111 On gravity duals for NRCFTs KEK Jun. 9, 2009 Kentaroh Yoshida Based on the works, Sean Hartnoll, K.Y, arXiv: , Sakura Schäfer Nameki, Masahito Yamazaki, K.Y, arXiv: Dept. of Phys. Kyoto Univ. Gravity

INTRODUCTION Application of AdS/CFT: quark-gluon plasma (hydrodynamics) 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 [Gubser, Hartnoll-Herzog-Horowitz] [Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi] EX. Most of condensed matter systems are non-relativistic (NR) AdS/CFT correspondence

333 NR limit in AdS/CFT Gravity (string) on AdS space CFT ? ? ? What is the gravity dual ? NRCFT NR conformal symmetry Let us discuss gravity solutions preserving Schrödinger symmetry Schrödinger symmetry EX fermions at unitarity Today (or other NR scaling symmetry)

444 Plan of the talk 1. Introduction (finished) 2. Schrödinger symmetry 3. Coset construction of Schrödinger spacetime 4. String theory embedding 5. Summary and Discussion [Hartnoll-K.Y] [S.Schafer-Nameki-M.Yamazakil-K.Y]

Schrödinger symmetry

666 What is Schrödinger algebra ? Non-relativistic analog of the relativistic conformal algebra Conformal Poincare Galilei Schrödinger algebra = Galilean algebra + dilatation + special conformal EX Free Schrödinger eq. (scale inv.) Dilatation (in NR theories) [Hagen, Niederer,1972] dynamical exponent

77 Special conformal trans. The generators of Schrödinger algebra C has no index = Galilean algebra a generalization of mobius tras.

88 The Schrödinger algebra Dynamical exponent Galilean algebra SL(2) subalgebra Dilatation Special conformal (Bargmann alg.)

99 Algebra with arbitrary z Dynamical exponent Galilean algebra Dilatation + M is not a center any more. conformal trans. C is not contained.

10 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 FACT A relativistic conformal algebra in (d+1)+1 D The generators: This is true for arbitrary z.

11 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)

12 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.

13 3. Coset construction of Schrödinger spacetime [Sakura Schafer-Nameki, M. Yamazaki, K.Y., ]

14 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)

15 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., ] 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.

16 1. MC 1-from vielbeins 2. Contaction of the vielbeins: vielbeinsspin connections Coset 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.

17 Nappi-Witten’s argument 2D Poincare with a central extension Killing form (degenerate) P1P2JTP1P2JT Most general symmetric 2-form The condition for the symm. 2-form PP-wave type geometry [Nappi-Witten, hep-th/ ] right-G inv.

18 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 ? Problems Killing form is degenerate Is it possible to apply the NW argument straightforwardly?

19 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

20 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?

21 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

22 Structure constants: Let’s consider the following case: D M D

23 vielbeins: Similarly, we can derive the metric for the case with an arbitrary z. where 2-form: ( has been absorbed by rescaling. ) metric: coordinate system

24 Gravity dual to Lifshitz fixed point Let’s consider algebra: Take 2-form: vielbeins: metric: [Kachru-Liu-Mulligan, ] : a subalgebra of Sch(2) Unique!

25 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.

26 4. String theory embedding of Schrödinger spacetime [S.A. Hartnoll, K.Y., ]

27 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

28 2. brane-wave deformation [Hartnoll-KY] Allow the coordinate dependence on the internal manifold X 5 The function has to satisfy For we know the eigenvalues: Thus the spherical harmonics with gives a Schrödinger inv. sol. Our idea The sol. preserves 8 supertranslations (1/4 BPS) super Schrödinger symm. Only the (++)-component of Einstein eq. is modified. EX

29 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

30 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

31 Scalar field fluctuations S 5 part Laplacian for S 5 : 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

32 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]

33 5. Summary and Discussion

34 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

35 Thank you!

36 DLCQ and deformation DLCQ (x - -cpt.) pp-wave def. Sch. symm. [Son, BM] [Goldberger et.al] in the context of pp-wave Historical order -compactification is important for the interpretation as NR CFT