On the geometry of supersymmetric AdS solutions
Nakwoo Kim
Physics Department
Kyung Hee University, Seoul Korea

IPMU
Based on works with
J. Gauntlett, D. Waldram, A. Donos, J.D. Park

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geometrySeoulof supersymmetric
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Table of Contents
1

Introduction and Summary

2

The Killing spinor analysis
AdS3 from D3
AdS2 from M2
Summary of wrapped D3/M2

3

Solving the master equation
Products of KE
Complex line bundle over KE

4Generalization to arbitrary dimensions
The geometry
Solutions and Reductions

5

Extension to inhomogeneous case

6

Conclusion

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References• NK, hep-th/0511029, AdS3 solutions of IIB supergravity from D3branes, published in JHEP
• NK, J.-D. Park, hep-th/0607093, Comments on AdS2 solutions of

D=11 Supergravity , published in JHEP
• J.P. Gauntlett, NK, D. Waldram, hep-th/0612253 , Supersymmetric

AdS3, AdS2 and Bubble Solutions published in JHEP
• J.P. Gauntlett, NK, arXiv:0710.2590, Geometries with Killing Spinors

and Supersymmetric AdS Solutions, published in Comm. Math. Phys.
• A. Donos, J.P. Gauntlett, NK, arXiv:0807.4375, AdS solutions

through transgression, published in JHEP

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Introduction and Summary

Introduction and Motivation

AdS/CFT correspondence :Weakly coupled gravity provides a holographic dual description to strongly
coupled gauge field theory.
More concretely,
IIB strings in AdS5 × SE5 ←→ D = 4, N = 1, Super YM
and it is the geometry of the Sasaki-Einstein space which determines thematter content and interactions of the dual field theory.
Sasaki-Einstein space ∼Vacuum moduli space of SUSY gauge theory

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Introduction and Summary

Introduction and Motivation

AdS/CFT correspondence :Weakly coupled gravity provides a holographic dual description to strongly
coupled gauge field theory.
More concretely,
IIB strings in AdS5 × SE5 ←→ D = 4, N = 1, Super YM
and it is the geometry of the Sasaki-Einstein space which determines thematter content and interactions of the dual field theory.
Sasaki-Einstein space ∼Vacuum moduli space of SUSY gauge theory

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Introduction and Summary

D-brane description

We put D3-branes at the tip of singular CY3, and take the near-horizon
limit. The backreacted geometry is AdS5 × SE5 , which is dual to thegauge theory living on D3.

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Introduction and Summary

Sasaki-Einstein space• Einstein space : Rab = λgab
• Sasakian : The metric cone is Kahler.
• When combined, the metric cone of SE is Ricci-flat and Kahler(Singular Calabi-Yau)• Thus allows nontrivial Killing spinor, satisfying ∇a η = iγa η• Examples : S 5 , T 1,1 , Y p,q , Lp,q,r ... (with explicitly known metric)

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Introduction and Summary

Sasaki-Einstein space• Einstein space : Rab = λgab
• Sasakian : The metric cone is Kahler.
• When combined, the metric cone of SE is Ricci-flat and Kahler(Singular Calabi-Yau)• Thus allows nontrivial Killing spinor, satisfying ∇a η = iγa η• Examples : S 5 , T 1,1 , Y p,q , Lp,q,r ... (with explicitly known metric)

Nakwoo Kim( Physics Department Kyung On
HeetheUniversity,
geometrySeoulof supersymmetric
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18 Dec 2008

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Introduction and Summary

Sasaki-Einstein space• Einstein space : Rab = λgab
• Sasakian : The metric cone is Kahler.
• When combined, the metric cone of SE is Ricci-flat and Kahler(Singular Calabi-Yau)• Thus allows nontrivial Killing spinor, satisfying ∇a η = iγa η• Examples : S 5 , T 1,1 , Y p,q , Lp,q,r ... (with explicitly known metric)

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Introduction and Summary

KE, SE and Singular CY• SE by definition is the base manifold of Calabi-Yau with conicalsingularity.
ds 2 = dr 2 + r 2 (Sasaki-Einstein)∂• SE is odd-dimensional, and (r ∂r)b J ba provides a Killing vector of SE.

• Locally, SE is always written as a Hopf-fibration over Kahler-Einsteinspace.
2
2
dsSE= (dψ + A)2 + dsKE,

dA = RKE• The most well-known nontrivial example: T 1,1 = SU(2)×SU(2)or U(1)U(1)fibration over S 2 × S 2

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Introduction and Summary

Wrapped D3
But more generally, we can consider wrapped branes which are alsosupersymmetric. (They are useful when we want product gauge groups
with different ranks.)

Then the worldvolume is 1 + 1 dimensional and we expect to get AdS3 in
the near horizon limit.
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Introduction and Summary

Wrapped D3 and M2

In the same way one can consider wrapped branes of different kinds. But
here we are particularly interested in D3 and M2 wrapping 2-cycles in CY.
And their near-horizon geometry, we expect• D3 wrapped on 2-cycle : AdS3 ×w M7• M2 wrapped on 2-cycle : AdS2 ×w M9Preserving SUSY: in general 1/8-BPS.

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Introduction and Summary

Main Result

For both D3 and M2 wrapped on 2-cycles, or AdS3 from D3, and AdS2
from D2, the geometry is built on Kahler space satisfying the following eq:
1∇2 R − R 2 + Rij R ij = 02
and it will be explained:• How this result is obtaind.
• What kind of ansatz can provide explicit solutions to it.
• Extension/generalization.

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The Killing spinor analysis

AdS3 from D3

AdS3 in IIB

Ansatz for the IIB solution :2ds10= e 2A (ds 2 (AdS3 ) + ds 2 (M7 ))F (5) = (1 + ∗)Vol(AdS3 ) ∧ F
and turn off all the remaining fields. Then the Killing spinor equation for
M7 is :
1[γ a ∂a A − i + e −4A Fab γ ab ]η = 0
2
i
1 −4A
[∇a + γa − e
Fab γc ab ]η = 0
2
2

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The Killing spinor analysis

AdS3 from D3

Killing spinor analysis• The next step is to consider all spinor bilinears, η † γ... η, and study thealgebraic and differential relations between them. One makes use ofFierz identities and the Killing spinor equation.
• We find, for instance, η † η is constant, η † γ a η is a Killing vector withconstant norm etc.
• Then the M7 metric can be written1
ds 2 (M7 ) = (dz + P)2 + e −4A ds 2 (K6 )4

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The Killing spinor analysis

AdS3 from D3

BPS relations

The two-form Jab = η † γab η provides a complex structure to K6 , which isin fact Kahler. The remaining field equations are summarised as1
e −4A = R
81
1
F = J − d(e 4A (dz + P))
2
8
1∇2 R − R 2 + Rij R ij = 0(∗)
2
once we find a solution to (*), we can construct the whole 10d solution.

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The Killing spinor analysis

AdS2 from M2M2 brane analysis
One can perform a similar analysis for M2-branes.
2ds11= e 2A [ds 2 (AdS2 ) + ds 2 (M9 )]G (4) = Vol(AdS2 ) ∧ F
After a similar procedure with 9d Killing spinor equation for M9 , constantnorm Killing vector in M9 etc.
ds 2 (M9 ) = (dz + P)2 + e −3A ds 2 (K8 )1e −3A = R,
2

F = −J + d[e 3A (dz + P)]

1∇2 R − R 2 + Rij R ij = 0
2

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geometrySeoulof supersymmetric
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The Killing spinor analysis

Summary of wrapped D3/M2

Summary so far• For D3 (or M2) wrapped on 2-cycles, in the AdS limit the internal

geometry takes locally a form of U(1)-fibration over 6(8) dimensional
Kahler geometry, satisfying the higher-order differential equation (*).
• This is very similar to SE, which in canonical form is a U(1) fibration

over Kahler-Einstein space.
• Through analytic continuation, one gets IIB(M) theory with S 3 (S 2 )parts as well. The same master equation.

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The Killing spinor analysis

Summary of wrapped D3/M2

Questions• Can we find a nice interpretation of (*)?

∇i Ji = 0, with J = dR + 2 ∗ P ∧ R ∧ J• Is it plausible to solve (*)?
• Generalization to higher dimensions?
• What is the cone geometry? Other constructions like GLSM?

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Solving the master equation

Products of KE

Kahler base as products of Kahler-Einstein• For simplicity, let us first consider the Kahler base as (product of)

Kahler-Einstein space. Like S 2 × H 2 × T 2 etc. (*) becomes algebraic.
• For a single KE, (*) holds only if d = 2. We can thus take
K6 = S 2 × T 4 , K8 = S 2 × T 6 . They lead to• IIB theory : AdS3 × S 3 × T 4 (1/2-BPS)• M theory : AdS2 × S 3 × T 6 (1/4-BPS)• More generally, if ds 2 (K ) =(i)

ds 2 (KE2 ), with R =
X
X
`i )2
`2i = (

Nakwoo Kim

P( Physics Department Kyung On
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P`i J i ,18 Dec 200817 / 32

Solving the master equation

Products of KE

Kahler base as products of Kahler-Einstein• For K6 , (`1 , `2 , `3 ) = (`, −`/(` + 1), 1).
• Special cases are (0, 0, 1) and (−1/2, 1, 1).
• The latter, for K6 = H2 × CP 2 , has been known, from the gauged
sugra solution of wrapped D3-brane (M. Naka)• For K8 , (`1 , `2 , `3 , `4 ) = (`1 , `2 , −(`1 `2 + `1 + `2 )/(`1 + `2 + 1), 1).
• special cases are (0, 0, 0, 1) and (−1, 1, 1, 1).
• The latter, for K8 = H2 × CP 3 , also found from gauged sugra(Gauntlett et. al)

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Solving the master equation

Complex line bundle over KE

Calabi-ansatz• It is well-known that the following metric is always Kahler(U = U(ρ)).
dρ2+ Uρ2 (Dφ)2 + ρ2 ds 2 (KE2n )U• For our purpose KE is positively curved, and Dφ = dφ + B with
dB = 2JKE .
2ds2n+2=• Now one can compute the Ricci tensor and then the master equation

becomes nonlinear ODE.

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Solving the master equation

Complex line bundle over KE

Solution of the Calabi-ansatz• One can find a polynomial solution for general n.
• U = 1 − αx n−1 (x − β)2 ,x = 1/ρ2• The issue is whether this solution can be made into a compact

M7 (M9 ), regular and smooth.
• It turns out, the IIB solution is equivalent to the AdS3 solution foundby Gauntlett, Mac Conamhna, Mateos and Waldram(hep-th/0606221, PRL)
• K6 is not smooth. But upon including (Dz)2 , M7 can be maderegular if we impose periodic b.c. for 3φ + z and z.

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Generalization to arbitrary dimensions

The geometry

Geometry in 2n + 2 dimensions• We consider dimensional reduction of IIB sugra on R 1,1 , or 11d sugraon R. From the effective action and Killing spinor equation, in 8d and10d, we can find the generalization to arbitrary higher dimensions(d = 2n + 2).
• We have a system consisting of metric, scalar φ, 2-form gauge field b.
• One can construct: bosonic action, and the associated Killing spinor

system.

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Generalization to arbitrary dimensions

The geometry

Action and the Killing spinor equations• Action : (f = db)

1
L = e 2(n−1)φ [R + 2n(2n − 3)(∇φ)2 + e −4φ f 2 ]2• Killing spinor equations[γ a ∇a φ +i −2φefabc γ abc ] = 0
12

i −2φ
e
fbcd γa bcd ] = 0
24• They are consistent, in the sense that any susy configuration
satisfying the gauge field equation and Bianchi identity, solves thefield equation.
[∇a −Nakwoo Kim( Physics Department Kyung On
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Generalization to arbitrary dimensions

The geometry

G-structure

Since we have 2n + 2-dimensional space, susy solution should come with
SU(n + 1) structure. For (1, 1)-form J and (n + 1, 0)-form Ω,d[e nφ Ω] = 0d[e 2(n−1)φ J n ] = 0d[e 2φ J] = f
From above, we see the space is complex, but not Kahler.

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Generalization to arbitrary dimensions

The geometry

Geometry in (2n+1) dimensions

Now consider reducing on the radial direction.
2(n−1)

e −2φ = r n−2 e B ,n

f = r 2−n dr ∧ F

We again find action and the associated Killing spinor system, for(2n + 1)d metric, scalar B, and 2-form f.s. F .
L = e (1−n)B [R +12nn(2n − 3)(∇B)2 + e 2B F 2 −]24(n − 2)2

again, any susy configuration, combined with Bianchi identity and theequation of motion for F , satisfy the entire field equations.

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Generalization to arbitrary dimensions

The geometry

SUSY in (2n + 1) dimensions

Following the usual Killing spinor analysis, one can show that, any susy
solution of the above (2n + 1)-dim system, can be written as follows.
(c = (n − 2)/2)22
ds2n+1= c 2 (dz + P)2 + e B ds2n
e B = c 2 R/2,

F = −J2n /c + cd[e −B (dz + P)]

2 is Kahler, and satisfies
where the base space ds2n

1
∇2 R − R 2 + Rij R ij = 0
2

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Generalization to arbitrary dimensions

Solutions and Reductions

Calabi-ansatz• For general n, we use the solution U = 1 − αx n−2 (x − 1)2• For large enough α, we have two positive roots 0 < x1 < 1 < x2 .
• Upon coordinate transformation φ = (ψ − z)/n,

1 2
R 2
ds2n+1 = (dz + P)2 + ds2n
2
c
2
RU
R
R
+
= wDz 2 + 2 Dψ 2 + 3 dx 2 + ds 2 (KE2n−2
)
2n wx
8x U
2x
with R = 8αx n−1 .
• At x1 , x2 , we have potential conical singularities. But in the form

given above, giving 2π periodicity to ψ can make the base regular atboth ends. Finally α should take discrete values to make Dz goodU(1) fibration. (One demands d(Dz) integrated over 2-cycles be
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Generalization to arbitrary dimensions

Solutions and Reductions

Solutions from the Calabi ansatz

DU
dx
df
R = 4(n − 1)xf − 4x 2
dxRU
w = (1 − f /n)2 + 2
2n x
1
RU
Dz = dz + gDψ, g = 2 (nf − f 2 −)n w
2x
f = n(1 − U) + x

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Generalization to arbitrary dimensions

Solutions and Reductions

LLM inspired ansatz• The 1/2-BPS fluctuations of max susy AdS backgrounds have beenstudied by Lin, Lunin and Maldacena.
• IIB with S 3 × S 3 , or M-theory with S 5 × S 2 . The reduced 4d systems

have been studied.
• Being 1/2-BPS, they are special cases of our 1/8-BPS geometries.
• We try to find similar reduction of (*) equation to 4 dimensions.

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Generalization to arbitrary dimensions

Solutions and Reductions

Metric ansatz and BPS relation• Metric ansatz

ds 2 = dy 2 /U + y 2 U(Dψ)2 + f /U(dx12 + dx22 ) + y 2 ds 2 (KE2n−4 )with Dψ = dψ + σ + V .
• Choose the Kahler form

J = ydy ∧ Dψ + (f /U)dx1 ∧ dx2 + y 2 JKE• dJ = 0, provided

dσ = 2JKE ,d2 V = 1/y ∂y (f /U)dx1 ∧ dx2• Kahler in fact, if we impose ∂y V = 1/y ∗2 d2 (1/U)

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Generalization to arbitrary dimensions

Solutions and Reductions

Reduction to 3d• Integrability condition leads to (∆ = ∂12 + ∂22 )

∆

1
1
f
+ y ∂y [ ∂y ( )] = 0UyU• Now assuming a BPS relation through a new field D, asy1
U = 2 ∂y D,

f = y 2p e qD the (*) equation is satisfied, if (k is scalar
curvature of KE)
p = 3 − n,(q − k)[(q(n − 1) − k(n − 3)] = 0• When n = 3, p = q = 0 we recover ∆D + y1 ∂y (y ∂y D) = 0.
n−4

• For n > 3, after some coordinate change, ∆D + x n−3 ∂x2 e D = 0.

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18 Dec 200830 / 32Extension to inhomogeneous caseWith 3-form flux in IIB

One can in fact turn on 3-forms, for instance to IIB construction.
1
2∇2 R − R 2 + Rij R ij + G ∗ijk Gijk = 0
23
where G is the complexified 3-form, imaginary self-dual in 6d. G = i G̃ .

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Conclusion

conclusion• We have found an interesting new class of complex geometry related

to wrapped brane solutions. (Or, giant gravitons, BPS fluctuations ofgauge theory etc.)• Quite similar to the hierarchy of singular CY, SE, KE. Extraordinary

higher-order equation in place of Einstein condition, for Kahler base.
• Presented several classes of explicit solutions. More systematicconstruction? (e.g. Toric data?)

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