H+G), we obtain

F₍₂₎ = −de^2(H+G) ∧ (dt + V ) − h²e^3G ∗₃ de^2(H−G)

where we may substitute in e^H = y and h⁻² = 2y cosh G

The metric vector V

• Finally, note that the antisymmetric part of the differential identity ∇_µK_ν

yields

dK = ¹₂e^{− 32(H+G)}(f₂F₍₂₎ − f₁ ∗ F₍₂₎)

This may be combined with ^{K = −h−2(dt + V )} to give

dV = −2h⁴e^H ∗3 dG or dV = −¹₂y⁻¹ ∗3 d tanh G

• Define z = ¹₂ tanh G so that dV = −y⁻¹ ∗₃ dz

• We now obtain the consistency condition d²V = 0 or

d „ 1

y ∗3 dz

«

= 0

Interpretation of the solution

• What have we learned?

ds² = −h⁻²(dt + V )² + h²(dx²₁ + dx²₂ + dy²) + y(e^GdΩ²₃ + e^−GdeΩ²₃) F₍₂₎ = −h

d(y²e^2G) ∧ (dt + V ) + h²e^3G ∗3 d(y²e^−2G)i with

h⁻² = 2y cosh G, z ≡ ¹₂ tanh G, dV = −y⁻¹ ∗3 dz where

h

∂₁² + ∂₂² + y ∂_y1 y∂_yi

z(x₁, x₂, y) = 0 harmonic in H3

or

h

∂₁² + ∂₂² + 3

y∂_y + ∂_y²i „ z y²

«

= 0 harmonic in R² × R⁴

Linear superposition and bubbling

• Underlying the 1/2 BPS solution is a linear system

⁶ „ z(x₁, x₂, y) y²

«

= 0

• This admits a Green’s function solution

z(~x, y) = y² π

Z 1

(|~x − ~x⁰|² + y²)²z(~x⁰, 0)d²~x⁰ where boundary conditions are imposed at y = 0

• Because ds² = · · · + y(e^GdΩ²₃ + e^−GdeΩ²₃) boundary conditions must be chosen to ensure a regular solution

Regularity of the geometry at y = 0

• When y → 0 the volume of ds²₆ = e^GdΩ²₃ + e^−GdeΩ²₃ goes to zero

• To be smooth, only a single S³ can collapse

Either e^G → 0 or e^−G → 0

⇒ G = ±∞ or z ≡ ¹₂ tanh G = ±¹₂ as y → 0

Suppose ^{z → 1}₂ as ^{y → 0}

Solving the harmonic equation gives an expansion e^G ∼ y⁻¹

so that h²dy² + y(e^GdΩ²₃ + e^−GdeΩ²₃) ∼ dy² + y²dΩ²₃ + deΩ²₃

• Boundary conditions: z(x₁, x₂, 0) = ±¹_{2 z= −1/2} ^x1

x z

2

= 1/2

Example: The AdS

× S

background

• We recover AdS₅ × S⁵ by filling the ‘Fermi sea’

z ≡ ¹₂ tanh G = r² + y² − `²

2p(r² + y² − `<sup>2</sup>)<sup>2</sup> + 4y<sup>2</sup>`^{2 z = −1/2}

1

x ₂

x = r cosθ

= r sin z

θ

= 1/2

V = − r² + y² + `²

2p(r² + y² − `<sup>2</sup>)<sup>2</sup> + 4y<sup>2</sup>`²dφ

• Make the change of coordinates y = ` sinh ρ sin θ r = ` cosh ρ cos θ

→ z = 1 2

sinh²ρ − sin²θ

sinh²ρ + sin²θ e^G = sinh ρ sin θ and h⁻² = `(cosh²ρ − cos²θ) V = −1

2

cosh²ρ + cos²θ cosh²ρ − cos²θ

Example: The AdS

× S

background

• Look at the metric

ds² = −h⁻²(dt + V )² + h²(dr² + r²dφ² + dy²) + y(e^GdΩ²₃ + e^−GdeΩ²₃)

= `



−(cosh²ρ − cos²θ)

„

dt − 1 2

cosh²ρ + cos²θ cosh²ρ − cos²θdφ

«²

+ dρ² + dθ²

+ cosh²ρ cos²θ

cosh²ρ − cos²θdφ² + sinh²ρ dΩ²₃ + sin²θ deΩ²₃ ff

= `



− cosh²ρ(dt − ¹₂dφ)² + dρ² + sinh²ρ dΩ²₃ AdS₅ + cos²θ(dt + ¹₂dφ)² + dθ² + sin²θdeΩ²₃

ff

S⁵

• Note the mixing between t and φ

motion of the giant gravitons along the equator of S⁵

Some additional resources

• A rather incomplete list. . .

M. Duff, B. Nilsson and C. Pope, Kaluza-Klein supergravity, Phys. Rep. 130, 1 (1986).

P. Candelas and X. de la Ossa, Comments on Conifolds, Nucl. Phys. B342, 246 (1990).

M. Duff, R. Khuri and J. Lu, String solitons, Phys. Rep. 259, 213 (1995).

P. Aspinwall, K3 surfaces and string duality, hep-th/9611137.

K. Stelle, BPS branes in supergravity, hep-th/9803116.

J. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS Backgrounds in String and M-theory, hep-th/0411194.