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On gravity duals for NRCFTsOn gravity duals for NRCFTs

Seminar @ KEKSeminar @ KEKJun. 9, 2009Jun. 9, 2009

Kentaroh YoshidaKentaroh Yoshida

Based on the works,Based on the works,

Sean Hartnoll, K.Y, arXiv:0810.0298,

Sakura Schäfer Nameki, Masahito Yamazaki, K.Y, arXiv:0903.424

5

Sean Hartnoll, K.Y, arXiv:0810.0298,

Sakura Schäfer Nameki, Masahito Yamazaki, K.Y, arXiv:0903.424

5

Dept. of Phys. Kyoto Univ.

Gravity

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1. INTRODUCTION

Application of AdS/CFT: 　 quark-gluon plasma (hydrodynamics)Application of AdS/CFT: 　 quark-gluon plasma (hydrodynamics)

condensed matter systems (superfluidity)condensed matter systems (superfluidity)

Gravity (string) on AdS space CFT

Quantum gravity, non-perturbative definition of string theory

Application of classical gravity to strongly coupled theory

superconductor

quantum Hall effect

superconductor

quantum Hall effect

[Gubser, Hartnoll-Herzog-Horowitz][Gubser, Hartnoll-Herzog-Horowitz]

[Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi][Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi]

EX.EX.

Most of condensed matter systems are non-relativistic (NR)

AdS/CFT correspondenceAdS/CFT correspondence

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NR limit in AdS/CFT NR limit in AdS/CFT

Gravity (string) on AdS space CFTGravity (string) on AdS space CFT

? ? ? What is the gravity dual ? NRCFT ? ? ? What is the gravity dual ? NRCFT

NR conformal symmetry NR conformal symmetry

Let us discuss gravity solutions preserving Schrödinger symmetry Let us discuss gravity solutions preserving Schrödinger symmetry

Schrödinger symmetry EX fermions at unitarityEX fermions at unitarity

Today Today

(or other NR scaling symmetry)

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Plan of the talkPlan of the talk

1. Introduction (finished)

2. Schrödinger symmetry

3. Coset construction of Schrödinger spacetime

4. String theory embedding

5. Summary and Discussion

1. Introduction (finished)

2. Schrödinger symmetry

3. Coset construction of Schrödinger spacetime

4. String theory embedding

5. Summary and Discussion

[Hartnoll-K.Y][Hartnoll-K.Y]

[S.Schafer-Nameki-M.Yamazakil-K.Y][S.Schafer-Nameki-M.Yamazakil-K.Y]

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2. Schrödinger symmetry2. Schrödinger symmetry

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What is Schrödinger algebra ?What is Schrödinger algebra ?

Non-relativistic analog of the relativistic conformal algebraNon-relativistic analog of the relativistic conformal algebra

Conformal 　　　　 Poincare GalileiConformal 　　　　 Poincare Galilei

Schrödinger algebra

= Galilean algebra + dilatation + special conformal

EX

Free Schrödinger eq. (scale inv.)

Dilatation (in NR theories)Dilatation (in NR theories)

[Hagen, Niederer,1972]

dynamical exponentdynamical exponent

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Special conformal trans.

The generators of Schrödinger algebra

C has no index

= Galilean algebra

a generalization of mobius tras.

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The Schrödinger algebra

Dynamical exponent

Galilean algebra

SL(2) subalgebra

Dilatation

Special conformal

(Bargmann alg.)(Bargmann alg.)

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Algebra with arbitrary z

Dynamical exponent

Galilean algebra

Dilatation

+

• M is not a center any more.

• conformal trans. C is not contained.

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A Schrödinger algebra in d+1 D is embedded into

a ``relativistic’’ conformal algebra in (d+1)+1 D as a subalgebra.

EX. Schrödinger algebra in 2+1 D can be embedded into SO(4,2) in 3+1 D

FACTFACT

A relativistic conformal algebra in (d+1)+1 DA relativistic conformal algebra in (d+1)+1 D

The generators:

This is true for arbitrary z.

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A light-like compactification of Klein-Gordon eq.

with

The difference of dimensionality

Rem: This is not the standard NR limit of the field theory

(d+1)+1 D

d+1 D

The embedding of the Schrödinger algebra in d+1 dim. spacetime

LC combination:

KG eq.

Sch. eq.

Remember the light-cone quantization

(Not contained for z>2)

(For z=2 case)

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Application of the embedding to AdS/CFT

The field theory is compactified on the light-like circle:

with -compactification

[Goldberger,Barbon-Fuertes ]

DLCQ description

But the problem is not so easy as it looks.

What is the dimensionally reduced theory in the DLCQ limit?

CFT

Gravity

Symmetry is broken from SO(2,d+2) to Sch(d) symmetry

NRCFT = LC Hamiltonian

The DLCQ interpretation is applicable only for z=2 case.

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3. Coset construction of

Schrödinger spacetime

3. Coset construction of

Schrödinger spacetime

[Sakura Schafer-Nameki, M. Yamazaki, K.Y., 0903.4245]

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Schrödinger spacetime

deformation term AdS space

This metric satisfies the e.o.m of Einstein gravity with a massive vector field

[Son, Balasubramanian-McGreevy]

There may be various Schrödinger inv. gravity sols.

other than the DLCQ of AdS backgrounds.

Deformations of the AdS space with -compactification

preserving the Schrödinger symmetry

(degrees of freedom of deformation)

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Coset construction of Schrödinger spacetime

A homogeneous space can be represented by a coset

EX

: isometry, : local Lorentz symmetry

[S. Schafer-Nameki, M. Yamazaki, K.Y., 0903.4245]

We want to consider degrees of freedom to deform the AdS metric

within the class of homogeneous spacetime by using the coset construction.

As a matter of course, there are many asymptotically Schrödinger inv. sol.

but we will not discuss them here.

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1. MC 1-from

vielbeins

2. Contaction of the vielbeins:

vielbeins spin connections

Coset construction of the metricCoset construction of the metric

symm. 2-form

If G is semi-simple straightforward (Use Killing form)

But if G is non-semi-simple, step 2 is not so obvious. (No non-deg. Killing form)

NOTE: MC 1-form is obviously inv. under left-G symm. by construction.

The remaining is to consider right-H inv. at step 2.

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Nappi-Witten’s argument

2D Poincare with a central extension

Killing form (degenerate)

P1

P2

J

TMost general symmetric 2-form

The condition for the symm. 2-form

PP-wave type geometry

[Nappi-Witten, hep-th/9310112]

right-G inv.

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NW-like interpretation for Schrödinger spacetime ?

G :Schrödinger group is non-semisimple

Q1. What is the corresponding coset ?

Q2. What is the symmetric 2-form ?

ProblemsProblems

Killing form is degenerate

Is it possible to apply the NW argument straightforwardly?

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Ans. to Q1. Physical assumptions

A candidate for the coset

Assump.1　　 No translation condition. doesn’t contain

Assump.2 Lorentz subgroup condition. contains and

Q1. What is the corresponding coset ?

Due to , is not contained in the group H

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Ans. to Q2. NW argument?

However, the Schrödinger coset is NOT reductive.

Reductiveness :

Nappi-Witten argument is not applicable directly.

Q2. What is the symmetric 2-form ?

It is possible if the coset is reductive

EX pp-wave, Bargmann

How should we do?

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The construction of symm. 2-form for the non-reductive case [Fels-Renner, 2006]

The condition for the symm. 2-form

A generalization of NW argument

The indices [m],… are defined up to H-transformation

The group structure const. is generalized.

H-invariance of symm. 2-form

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Structure constants:

Let’s consider the following case:

D

M D

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vielbeins:

Similarly, we can derive the metric for the case with an arbitrary z.

where

where

2-form:

( has been absorbed by rescaling . )

metric:

coordinate system

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Gravity dual to Lifshitz fixed point

Let’s consider

algebra:

Take

2-form: vielbeins:

metric: [Kachru-Liu-Mulligan, 0808.1725]

: a subalgebra of Sch(2)

Unique!

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Lifshitz model (z=2)

4th order

scale invariance with z=2

2nd order

The theory, while lacking Lorentz invariance, has particle production.

But the symmetry is not Schrödinger.

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4. String theory embedding of

Schrödinger spacetime

4. String theory embedding of

Schrödinger spacetime

[S.A. Hartnoll, K.Y., 0810.0298]

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String theory embedding

Known methods:

1. null Melvin Twist (NMT) TsT transformation

2. brane-wave deformation

[Herzog-Rangamani-Ross]

[Hartnoll-K.Y.]

[Maldacena-Martelli-Tachikawa] [Adams-Balasubramanian-McGreevy]

1. null Melvin Twist extremal D3-brane

EX

NMT, near horizon

Non-SUSY

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2. brane-wave deformation [Hartnoll-KY]

Allow the coordinate dependence on the internal manifold X5

The function has to satisfy

For we know the eigenvalues:

Thus the spherical harmonics with gives a Schrödinger inv. sol.

Our ideaOur idea

The sol. preserves 8 supertranslations (1/4 BPS) super Schrödinger symm.

Only the (++)-component of Einstein eq. is modified.

EX

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The solution with an arbitrary dynamical exponent z

Dynamical exponent appears

The differential eq. is

For case = a spherical harmonics with

The moduli space of the solution is given by spherical harmonics

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The solution with NS-NS B-field

Here is still given by a spherical harmonics with

The function has to satisfy the equation

By rewriting as

has been lifted up due to the presence of B-field

non-SUSY

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Scalar field fluctuations

S5 part

Laplacian for S5 :Eigenvalues for S5

Eq. for the radial direction:

The solution: (modified Bessel function)

: large negative : pure imaginary (instability) (while is real)

The scaling dimension of the dual op. becomes complex

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Characteristics of the two methods

Applicable to the finite temperature case. Schrödinger BH sols.

Non-SUSY even at zero temperature.

SUSY backgrounds (at most 8 supertranslations, i.e., 1/4 BPS)

Difficult to apply it to the finite temperature case

1. null Melvin Twist

2. brane-wave deformation

Instability

Only for z=2 case (?)

Applicable to arbitrary z spherical harmonics

Generalization of our work: [Donos-Gauntlett] [O Colgain-Yavartanoo] [Bobev-Kundu]

[Bobev-Kundu-Pilch] [Ooguri-Park]

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5. Summary and Discussion5. Summary and Discussion

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Summary

1. Coset construction of Schrödinger spacetime

[Hartnoll-KY]

[Schafer-Nameki-Yamazaki-KY]

2. Supersymmetric embedding into string theory (brane-wave)

applicable to other algebras

1. If we start from gravity (with the embedding of Sch. algebra)

Difficulty of DLCQ (including interactions)

2. If we start from the well-known NRCFTs (with the conventional NR limit)

What is the gravity solution?

Discussion

No concrete example of AdS/NRCFT where both sides are clearly understood.

NR ABJM gravity dual

Gravity dual for Lifshitz field theory

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Thank you!Thank you!Thank you!Thank you!

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DLCQ and deformation

1

2

3

DLCQ (x- -cpt.) pp-wave def.

Sch. symm.

[Son, BM]

[Goldberger et.al]

2002- in the context of pp-wave

Historical order

-compactification is important for the interpretation as NR CFT