BPS deformations of
AdSp × S
q
Oleg Lunin
Institute for Advanced Study
H. Lin, O. L., J. Maldacena, hep–th/0409nnn
Outline
• 1/2 BPS states in field theory & Fermi liquid.
• Technique for constructing gravity solutions.
• Solutions in IIB SUGRA and Laplace equation.
• Solutions in 11D SUGRA and Toda equation.
• Summary.
Half–BPS states in N = 4 SYM
• N = 4 SYM on S 3 × R:
– chiral primaries:
) . . . Tr(Z nk ),
– symmery: S 3 × SO(4)
Tr(Z
n1
Z = φ1 + iφ2
• Matrix model description: harmonic oscillator
†
= Tr[(a† )n ]
– set of harmonic oscillators: αn
– eigenvalue basis & Fermi liquid
Berenstein ’04
pλ
λ
• Brane probe approximation
– giant gravitons expanding on S 3 or S̃ 3 .
McGreevy, Susskind, Toumbas ’00
–
AdS7 × S 4 : giant gravitons with S 5 × S 2 .
Technique for constructing gravity
solutions
• Assumptions
SO(4) × SO(4)
– bosonic fields: mertic and F (5)
– bosonic symmetries:
– existence of Killing spinor:
i M1 M2 M3 M4 M5 (5)
∇M η+
FM1 M2 M3 M4 M5 ΓM η = 0
Γ
480
• Reduction on S 3 × S 3 : spinor in 4D interacting
with gauge field and 2 scalars
• Using bilinears of Killing spinor
Gauntlett, Gutowski, Martelli, Pakis,
Kµ = −ε̄γµ ε
5
Lµ = ε̄γ γµ ε
–
Reall, Sparks, Waldrum ’02–’04
2
2
K · L = 0, L = −K
L is an exact form, K µ is a Killing vector
ds2 = h2 dy 2 − h−2 (dt + Vi dxi )2 + h̃ij dxi dxj
1/2 BPS geometries in Type IIB
SUGRA
• Explicit geometry and Laplace equation
ds2 = −h−2 (dt + Vi dxi ) + h2 (dy 2 + dxi dxi )
+yeG dΩ23 + ye−G dΩ̃23
F(5) = Fµν dxµ ∧ dxν ∧ dΩ3 + F̃µν dxµ ∧ dxν ∧ dΩ̃3
F = dBt (dt + V ) + Bt dV + dB̂
– functions appearing in the solution:
h−2 = 2y cosh G,
1 2 2G
Bt = − y e ,
4
1
B̃t = − y 2 e−2G
4
ydV = ∗3 dz
z + 12
1 3
dB̂ = − y ∗3 d( 2 )
4
y
1
z
−
1
ˆ = − y 3 ∗ d(
2
dB̃
)
3
2
4
y
– solution is paramaterized by one function z
z=
1
tanh(G),
2
∂i ∂i z + y∂y (
∂y z
)=0
y
• potential singularities: RR̃ = y = 0
Regular solutions:
general description
• Laplace equation and boundary conditions
– 6D Laplace equation for Φ = yz2
– regularity at y = 0: z = ± 12
1
(dy 2 + y 2 dΩ̃23 )
c(x)
• Boundary condition for a generic state
h2 dy 2 + ye−G dΩ̃23 ∼
x2
x1
• Plane y = 0 ↔ phase space of the oscillator
Examples
• AdS5 × S 5
z=
2
2
2
r
−r
+y
0
√ 2 2 22 22
2 (r +r0 +y ) −4r r0
• (Giant) gravitons
• PP wave and its excitations
z = √x22
2
x2 +y 2
Topology and fluxes
• Two types of closed five–manifolds
y
• Different topologies: non–contractible spheres
• Quantization of fluxes:
(Area)z=− 1
1 ˆ
2
Ñ = − 2 4 dB̃ =
2π lp
4π 2 lp4
• Energy and higher moments
∆=J =
2
dx
2πh̄
1
(x21
2
+
h̄
x22 )
−
1
2
2
D
dx
2πh̄
2
1/2 BPS geometries in M theory
• Bosonic symmetries: SO(6) × SO(2)
2λ
2 −4λ
e
e
y
2
2
2
dΩ
+
d
Ω̃
ds11 =
5
2
m2
4m2
e2λ h2
e−4λ 2
i 2
D
2
−
(dt + Vi dx ) +
dy + e dx
2
2
2
m
4m h
∂y D
h = 1 + y 2 e−6λ ,
e−6λ =
y(1 − y∂y D)
• Solution is parameterized by one function D
which satisfies 3D Toda equation:
∆D + ∂y2 eD = 0
• Boundary conditions at y = 0
∂y D = 0,
R2 → 0
D ∼ log y, R5 → 0
x2
• Boundary condition at infinity
x1
Solutions of Toda equation
• AdS7 × S 4
r2 L−6
r2
−3 2
e =
,
x
=
(1
+
r sin θ
)
cos
θ,
4y
=
L
4 + r2
4
D
• AdS4 × S 7
e = 4L
D
x= 1+
2
r
4
−6
1/4
r2
1 + sin2 θ
4
cos θ, 2y = L−3 r sin2 θ
• PP wave
r52
r52 r22
1 2
e = , y = r5 r2 , x 2 =
−
2
4
4
2
D
• Translational invariance in x1 : linear equation
eD = ρ2 , ρ∂ρ V = y, ∂η V = x2
1
Ward ’90
∂ρ (ρ∂ρ V ) + ∂η2 V = 0
ρ
• Compactification of x1 : type IIA gravity duals of
the BMN matrix model
Solution of gauged SUGRA
• M theory on S 4 : gauged SUGRA in 7D
– field content: SL(5, R)/SO(5) coset,
SO(5) gauge field, five 3–forms
Perini, Pilch, van Nieuwenhuizen ’84
– 1/2 BPS black hole: symmetry group
SO(6) × SO(3) × SO(2) × U (1)
Liu, Minasian ’99
• Our goal: regular supersymmetric solution
– symmetry SO(6) × SO(3) × U (1)
– excited fields:
⎡
VI i = ⎣
e
−3χ
g
03×2
02×3
e2χ 13×3
⎤
⎦,
⎡
AµI J = ⎣
iAµ σ2
02×3
03×2
03×3
g = exp(iθσ2) exp(−ρσ3 ) ∈ SL(2, R)/U (1)
• Regular solution exists & gives eD
⎤
⎦
Relation to known solutions
• N = 2 superconformal field theories
– 16 supercharges, SO(4, 2) × SU (2) × U (1)
– double analytic continuation of 11D solutions:
dΩ25 → −ds2AdS5 ,
t→ψ
– different boundary conditions
– example of a solution
1 1
D
e = 2 ( − y2 )
x2 4
Maldacena, Nunez ’00
– new feature: space ends at y = 1/2:
ψ ∼ ψ + 2π
• M2 brane with mass deformation
– IIB with x1 isometry → IIA → M theory
– Bena & Warner solutions with regular BC
Summary
• Geometries dual to chiral primaries: no
singularities & horizons
• All 1/2 BPS gravity solutions for type IIB
– reduction to 3D Laplace equation
– boundary conditions and free fermions
– explicit solutions in terms of integrals
– fluxes and topologies
• All 1/2 BPS solutions of 11D SUGRA
– reduction to 3D Toda equation
– specific boundary conditions
– examples: AdS, pp wave
– new regular solution of gauged SUGRA
• Future directions
– properties of the new geometries
– 1/4 BPS states