On String Theory Duals of Lifshitz-like Fixed Point

1. On String Theory Duals of Lifshitz-like Fixed Point Tatsuo Azeyanagi (Kyoto University) Based on work arXiv:0905.0688 (to appear in JHEP) with Wei Li (IPMU) and Tadashi Takayanagi (IPMU) YITP workshop July 2009

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

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

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

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

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

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

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

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

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

11. Properties of the scaling solution(2-1) Linear perturbation Linear perturbation around the scaling solution provides us valuable informations: - 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 ).

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

13. 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 ) Bound for stability Actually there is an unstable mode : k=2 is unstable

14. Relation to AdS5 Dilaton diverges when approaching the boundary Actually, there exists a solution interpolating AdS5 and the scaling solution a la holographic RG flow. Simplifying the ansatz ← realized by redefinition ofand 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.

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

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

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

18. 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

19. Conclusions We embed gravity duals of QFT with spatially anisotropic scaling symmetry into string theory. We also analyzed properties of the scaling solution by calculating some physical quantities and analysing the stability. We derived a scaling solution relating AdS5 (UV) to the scaling solution a la holographic RG flows 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