ゲージ重力対応における「バブリング」の考え方
土屋麻人（静岡大学）

Introduction•Candidates for nonperturbative definition of superstring~ matrix model, large-N gauge theory•
Proposed models~reduction of 10D
SYM to lower dimensions
1. 0D IIB matrix model
IKKT
type IIB superstring
2. 1D Matrix theory
BFSSM theory
3. 2D Matrix string theory DVV
type IIA superstring
4. 4D
••
•
•

SYM

type IIB superstring
onIn 1 and 2, space-time is dynamically generated throughdistribution of matrix eigenvalues `emergence of space-time’fluctuation of eingenvalue dist.~ particle talks by Kawai, Aoki,
Umetsu , Nishimura andA similar interpretation for 4?
Suginois directly derived from SYM?
This partially succeeds in bubbling AdS
Lin-Lunin-Maldacena
Maldacena

Contents
12
34
56

Introduction
Analogy with c=1 string
Bubbling AdS
Superstar
Berenstein’s model
Conclusion and outlook

Analogy with c=1 stringmatrix quantum mechanics(open string picture)

2D string with linear dilaton b.g.
(closed string picture)open-closed
duality: N free fermionsFermi surface

Das-Jevicki
Gross-Klebanov
PolchinskiMcgreevy-Verlinde

N-1eigenvalue distribution ~ fermi surface
fluctuation of eigenvalue distribution~waveson fermi surface
isolated eigenvalue

Kuroki’s talkone dimension
closed string tacyon
ZZ brane(open string picture)(closed string picture)

N=4 SYM~theory on D3branes

type IIB string on AdS5xS5open-closedduality

AdS5xS5~near horizon limit of D3-brane
solutionD3-brane in AdS5xS5

N-1

chiral primary states
matrix model ~ free fermions

eigenvalue distribution
fluctuation of eigenvalue distribution~waveson fermisurface
isolated
eigenvalue

part of S5
KK graviton(AdS) giant graviton (D3-brane)

Bubbling AdSLandau problem• charged particle in magnetic field in 2D

2D harmonic oscillator
angular momentum• classical ground state (lowest Landau level)x2

x1

•complex coordinate•creation-annihilation operators•eigenstates• Lowest Landau level
J=nholomorphic except for
exponential factor

system of N fermionssubtract the minimum
momentum and rename it as Jstate with J=0
AdS5 x S5

from the total angularclassification of states with the total angular momentum J

Young tableau specifies irreducible
representations of GL(N,C) and SJ
simultaneously
Schur polynomial

Kimura’s talkCorley-Jevicki-Ramgoolam

giant graviton

AdS giantgraviton

KK graviton

relation to phase space of 1D harmonic oscillator
Wigner phase space distribution for 1D harmonic oscillator

relation between charge density and WPSD

in the classical limit

Iso-Karabali-Sakita

Chiral primary operators and giant gravitons
SYM on• chiral primary operators of type• half-BPS• extremal correlator• non-renormalization theorem
extremal correlators is independent ofcan be calculated using the free partEden-Howe-Sokatchev-Westpropagatorit•KK graviton with angular momentum J~O(1)• giant graviton = D3-brane wrapped on S3 in S5
McGreevy-Suskind-Toumbas
Balasubramanian-Berkooz-Naqvi-Strassler

J~O(N)size

R: radius of S5•AdS giant graviton = D3-brane wrapped on S3 in AdS5
Hashimoto-Hirano-Itzhaki
Grisaru-Myers-Tafjord
Corley-Jevicki-Ramgoolam

J~O(N)

×no restriction on J×
KK graviton

×
giant graviton

dual giantOrhogonal basis

SYM on• global coordinates of AdS5boundary• relation betweenandconformallyequiv.

on• operator on

onstate on• chiral primary statess-wave of KK decompositions
only t-dependence• preserve 16 susy ×R ×SO(4)×SO(4)R-symmetry SO(6)•free part of~ complex matrix model

kinetic terms-wave on S3
Hashimoto-Hirano-Itzhaki
Corley-Jevicki-Ramgoolam
Berenstein
Takayama-A.T.

coupling of conformal matter tocurvature of S3

operator-state correspondence

giant graviton

AdS giantgraviton

KK gravitonBubbling AdS

Lin-Lunin-Maldacena• general form of half-BPS sof type IIB sugra that preserve
R ×SO(4)×SO(4) isometry

characterized by a single function z(x1,x2, y)• differential eq. satisfied by z
• z must take ½ (white) or -½ (black)on the plane y=0 for the solution to be smoothconfiguration of droplet

geometry• shrinking S3
y=0

topology of droplet(bubble)topology of space-time• fluxArea is quantized• example

x2r0

geometry
AdS/CFT duality
quantized !

area of droplet

x2droplet and S5r0x1x1

Identification with Fermi droplets• quantization of area
area of droplet~

flux

area of dropletx2•enegy and angular momentumx1

total energy of N fermions• ground state

ground state energyx2

r0 x1

ground state offermionsKK graviton and (AdS) giant graviton• KK graviton with angular momentum Jx2~KK graviton with angularx1

momentum J
Grant-Maoz-Marsano-Papadodimas-Rychkov• giant graviton with J～O(N) • AdS giant graviton with J～O（N）
x2x2x1x1

Further developments• General half-BPS operators
YoneyaCuntz algebra+free fermions• Bubbling Wilson loop
Yamaguchi, Kuroki’s talk
• IIA bubbling~ SU(2|4) symmetric solution of IIA sugra
Lin-Maldacena

Electrostatics in an axially symmetric system in 3D
gravity duals of Vacua in SU(2|4) symmetric theories
Relations between SU(2|4) symmetric theories
Ishiki-Shimasaki-Takayma-Tsuchiya, Shimasaki’s talk…………

Superstar

extremal limit of black hole with one charge in 5D N=2 gauged
SUGRA → lifted to ten dimensions
Behrndt-Chamseddine-SabraSuperstar (ten-dimensional form)

naked singularity at

Behrndt-Cvetic-Sabra
Cvetic et. al.

LLM geometry

CTC

regularregion
naked singularity

chronology protection

Caldarelli-Klemm-Silva
Milanesi-O’Loughlin

Pauli exclusion principlesmooth geometry

pure statesuperstar,
hyperstar

mixed states
Ensemble of Young diagrams~ limit shape
entropy

entropy?

Balasubramanian-de Boer-Jejjala-Simon
Buchel, Suryanarayana, ,,,,,,,,,

Berenstein’s model
Berenstein, Berenstein-Correa-Vazquez, Berenstein-Vazquez,
Berenstein-Cotta, Berenstein-Cotta-Leonarditruncation
keep only s-wave on S3 of six scalars

andsupersymmetric states excite only these modes in the free limit
and supersymmetry should account for cancelations of quantum
corrections

integration overhamiltonian

large ‘t Hooft coupling

Gauss law constrainthamiltonian

inner productwave function for ground state

redefinition of wave function

probability distribution for N particlescf.) d=2

N →∞ limit
constraint

saddle point approximation

contradiction

forforConfirmed by numerical simulation

OKhamiltonian for off-diagonal partBMN-type stateenergyGiant magnonii’Summary• Emergent geometry in AdS/CFT• Analogy with c=1 string• Bubbling AdS ~ chiral primary states
emergence of part of S5• chronology protection, mixed states• Attempt for emergence of full S5outlook• emergence of AdS5xS5• thermodynamics• 1/N correction (string correction)• Implication for IIB matrix model•例1

pp wave geometryx2

x1