Emergent bubbling geometries
in the plane wave matrix model
Yuhma Asano (Kyoto U.)
March 6, 2014

Collaboration with Goro Ishikib, Takashi Okadabc and Shinji Shimasakia( aKyoto U., bYITP and cKITP )Ref) JHEP1302(2013)148 (arXiv:1211.0364),
arXiv:1401.5079

Gauge/Gravity Correspondence for SU(2|4) symmetric theories

Plane Wave Matrix Model(1D Quantum Mechanics)

IIA SUGRA solution
This gravity solution was constructedby Lin and Maldacena in ’05.

Q: How does the space-time geometry emerge in the framework ofthe corresponding gauge theory?Gauge/Gravity Correspondence for SU(2|4) symmetric theories

Plane Wave Matrix Model(1D Quantum Mechanics)

IIA SUGRA solution

Eigenvalue dist. of matrices

Geometry

Range of the eigenvalues

Typical geometric scale

I’m going to show

how the geometry can be constructedfrom the Plane Wave Matrix Model!

Plan of Talk
1) Gravity Side
2) PWMM (Gauge Theory)3) Geometry from PWMM
4) Summary, Some Notes and Future Works

Gravity SideThe IIA SUGRA solutions dual to SU(2|4) symmetric theories
have R×SO(3)×SO(6) isometry.
The metric can be written by a single function V(r,z):[Lin-Lunin-Maldacena ’04, Lin-Maldacena ’05]

ds210 =
C3 =V̈2V̇

1/2V̈VV V̇
2224
dt 2(dr +dz )+4d 5 +2d 22 ,VV̈ 2V̇
V̇!
⇣(V̇ 2 )0V̇ 2 V 00
4dt ^ d⌦2 , B2 =+ 2z d⌦2 , · · ·= (V̈ 2V̇ )V 00
2(V̇ 0 )2⌘

EoM and Killing spinor eq. for the geometry
V satisfies the Laplace eq. in a 3D axisymmetric system.
Reducing to axisymmetric electrostatic problems
with “conducting disks.”

Gravity Side
General geometry:
The geometries arelabeled by {N2(s),N5(s)}s=1,...,¤¤=1 case:✓V (r, z) = V0 r2 z◆

2 3z + ()V0 R33Z RRdt

11p+pr2 + (z + d + it)2
r2 + (z d + it)2

V0: constant, R: radius of disk, ∙=d/R, ¯(∙): function of ∙f (x)1⇡Z 12
dy 2
4 + (x1f (y) = 12y)2 2x()!

f (t).
⇡PWMM(Gauge Theory Side)
PWMM is a 1D Matrix Theory with SU(2|4) symmetry.
1
S= 2gZ[Berenstein-Maldacena-Nastase ’02]

⇣1

12 1
2 1
[Xa , Xm ]2d⌧ Tr (D⌧ Xa ) + (D⌧ Xm ) + (2✏abc Xc i[Xa , Xb ])
2
242
⌘
1
1
D⌧ = @⌧ i[A⌧ , ⇤][Xm , Xn ]2 + Xm X m + fermions
42
2

a,b=1,2,3, m,n=4,...,9The vacua are given by SU(2) rep.
(s)X̂a =2(1N (s)s=12[N5 ]La)

La[N5(s)]: spin (N5(s) — 1)/2 rep. mat.
N2(s): multiplicity of the spin (N5(s) — 1)/2 rep.

・・・labeled by {N2(s),N5(s)}s=1,...,¤

So as to construct geometries
Consider simple vacua (¤=1) from now on.

Then apply the localization method to PWMM
and get the same equation as that obtained on the gravity side:f (x)

1
⇡Z 1

2
dy 2
4 + (x1f (y) = 12y)2 2x()Localization
Let us apply the localization method to PWMM.

[Y.A.-Ishiki-Okada-Shimasaki ’12]

- SUSY: quarter BPS sector such that(⌧ ) = 2( X3 (⌧ ) + sinh(⌧ )X8 (⌧ ) + i cosh(⌧ )X9 (⌧ )) is invariant.

- B.C.: all fields approach to the vacuum configurations at ¿→∞.
- V : ±sV is SUSY-invariant and positive-definite.

V=d Tr[s+ Vghost ]Then,Z(t) :=ZdZ(t)=0dt

DXe S[X] t s VX̂a (⌧ ) =Z

DXe S[X] = Z(0) = Z(1)Localized around ±sV=0 !
2⇤
M
s=1(s)[N5 ](1N (s) ⌦ La)
2MX̂9 (⌧ ) =cosh(⌧ )([La , M ] = 0)

Localization
Hence, expectation values of any supersymmetric operators can beobtained exactly by 1-loop calculations.
hYaTrfa ( (⌧a ))i = hY

Trfa (4L3 + 2iM )iM MaWhen ¤=1, ⟨ ⟩MM is evaluated by the following partition function:
N5 1 N2Z=dqii

J=0{(2J + 2)2 + (qi qj )2 }{(2J)2 + (qi2 + (q2 }2{(2J+1)q)iji>jqj ) 2 }e2N5g2i qi2.

qi: eigenvalues of Q, where M = Q ⌦ 1N5

Interacting Fermi Gas System
In the large-N5 limit, the partition fn. can be written as
Z=Z YidqiN2Yi>jtanh

2 ⇡(qiqj )2

exp

Repulsive force(2N52N5 X 2q+g2 i i(2N5 )2 + (qiCentral forceqj ) 2···)

Attractive force

By using the Cauchy id.,N2tanh(qii=jqj )2

N2=( )
SN 21( )i=1 cosh(qi q (i) )2[Marino-Putrov ’12]

Interacting Fermi gas system with hamiltonian
H=N2X
i=1log cosh pi +N2
X
2N5g2

i=1

qi2

N2X
12i6=j

2N5(2N5 )2 + (qiqj ) 2

Interacting Fermi Gas SystemWe consider the limit in which the SUGRA approx. is valid:
large-N2 , large-N5 , ¸:=g2N2 ≫N5・・・Thomas-Fermi approximation at zero temperature

The system is described by mean-field density ½(q).
=eigenvalue density

It is determined by the following saddle point eq.:µ=2N5 2(q) + 2 q
gqm

2N5dq
2 + (q(2N)5qmµ : chemical potential
qm: upper limit of the support of ½(q)

q)

(q)
2

Construction of geometry⇡⇢(qm x)µZ 1f (x)2N5 /qmdy(2N5 /qm )2 + (x11⇡Z 1

2dy 24 + (x1⇢(qm y)=12y)µf (y) = 12y)2
2N5 qm2xµg 22 2x()Construction of geometry⇡⇢(qm x)µZ 1

2N5 /qmdy(2N5 /qm )2 + (x1⇢(qm y)=12y)µ2
2N5 qm2xµg 2

Identification to obtain the gravity pictureµ⇢(q) = f (q/qm ) ∝ charge density on the disk
⇡By this identification, the range of the eigenvalue dist. can bewritten by the quantities on the gravity side:
2qm = R
⇡so RS2 = 2⇡qm
5The gravity solution corresponding to a simple vacuum inPWMM has been constructed from the gauge theory side!

D2-and NS5-brane limit[Ling-Mohazab-Shieh-Anders-Raamsdonk ’06]
On the gravity side,these two double scaling limits are known:

D2-brane limit

NS5-brane limit

d
=R

!1

!0

RS2 5 = 2⇡(8122RS 5 = ⇡(3gS 2 N2 ) 3)

14

On the gauge theory side, solutions in the corresponding limits are"✓µ⇢(q) =1⇡qqm◆2 #,qm =✓3⇡
8N5◆ 31"µqm1⇢(q) =3⇡N5✓qqm◆2 # 32Then, RS2 = 2⇡qm
reproduce the same results as those on the gravity side.
5, qm = (8 )14Summary- The mean-field density, that is, the eigenvalue distribution in thematrix integral determines the geometry.
- The range of the eigenvalue dist. corresponds to S5 radius.

Some Notes and Future Works- We can also reconstruct geometries dual to general vacua (¤≠1)
in the same way.
- This method also works for other SU(2|4) symmetric theories
such as SYM on R×S2 and R×S3/Zk.
- Exact proof for the NS5-brane limit. (Work in progress)