On String Theory Duals of Lifshitz-like Fixed Point Tatsuo Azeyanagi (Kyoto University) Based on work arXiv: (to appear in JHEP) with Wei Li (IPMU) and Tadashi Takayanagi (IPMU) YITP workshop July 2009

Introduction (1) Generalization of AdS/CFT correspondence to deformed AdS spacetimes Gravity dual of QFT with Schrodinger symmetry Gravity dual of Lifshitz(-like) fixed point [Son, Balasubramanian-McGreevy ‘08] [Kachru-Liu-Mulligan ‘08] our scope Yoshida-san’s talk Motivation Application to condensed matter physics (not Lorentz inv.), Example of gravity duals with non-relativistic symmetry To find gravity duals with non-relativistic symmetry

Introduction (2) Very brief review of Lifshitz point Characterized by anisotropic scaling symmetry (along the temporal or/and spatial directions) Example ) anisotropic spin system [Hornreich-Luban-Shtrikman ‘75] [Becerra-Shapira-Oliveira-Chang ‘80] Scaling symmetry at the Lifshitz point

Introduction (3) Gravity dual of Lifshitz fixed point [Kachru-Liu-Mulligan ‘08] (anisotropic in the temporal direction) In order to understand microscopic interpretation of the system, the embedding into string theory is indispensable. Preceding works and current status Scaling symmetry We derive gravity duals of a kind of Lifshitz-like fixed point within the framework of string theory. ( generalization to more general scaling symmetry [Pal ’08,09] )

Outline 1. String theory duals of Lifshitz-like fixed points 3. Relation to AdS5 2. Properties of the scaling solution 0. Introduction 4. Conclusions

Lifshitz-like fixed point Spatially anisotropic scaling symmetry We realize anisotropic behavior along direction Since this geometry is almost the same as AdS5, we try to derive the scaling solution as a deformation of it. Our target : gravity dual (scaling solution) of QFT with → gravity dual expected :

Brane configuration Realized as a D3-D7 system D3 D7 ×××× ×××××××× Supersymmetry expected to be broken completely is in this direction Cf. [Fujita-Li-Ryu-Takayanagi ‘09]

Ansatz Anisotropic scaling in this direction Axion flux along direction Einstein manifold Ansatz We start with Type IIB supergravity in string frame with field contents : metric fluxes

Gravity Dual A new Solution rewritten in Einstein frame Scaling symmetry ( z=3/2 ) (Boundary: ) → r-dependent, diverges at the boundary (discuss later) Lifshitz-like scaling is available only in the Einstein frame

Generalization to black hole the other fluxes is the same as ones with zero temperature Thermodynamic quantities Generalization to BH solution is straightforward: fractional behavior Cf. scaling dimension

Properties of the scaling solution(2-1) Linear perturbation From the asymptotic behavior of the fluctuation modes, we can determine the scaling dimensions of operators in field theory side. We can determine the gravitational stability of the scaling solution ( for non-supersymmetric solution the stability is non-trivial ). Linear perturbation around the scaling solution provides us valuable informations: - -

Properties of the scaling solution (2-2) 1. Gauge fixing of the fluctuations 2. Expansion by the spherical harmonics 3. Analyze the asymptotic behavior of the fluctuations Step for the linear perturbation(Assuming ) ・・ ・ Cf. [Kim-Romans-Nieuwenhuizen ‘85]

Properties of the scaling solution (2-3) Stability of the scaling solution Scaling dimension for a mode with mass m For unstable modes, this part becomes negative (cf. BF bound for AdS ) Actually there is an unstable mode : k=2 is unstable Bound for stability

Relation to AdS 5 Dilaton diverges when approaching the boundary Actually, there exists a solution interpolating AdS 5 and the scaling solution a la holographic RG flow. Simplifying the ansatz ← realized by redefinition of and Imposed such that the radius of the Einstein manifold is constant in the Einstein frame. AdS5 and the scaling solution are included in this class.

Holographic RG flow (1) → Two fixed points Physical condition (such that the axion flux is real) Two equations to solve e.o.m reduce to 2 equations with one physical condition

Holographic RG flow (2) Scaling solution (UV) (IR)

Physical interpretation UVIR Scaling solution( z=3/2 ) AdS5 solution ( z=1 ) Dilaton asymptotes to a constant at the boundary. No divergence !

Interpretation of RG flow Our system is realized as a D3-D7 system Axion flux in the bulk ⇔ the theta-term is induced in the YM side We can obtain the QFT at the Lifshitz-like fixed point from N=4 SYM via RG flow triggered by the theta-term perturbation

Conclusions We embed gravity duals of QFT with spatially anisotropic scaling symmetry into string theory. We derived a scaling solution relating AdS5 (UV) to the scaling solution a la holographic RG flows We also analyzed properties of the scaling solution by calculating some physical quantities and analysing the stability. Future direction To realize string embedding of KLM solution (or No-go theorem?). To derive a constant dilaton scaling solution within string theory To understand physical meaning of the instability

Properties of the scaling solution (1-1) Entanglement entropy (EE) A B A B Holographic derivation of entanglement entropy [Ryu-Takayanagi ‘06] : minimal surface with boundary Hilbert space : Density matrix for the subsystem A : a measure for D.o.F., useful to characterize zero temperature system … By holography, EE is derived geometrically

Properties of the scaling solution(1-2) Entanglement Entropy Fractional scaling (divide along x direction) Cf. scaling dimension

Properties of the scaling solution(1-3) Entanglement Entropy (divide along w direction) Cf. scaling dimension Deviation due to anisotropy

Kachru-Liu-Mulligan fluxes metric where A solution with Lifshitz scaling symmetry