My talk Two different points of view:

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1 Shin Nakamura (Dept. Phys. Kyoto Univ.) Reference: S.N., Hirosi Ooguri, Chang-Soon Park, arxiv: [hep-th] (to appear in Phys. Rev. D) ( k B = h= c=) My talk Two different points of view: rom the viewpoint of general relativity: Stability analysis of a charged black hole and find a new instability. rom the viewpoint of condensed-matter physics: The gravitational system may be dual to a system with spontaneous generation of spatially modulated, helical current (of global charge).

2 Black hole and holography dm = T H d (area) G If we want to make a correspondence with the thermodynamics in the 3+ dimensional world, the horizon has to be 3+ dim. surface. The gravitational theory has to have at least one extra direction(the radial direction): 5d gravity Black holes in asymptotically flat spacetime has negative specific heat. If they are in the asymp. dsspacetime, the specific heat is positive. 5d ds-bh If we include Maxwell field in addition to gravity, the black hole can carry the charge: dm = T H d(area) + dq 0 Introduction of finite density G 5 d asymptotically dscharged black hole (ds-reissner-nordström black hole) The analogy was promoted to be a duality: ds/ct Dual to d N= SYM theory with R-charge. global charge But, Einstein+Maxwell is not very enough.

3 The gravity side 5-dim. Einstein (Λ<0)+ Maxwell theory with a CS term. M 6πG5 L= g R+ M + Why CS term? α ε 3! IJKLM The U() truncation of the S 5 reduction of 0d type IIB supergravity has a CS term with α = 3. ( ) Gauge-theory side: The CS term holographicalyrepresents the R-sym. anomaly of SYM. D. Son s talk on Wednesday I JK LM What does the CS term? Equations of motion: R M g α + ε L g ( + ) = M R ML gm M M IJKL IJ KL = 0. If we consider only electrically charged solutions, the CS term does not affect the solutions. The solution is just an ordinary ds-reissner-nordström black hole. However, if consider the fluctuations around the solution, the CS term plays animportant role.,

4 flat-space excercise Maxwell + CS on the 5d Minkowski(without gravity) M M α + ε IJKL IJ KL = 0. If we have a background electric field 0 =E, the e.o.m. under the Lorenz gauge *is + αeε 0 KL K L + O ( ) = 0. iωt+ ikx or linear perturbations of = a e, 3 αe( ik) + αe( ik) 3 = 0, = 0. (* we can also formulate in a gauge invariant way.) 3 αe( ik) + αe( ik) 3 ( ) ± i = 0, = 0. () () () ( ω k ) ± i) mαek( ± i) = 0 The dispersion relation is ω ( 3 3 ( k± αe) = (αe) tachyonic for the circular-polarized modes. This does not always mean the instability, but the system is unstable in the following momentum range: 0 < k <αe

5 Dispersion relation The dispersion relation is non-standard: roton-like dispersion ω unstable range k The minimum of the spectrum is located at k=αe 0. The instability is proportional to αand E. Inhomogeneous condensation The instability occurs at finite momenta. The resulting condensation of the gauge field is spatially modulated (and circularly polarized). Spontaneous breaking of translationaland rotational symmetry.

6 Cartoon of the unstable mode momentum k Helical strudture Spontaneous breaking of the rotational and the translational symmetries What happens in the black-hole geometry? The background spacetime is asymptotically ds: slightly tachyonic mode is still stable if it satisfies the Breitenlohner-reedman bound. Breitenlohner-reedman, nn. Phys. (98) ; Phys. Lett. (98) 97. We need to check whether the effect of the CS term breaks the B bound or not.

7 Breitenlohner-reedman bound In d+-dimensional ds spacetime, ds = L η µν dx µ dx z ν + dz a scalar fluctuation is stable if its mass m satisfies: d m L Breitenlohner-reedman bound Stable means The amplitude of the field fluctuation does not grow along the time evolution. 5d ds-reissner-nordström black hole BH ds R 3 horizon 5th diresction boundary The charge at a given mass has an upper limit. Consider the maximum charge: extremal, T=0 Consider near the horizon where the electric field is biggest. The near-horizon geometry is much simpler.

8 The near-horizon geometry In my convention, ds = E z B bound: ( dt + dz ), z = 6 g. m ( α ) = 96α E= 6 α = = -8 0 = 3 ( ) 3 (for SUGR) Breaks the B bound. ( ds ) nother point in the BH analysis The Maxwell field couples with gravitons: g g M M = g µν h i µ f νi +... = g + h, f =,( µ, ν) (0,) µ i µ i µ i νi ν i iν The Maxwell field couples to the off-diagonal part of the metric. We need to diagonalize the possible fluctuations.

9 rom the viewpoint of ds The Maxwell field couples to the off-diagonal part of the metric. KK-gauge field from the d (ds ) point of view. ds R 3 (x μ=0,, x, x i=3, ) ds (x μ=0, ) i Maxwell scalar h μi off-diagonalgraviton KK gauge field h i off-diagonal graviton Stückelberg field Diagonalizethese modes and find the lowest mass eigenvalue. L eff g (d) = K ij g ( i) µν K ij (d) ( i) µν Eε µν µ j µ j µ j h h ( i) µ ν j ikh ( i) µ α 6 Eε 3! ijk i jk graviton Maxwell+CS coupling m The equations of motion reduce to j ( ds + ) 0, j fi αeεijkjfk + EεijkjKk = j E f + ( + ) ε K = 0. ds ( α + ) i ds The mass eigenvalue should satisfy: j 6 ( 6α α + 6α ( 6α + + ) ) 3/ = Oops! slightly stable! > -3 (B bound) min + ijk j k fi = εijk det m k αek E = 0, Em m k or SUGR E= 6, α = 3 E = α jk, K i = K ( i) 0 g (d)

10 Result at the near horizon Einstein +Maxwell+CS theory near the horizon of the extremalrn-bhbecomes unstableif the CS coupling α is α> α c = The 5d SUGR has α=/ 3= and barely stable. (Notice: the SUSY is broken even at the extremal case.) ull-geometry analysis The near-horizon analysis provides a sufficient condition for instability but not the necessary condition. Numerical analysis on the full ds-rn-bh geometry: consider normalizable modes, impose in-going boundary condition at the horizon, see whether its amplitude decays or grows along time. The behavior depends on kand αand T. We found that the numerical analysis on the full RN-BH geometryyields the samecritical value α c =0.896.

11 Other geometries We have also analyzed the following cases at the near-horizon limit. 5d extremal ds-rn-bh with the boundary geometry S 3. 3-charge solutions s far as we have analyzed, the B bound is always barely satisfied (within 0.%) if we employ the CS coupling coming from type IIB super-gravity. re there any reason? (We do not know so far.) Can we find some unstable example from SUGR? CS coupling at arbitrary values It is worth while investigating the nature of Einstein+Maxwell+CS theory at arbitrary CS coupling. Presenceof CS term is quite general for gravity theories coming from superstring theory through various compactifications. Let me consider further with arbitrary CS coupling, being motivated by the possible correspondence to yet to known CT.

12 Unstable region The unstable range in k is broaderthan the result in the nearhorizon analysis, although the critical coupling is unchanged. The upper limit can be drawn like this. Unstable region Next slide T=0 near horizon Stable full SUGR Unstable region b: unstable region from the near-horizon analysis

13 ds ~ gµν( z) = holographic interpretation of the instability dx µ z dx ν + dz (0) µ ( z) µ + z () = µ non-normalizable mode source ( (0) 0 =μ) Condensation of the gauge field: +... normalizable mode <current> development of the non-zero expectation value of the current without the external source. inite momentum spatially modulated current (circularly polarized helical current) Spontaneous generation of helical current momentum k The current has a helical structure Spontaneous breaking of translational and rotational symmetries.

14 Brazovskii model Brazovskii, Sov. Phys. JETP (975) 85. Brazovskii model: a model for phase transitions to inhomogeneous phases. The non-standard dispersion relationin which the spectrum has the minimum at a finite momentum is postulated there. pplied to various physics, such as weakly anisotropic antiferromagnets cholesteric liquid crystals pion condensates in neutron stars Rayleigh-Bénard convection symmetric diblock copolymers In our model, the non-standard dispersion relation is realized by the CS term. Summary We found a new instability of the ds-rn black holes at sufficiently large CS interaction. The instability occurs at finite momenta: the condensation is spatially modulated and helical. The CS terms from SUGR are barely smallto achieve the instability, as far as we have analyzed. The circularly-polarized mode has non-standard dispersion relation even for stable cases. Einstein+Maxwell+CS theory is an interesting theory, which has a potential possibility to be a gravity dual of phase transitions to inhomogeneous phases.

15 Generalization Discussions re all the SUGR-origin models stable?. If so, can we prove it? Other generalization? introduction of angular momentum, consideration of non-abelian gauge field,.