JHEP03(2013)126 d Springer March 21, 2013 : January 10, 2013 February 18, 2013 : : Nilanjan Sircar, e , Published 10.1007/JHEP03(2013)126 Received Accepted doi: , [email protected] , Prithvi Narayan, Published for SISSA by d [email protected] , b,c Nilay Kundu, b,c [email protected] , ], we proposed a Bianchi classification of the extremal near- 1 [email protected] , and Huajia Wang Shamit Kachru, d 1212.1948 a Gauge-gravity correspondence, AdS-CFT Correspondence, Holography and Classifying the zero-temperature ground states of quantum field theories with [email protected] Tata Institute of Fundamental Research,Weizmann Mumbai Institute 400005, of India Science, RehovotE-mail: 76100, Israel [email protected] [email protected] Yukawa Institute for Theoretical Physics,Stanford Kyoto Institute University, for Kyoto Theoretical Physics, 606-8502,Stanford, Department Japan CA of 94305, Physics, U.S.A. StanfordTheory University, Group, SLAC National Accelerator Laboratory, Menlo Park, CA 94309, U.S.A. b c e d a Open Access others are forbidden by thegoverned Null by three-algebras Energy or Condition. four-algebras Extremal are horizons also inKeywords: discussed. four dimensions condensed matter physics (AdS/CMT) ArXiv ePrint: sification of homogeneous horizon geometriesfour-algebras where with three the dimensional natural sub-algebras. mathematicsnear-horizon This involves geometries, gives real where rise the to holographic awith radial richer field direction set is of theory possible non-trivially spatial intertwined coordinates.tems consisting We of find reasonably examples simple of matter several of sectors the coupled new to types gravity, while in arguing sys- that general, anisotropic. Here, weattractors can extend lead our to study newin in phases, a two and directions: generalize way the suggested weolation classification by show can of holography. that homogeneous naturally phases Bianchi be Inexamples incorporated the of “striped” into first horizons. the direction, In Bianchi we the horizons. show second that direction, We we also hyperscaling propose find vi- a analytical more complete clas- Abstract: finite charge density is a veryto interesting the problem. classification of Via holography, extremaltotics. this charged problem black is brane In mapped geometries a withhorizon recent anti-de geometries Sitter paper asymp- in [ five dimensions, in the case where they are homogeneous but, in Sandip P. Trivedi hyperscaling violation, stripes, andthe a homogeneous classification case for Norihiro Iizuka, Extremal horizons with reduced symmetry: JHEP03(2013)126 28 30 30 14 3 21 8 0 32 15 7 11 6 5 35 42 32 attractors 0 26 16 – i – 18 in 4d Bianchi attractors ,k 4 A 43 43 9 44 44 ) h 10 11 42 5 = 0) = 1) h 32 h 1 39 2 3 6 , , , a 4 4 a,b 4 A.2.1 Magnetic field A.2.2 Electric field A A A A.1 Ricci curvature A.2 Maxwell’s equations A.3 Einstein’s equations 10.1 Bianchi three-algebras for10.2 radial and Bianchi two-spatial four-algebras directions 9.2 9.3 9.4 Comment 7.2 Three-dimensional subalgebras 9.1 7.1 Real four-dimensional Lie algebras 4.1 Type VI4.2 (generic Type III4.3 ( Type V ( 3.1 General idea 3.2 Dimensional reduction3.3 of 5d type Hyperscaling III violating to type 4d III space-time metrics in 4d space-time 11 Discussion A Einstein gravity coupled to massive vector fields 10 4d metrics governed by three and four-algebras 8 Null energy condition for9 5d space-times with 5d four-algebras space-time avatars of four-algebras 6 Striped phases by Kaluza-Klein7 reduction of 5d Classification type of VII four dimensional homogeneous spaces 4 Hyperscaling violation in 5d Bianchi VI, III, V attractors 5 Hyperscaling violation in 5d Bianchi VII 3 Hyperscaling violation via Kaluza-Klein reduction Contents 1 Introduction 2 Kaluza-Klein reduction of gravity coupled to massive vectors JHEP03(2013)126 ]. 44 45 21 – 12 ], and more 36 – 31 ], stripe order [ 30 – 22 ] and hyperscaling violation [ ]. Related fascinating phenomena are 11 6 – 8 – 1 – 6 , 4 ]. The problem of exemplifying interesting ground A 7 ]. Some of the most interesting phenomena arise when one 5 – 2 ]. One largely open problem is to give simple analytical examples of 40 – 37 There have been interesting holographic studies of some such spatially modulated Studies of doped matter in AdS/CFT have yielded new near-horizon geometries with For the subset of field theories with weakly curved gravitational duals, holography tion ing violation with helical order)enough that — numerical many work is cases required. discussedgeometries But in is a the also systematic classification something literature of one are possibleclear could complicated emergent from hope the to success achieve. ofasymptotically the That flat Kerr-Newman this four-dimensional classification is space-time, of not and charged, always perhaps rotating hopeless more black is relevant holes in in this context, waves, stripe order, nematicmodulated order, phases and also other occur more in exotic the orders phase are diagram well ofphases known. — finite-density notably, QCD. Similar studies ofelaborate emergent orders helical [ order [ phases realizing or even combining the various properties mentioned above (e.g. hyperscal- novel properties includingThese dynamical are scaling dual to [ emergentdoes infrared in phases the which break UV),more Lorentz-invariance but (as typically, one which the expects respect doping to spatial findsymmetries. emergent rotation geometries In and which condensed break translation matter more invariance. of physics, the However, for space-time instance, phases with spin or charge density extremal charged black braneinteresting geometry. examples of, Therefore, or in perhapstotically holography, anti to the de classify, task Sitter extremal solutions is black ofto brane to Einstein be geometries gravity present typical coupled in to of asymp- various the matter content fields of (chosen low-energy string theory). maps these complicated questionscal about general quantum relativity. dynamics to Finitegeometries simple temperature with questions states in planar in classi- horizons the(and (“black field charge branes”), theory density) and are map states dualsystem to at to — charged black finite dual black hole chemical to branes. potential the The ground low-temperature state limit of of the such doped a field theory — is then governed by the thought to occur alsoand in number the of phase light diagramstates of of quark doped QCD flavors quantum as [ fieldbut one theory, well-motivated and varies one. perhaps the even chemical classifying potential them, is then a difficult 1 Introduction There has been considerabledensed recent interest matter in systems possible [ “dopes” applications of an holography insulator to with con- finite charge density [ C Null energy condition for 4d B Rosetta stone relating different nomenclatures for the Bianchi classifica- JHEP03(2013)126 ], 1 ] in several directions. 1 space. This gives rise to a 5 AdS Happily, the classification of such ] for the proper history. Here, we ]. The generators of the Lie algebra 2 ] that remains possible can be easily 45 1 43 , 42 – 2 – 1 ], we found that the basic strategy of the Bianchi classification in arise in solutions of theories that satisfy simple physical conditions, 1 ] are a degenerate case of this structure where there is a semi-simple Lie algebra ] was the symmetry structure of the 3d spatial slices (the spatial di- 1 ] for a clear exposition and also [ 1 44 cannot ] — very readable expositions appear in [ ], we also considered some examples where the three-algebra involves the time coordinate, which 1 41 The more general possibilities described above also arise in the four dimensional space- The basic generalization of the classification of [ In this paper, our goal is to apply and generalize the results of [ The focus in [ In [ The examples in [ 1 2 with a separate Abelian factor corresponding to the isometry of translations in the radial direction. and as a result,rics, we though generically static obtain metricspenultimate either section. occasionally time-dependent arise. metrics We or describe stationary several met- such examplesyield stationary in space-times. the time, implying thatdimensional one algebras can with find two-dimensionaltypes, sub-algebras homogeneous but — horizons now i.e., with there“field governed the by theory” governed the symmetries spatial by Bianchi acting coordinates. realin on four three- We the dimensional can space-times. “radial” also direction In potentially in such realize cases, addition the the to real time-direction the four-algebras is nontrivially involved relevant for holography in 5try bulk structures dimensions. We alsosuch show as that respecting a the few Null ofinteresting Energy the new Condition new (NEC), types symme- and that we do give not simple examples run of afoul some of the NEC. dimensional Lie algebrasthe with spatial a symmetry preferred groupalgebras in three (with the sub-algebra the dual (which, relevant fieldrecently subalgebras) roughly, theory). — has generates see also [ beenuse these accomplished, results though to much find more the more general classification of extremal black brane geometries explained. Even withsymmetry a structure focus that on couldraphy static be could solutions be relevant. in non-triviallymakes 5d The it intertwined clear gravity, additional with that there “radial” the the is relevant direction spatial symmetry a structure of field could more holog- theory more general generally coordinates. involve real four- This one can give simple analytical exampleslation. of Similarly, Bianchi we horizons also which exhibit findof hyperscaling analytical dimensional vio- examples reduction. of “striped” Andthe phases, finally, possible by we homogeneous, using anisotropic attempt techniques near-horizon to metricsfield give dual theories. to a doped more 3+1 general dimensional classification of classification based on realtion three-dimensional [ Lie algebras, thecorrespond original to Bianchi Killing classifica- vectorscorresponding in spatial the slices. geometry, which generate the isometryWe group show of that the by using standard ideas of dimensional reduction and the solutions of [ in a previous papercosmology can [ also be usedblack to brane classify geometries. homogeneous (but, in general, anisotropic) extremal mensions “where the field theory lives”) in asymptotically the success of the Bianchi classification of homogeneous cosmological solutions. In fact, JHEP03(2013)126 , ) i ( +1 and A (2.2) (2.3) (2.1) , d 4 , ) ··· B , + = 1 dz µ )( + 1)-dimensional x ( d ) i  ( , we change gears and χ 7 + Λ + ), where ˆ  )2 ) , z i ( µ x -th massive vector field ˆ ( x i , we describe generic Kaluza- A 2 i B in order to obtain hyperscaling 2 ) , m = ( x 6 2 ( 1 4 ˆ ) µ ) ]. In section i µ x ( 1 − ]. , we show that the Null Energy Con- χ dimensions, while hatless objects are dx 8 )2 48 ) i − − ( x ) ( ˆ x F µ ( 1 4 ) + 1) i B ( and section d − A + 3 ˆ  – 3 – R  dz is a Killing vector. Consider the reduction ansatz = ( | geometries of [ z ˆ ) g | 0 ∂ x dz ( ) φ p x 2 ˆ ( x +1-dimensions into α ) , we show that prototypical examples of the remaining i 2 , i.e. d ( +1 e 9 z d χ do arise as solutions of simple gravity coupled to matter d + + 8 2 µ Z ds 2 with a brief discussion of possible future work. Some additional dx ) 1 ) ˆ κ x ( x , we give examples of analytical solutions for “striped” phases, by 11 φ ( , we describe 4d static space-times where the radial direction is non- = 1 ) i 6 α ( µ 2 S is the field strength corresponding to the 10 +1 dimensional Einstein-Hilbert action coupled to massive vector fields e A ) d i . We focus on field configurations where none of the ( ( = ˆ A , d ) = 2 d ˆ s d ··· = x, z , ( ) ) i , we discuss how to obtain hyperscaling violating space-time metrics with reduced . In section i ( ( 5 ˆ 5 ˆ – F = 1 A , and by analyzing directly in 5d generalizations of the Bianchi types in section 3 µ Let us consider a split of the Consider the The organization of this paper is as follows. In section 3 dimensional. We now follow the calculation of [ − fields depend on the coordinate to be: and where we noteQuantities that with our a convention ˆ for correspondd Λ to requires fields positive in ( Λ to supportand AdS space. and a cosmological constant, where In this section, we reviewfields. the We Kaluza-Klein will use reduction this ofviolating result gravity later and coupled at striped-phase to section metrics massive vector in lower dimensions. four-algebras and also show howWe the conclude NEC in constrains section orhelpful forbids details some are of relegated these to space-times. appendices A–C. 2 Kaluza-Klein reduction of gravity coupled to massive vectors dition (NEC) gives interesting constraintsalgebraic and structures. rules out In severaltypes of section enumerated the potentially in relevant section theories. In section trivially intertwined with “field theory” space-time coordinates by either three-algebras or Kaluza-Klein reducing the 5d type VII discuss the four-dimensional real Liealso algebras and present their invariant three-dimensional vector-fields subalgebras. andthe We algebra. invariant one-forms One might which hope geometricallygraphic that one space-times, represent could where realize the all holographic offield radial these theory direction symmetry spatial structures is in coordinates. nontrivially holo- intertwined However, with in section Klein dimensional reduction oftions Einstein gravity coupled tospatial massive symmetries. vector This fields.tion is In done sec- by dimensionalsection reduction of 5d Bianchi metrics in sec- JHEP03(2013)126 (2.4) (2.6) (2.7) (2.8) (2.11) (2.12) (2.13) (2.14) (2.10) (2.17) (2.15) (2.16) . . 2 2 C χ φ ) φ (2.5) , (2.9) 2 ) ) ) 2 α 2 , α B − ). Now (we will drop B χ 1 ¯ − z φ 1 α  + 2 + . 2 α a, 2) dα 2 . ( dz − H dz − e ( d , 2) a ( φ e 2 = ( 1 (( χ . ) 2 C ∧ ∧ α − e φ m φ | ) A 2 B ) ] m a i 1) d g 1 4 2 ( α | 1 4 − α E + , 48 − 1 d ¯ . z − 1)( + e p dχ ) a − 2( 1 | 2 1 2 ˆ dz α F B g − 2 ) − + = ( ( | ) φ e ) d ¯ − φ − z  + 1 4 2 2 p e ˆ χB 2 A α 2( α B φ χB ) e , − dz + F = − 2 ( 1 ∧ φ − = 2 ) α ¯ ) z α ) = ) A ); ( 2 i ∧ a 2 1 + B ( A ( e ¯ ˆ 1 z α F a ( φ ∂φ ˆ ] + φ + E ( dα χB – 4 – + 2 dχ 1 E 1 ( ; α 1 2 a α b α e ; − − dz 2 + 4) C E ( ab 1 ) − − , α a = − i A α F e ( B d + ∧ | ( 1 ∧ E 2 φ R 2) ˆ ˆ φ g (( b ) a α 1 E φ | i 1 − e  dA 1 m ( α | E d α E | 2) 2 ∧ α  1 4 g p [( g − | e − ∧ ab | dχ e A e − + 1)-dimensions, and index − e ˆ = 2 F a . The vierbein basis is related as, ˆ p = E p d + d φ ) = i 1 4 m E ( = = B 1 a ) ( i a α 1 4 = 2 ˆ ˆ ( − AB ab − ∧ 2 E A ˆ ab ˆ ˆ A F e F A ˆ − ˆ ) d F R F i = 2 = | 1 2 1 2 1 2 ( ˆ g = 2 2 = = | m α ˆ dχ 2 = = = 1 4 ) F i | ˆ p ( A ˆ g ˆ − 2 F | − ˆ . The determinant of the metric satisfies, F ) i µ m ( p and to write the gravitational action in the above form, we have used the 1 4 1 4 dx | , dA µ ˆ g | − dB for convenience) B dχ = is the vierbein in ( p i ) = = i = A ( − E H C F B Now the gravitational part of the action reduces to [ So the kinetic term of the gauge field gives The field strength is given by, where identities so Similarly, the mass term satisfies The components of the potential are related as where where the index where where JHEP03(2013)126 r 2 ) be ) i r ( (3.1) (2.18) ∂χ ( φ 1 (which has ). Let α 0 2) ]. Let us con- − 2.17 d 49 2( e 1 2 . − 5 ]. The cases studied in )2 2 i ) ( 16 µ F φ dx 1 ) α r 2 ( . − µ ] e B φ 1 4 1 α + − 2 , we demonstrate this for type III so- e 2 dz ( H 3.2 r φ + Λ 0 are constants given by eq. ( 1 φ 2 α 2 2 χ α 1) φ 2 − , α – 5 – 1 e d 1 α α 2( + 1) − 2 − e d 1 4 ds 2( r e 0 − 2 φ ]. This is in part because, in some specific cases, they , we demonstrate that 5d Bianchi solutions (where the 1 2 m ) (and appropriate rescaling of the spatial coordinates). Let α ), where 4 21 2 1 4 – r e + 1)-dimensions, which will also appear as a coordinate after ∂φ 2.2 ( − 12 d = 2 1 2 we show that the type III solutions of this sort can be obtained ) 2 ˆ s − B d ) i 3.3 6= 0 [ R ( [ -dimensions. We will consider metric components to be functions of χ θ d | g | − ) i p ( ˆ x A d ( d 2 i : r m Z 0 1 4 6= 0. These can be duals to ordered phases with hyperscaling violation. 2 φ 1 κ − θ As a next step, in section Before proceeding let us note that in the gravity theory hyperscaling violation arises if We actually do this in two steps. As a first step, in this section, we show that di- In summary, the total dimensionally reduced action becomes: = ) = r ( S compactification to only, and will allow thetions, higher generated dimensional by metric a to shiftφ be of invariant under scale transforma- 3.1 General idea In this subsection wesolutions discuss from the scaling solutions general via prescriptionsider dimensional of reduction, the as metric obtaining described ansatz hyperscaling inthe eq. violating [ “radial” ( coordinate in ( the metric has a conformal killingformation vector, by i.e. the if the corresponding metric coordinate isand transformation. left in invariant section upto In a 4 the weyl and solutions trans- vectors section which of make 5, this it we section homogenoeus construct along examplesditional the conformal field where theory killing the directions vector metric while which has possessing leads genuine an to ad- killing a scaling transformation in the field theory. algebra acts only onviolation. the We give field several examples theorybeen there, “spatial” especially leaving popular the coordinates) in special can the case also of literature) type to enjoy VII its hyperscaling own section, section solutions with hyperscaling violation.alizing (By one a of 4d thedimensions Bianchi Bianchi governed solution, by types we the with mean algebra).lutions, the a and In radial in solution section section direction re- from included more as general one actions in of 4d the which spatial do not arise from dimensional reduction. the literature have all enjoyedwe standard demonstrate spatial that rotation and it translation isenjoy symmetries. simple to Here, find homogeneous but anisotropic solutions which also mensional reduction allows one to turn some of the 5d Bianchi solutions into 4d Bianchi 3 Hyperscaling violation viaThere Kaluza-Klein has reduction been considerableviolation recent interest exponent in theshare geometries many that of include the a hyperscaling properties of theories with a Fermi surface [ Einstein frame, and that the scalar field has a canonical kinetic term. These relations guarantee both that the Weyl rescaling puts the gravitational action in JHEP03(2013)126 . 3.1 (3.5) (3.6) (3.7) (3.8) (3.9) (3.2) (3.4) (3.10) .  2 A = 4. 2 , d m 2 2 ) (3.3) 1 4 . β 2 2 2 , − dx β 2 2 + 2  ) with β + − F 2 2 2 1 φ 2 β 1 1 dx ]. α 2.17 dρ r . 1 2 2 = ( + 2 ρ 2 − t 2 Λ = 1 e β 2 α ρ implies that the lower dimensional 1 4  dρ 2 by using the techniques of section = + r . − 2 z + ds + dt .  Λ r 2 r 0 dz from eq. ( log ˜ φ t φ r 1 β t 1 dt 2 3 α e r 1 → β α β t 2 t √ 2 2 e = 4 and only one vector field turned on. For β r e ) A 1 − 2 + − dt , φ – 6 – e e d + 2 r + p t = β 2 2 = − β 2.18 ) 2 = e 2 2 φ t dt = α r A A t ∇ , dr r ds ( β ) with , 3  2 p 1 2 1 ) , β √ e r 2 2 2 2.1 2 2 2 = 2 2 − β β − β e β = 4 2 A R − 1 − =  − dr α 1 2 g 2 is invariant under = − ds 2 2 = 2(1 = √ ˆ s t 2 ds x d 4 A m d ) shows that this is a hyperscaling violating solution of Bianchi type III. Z ]. Let us first recall the relevant results of [ 3.8 = 1 S It is possible to view this solution in different coordinates which make it more trans- The new 4d action is given by eq. ( Now, consider the dimensional reduction along The action is given by eq. ( Actually eq. ( parent that it istransformation a hyperscaling violating solution. We perform the following coordinate and the matter fields are The relevant solution can be read off from the 5d solution above. The metric is For this subsection we use In order to findshould be a related solution, via: the parameters in the action and in the metric/vector field convenience we do notspace-time dimension write of the the quantity. quantities The with metric ˆ in here, (4 + and 1) dimensions we takes explicitly the form mention the with vector field given by 3.2 Dimensional reductionIn of this 5d subsection, type we consider IIIeq. the to (4.38) dimensional of 4d reduction [ of space-time the type III metric as given in metric transform as So the lower dimensional metric isscaling invariant under function the in scale transformation frontwhich up of to exhibits the an hyperscaling overall metric, violation. a conformal factor. This corresponds to a metric Now, requiring that JHEP03(2013)126 6= 1, (3.15) (3.16) (3.11) (3.13) (3.14) (3.19) (3.20) (3.17) (3.18) (3.12) γ , , = 0 = 0 .

2 2  λ λ kδ . Λ − δφ 2Λ 2 = e + 2Λ − 2 2
+ 2 k dx 2 δ . 2 + A + ) ρ become 2 2 2 φ , β . 2 λ ) t 1 + 4 e dx . 4 dρ β α 2 ) − 2 2 + 4 r, t, ρ β r + − m 2 β k 2 2 r 2 δ ) has a hyperscaling violating type ( − + β dρ log( − + ˜ +1)-dimensions coupled to a massless k ( 2 2 3.7 d 2
k β = λ 2 F 2 2 4 ρ dt r β = )) γ αφ 2 4 δ 2 2 + = 1 + ˜ r e 6= 0. Therefore, we generically have + 2 + 2 2 − r , −
α dt β 2 – 7 – , 2 2 γ γ ) − dt , φ ˜ ) r 2 2 t ( t log ˜ θ ˜ r r d φ dt , β k kδ r β t 2 ) t ∇ − δ , θ t β ( + + A  t + 2 β 2 + γ β t 2 1 2 + 2 2 2 p β = 3. This action is a natural generalization of the one β β r + dr kα β 2 α − ˜ r ). In this subsection, we define a family of actions which ) of the previous subsection, and we exhibit more generic 2 d 1 2 β + = − t λ + 2(1 + β = R = 2 ( 3.7 . A A 2 β  2 γ 3.11 = λ 2 α g  p 2 2 3.2 √ + 2(1 m ds = = = x 2 − 2 λ φ A +1 2 t d kδ ,  ds d m A t Z ). ) + A = = 1 + 3.7 kδ γ S (2 + kδ 8 All of the equations become algebraic if we take the following relations among param- As an ansatz, let us take a hyperscaling violating type III metric of the form: Consider the general Einstein Hilbert action in ( For general mass parameter, we have − Plugging in these relations, the Einstein equations along Also, let the gauge field and the scalar field be eters: given by eq. ( scalar field, a massive vector field and a cosmological constant: In this section we will consider generalizes the 4d action eq.hyperscaling violating ( type III solutions ofparameters these here actions. do We not note thatgeneralize allow generic those an values of “uplift” of the section to a 5d solution, so these solutions considerably transformations, in the way which is characteristic of3.3 hyperscaling violating metrics. Hyperscaling violatingWe type saw III that metrics the inIII Kaluza-Klein 4d solution reduced space-time of action the eq. form ( eq. ( where which implies that this metric is not scale invariant. Instead, it Weyl rescales under scale In the new coordinates, the solution is given by JHEP03(2013)126 , (3.24) (3.21) (3.22) (3.23) 0. All o . , ] can give
. 1


0 this gives 2 , λ > . 2 = 0 t λ λ

2 → A
2 ) . 2 r δ λ 2 2 1+ λ δ δ − 4 Λ = 0 2 − 1+ αδ 2Λ α 2 δ 1 1+4 ) span an Euclidean ( − δ δλ − q +4 , 2 2 ρ, x − 2 1 + 4

δ k 2
+ 4 2 λ 2 λ − , λ
) λ 2 ) δ 4 + 3+ δ α < λ < 1+ − − 1 + + 2 ) = 0 − α because ( 2 4 α δ + 2 ( δ αδ λ 0, 0 ρ
2 α − 2 λ In addition we will mainly consider 2 = 1+ α 2+
)) + 8 ) + 2( 2 , which has been especially popular in δ δ
− m 1 + 4 0 3 λ ))( 2 2 2 + < δ < δ ) + λ ) δ 1 2 α ) + αδ α − δ , 4 − − . − αδλ Choosing the horizon to lie at α 1 + 2( ( 2 , θ − , ( α + 5 k − − λ
4 k
 – 8 – 2 α 7 + 4 2 2 ) 2
= λ − +2( δ λ √ δ 2 ( ( 2 1 + 3

λ δ − 2 1+ − kα − ( λ α + 2(1 + 1+ αδ k 8 +2 δ 1 + 2 2 ) + 4 − 2 δ λ αδ 1 +
2 , k δ + 2( + 2 2 2 4 ) − 1 2 2 2 1 + 4
λ 2 δ m α − 0. Finally, for a physically acceptable solution 2 δ 0 in our conventions. λ δ +4 2 2 − 4 t δλ α λ > 1 + α < m A + 2 α 2 + 4 ( + + 1+2( 2 − n − α 2 2 αδ λ 1 γ, β > 2 is the same as the equation along ) λ = 4 × k αδ 2 − x αλ kδ 2 − − 1 + ) ) − 2 + 2 1 δ − δ  α 1 + 4 α − 2 (2 + 2 αδλ 2 2 + 1 α α − kδ t α δ , γ , we have seen how dimensional reduction of the Bianchi solutions of [ 4( 8 + 4 1 A +2 3 − + δ λ − ( 2 + α = = k factor. = 2 2 t = ]. In this section, we exhibit hyperscaling violating solutions in 5d for Bianchi types 2 Throughout this paper we will take the cosmological constant to be negative, this The solutions to the above equations are The gauge field equation is Hopefully this will allow the near-horizon geometriesOf we course, study as the to extremal smoothly RN connect black brane with example asymptotic shows, this condition can be relaxed in some cases. 1 m A Λ =  β 3 4 VI, III and V.the We literature, leave the to discussion a of separate section, type section VII AdS space. We leave a study of such interpolating geometries for the future. In section rise to hyperscaling violatingare solutions. fairly easy This to leadssection find to by in an finding their expectation hyperscaling own thatin violating right, such [ also generalizations solutions in for 5d. many of We the confirm this solutions expectation found in this rise to the conditions these conditions can be metranges; for the one solution such found range above is in various open sets of4 parameter Hyperscaling violation in 5d Bianchi VI, III, V attractors corresponds to takinggeometries Λ where the horizon area vanishes. and the scalar field equation is The equation along AdS JHEP03(2013)126 (4.5) (4.4) (4.1) (4.2) (4.3) , , , , , , , . = 0 = 0 = 0 = 0 = 0 ) = 0



Λ = 0 kδ δ 4Λ = 0 ) . kα 2Λ 2Λ 2Λ 2Λ h − ) − , − − − − + 4
α 



2 2 2 2 2 2 2 kδ z h ) − β dz kδ kδ kδ + + (1 + δ kα + ( hx z h λ z 2 k = 4. We can achieve + 8 + 4 + 4 )(1 + − β z − hβ δ δ δ y β d e ) ) ) 1 + kδ β z z y z + + β β β β y 8 + 2 y β 2 y r = 1 + β (1 + + + 2 β ) for α
+ + y 2 β 2 kδ + 4(2 + kα + 4(2 + + 4 3.15 ) + 2 dy 2 k k 2( δ x k + 4 2 − + k k δ , θ δ − 2 2 α e ( + 4(3 + 2 + − y m ) + ) + 2 + 4 β z k y
α k 2 m β β 2  r k h t k = 2 + A + + (1 + (1 +
2 h − . In the discussion that follows we will find λ z y 2 appearing in its real three-algebra. z – 9 – ) 2 β ) + β β z Λ dx h m β 1 + x kδ + y β z β 2 + 8 4 + 4 4 + 4 β which characterizes the three-algebra, one can show z y + λr − β δk ,  and β h z ) β + + 2 2 + 2 since the equations simplify in this case. Furthermore, + + y 2 z m 2 + 8 λ δ β h λ β y 8 − dt β = 1 + α, δ,  + + γ 2 ) + ]. . ) ) 2 2 y 2 (1 + dt , α ) r ) + h h β λ k θ 14 2 m r kα 8 2 r – ) + 8( + − t m = − − 2 y )+ 2 10 2 γ A −  β 2 m log( ) 2 kδ 2 r dr m p k ) − kα δk , γ 2 − 1 + ) kα 2 = = = − − ) 4 2(1 + − 2 kα φ kδ A = − kα ( − ds kδ t − x +2 A β kδ kδ 2(1 + − 2(1 + ( t − 2(1+ ( A (2(1 + t − t ( Generically, it is difficult to find solutions to these nonlinear algebraic equations. How- The Einstein equations, gauge field equation, and scalar field equation then respectively All of the equations become algebraic if we choose Before we move on to individual solutions, it is worth noting that the dimensionless We do the calculations in 5d, although we expect similar results to hold in other di- A t A A ever fortunately, given athat choice the of following solutions exist: give: To begin with, let usmatter start fields with to a be Type of VI the metric. form We take the ansatz for the metric and constants in theit action convenient are to take sometimes the algebra governingfor the example, Bianchi type type VI will has have one some parameter free4.1 parameters too — Type VI (generic mensions too. Wehyperscaling begin violation with by the allowing actioncoordinate. the given scalar by This field eq. isgeometries to ( similar studied run in to logarithmically [ the with earlier the dilatonic radial realizations of hyperscaling violating JHEP03(2013)126 (4.6) ) by } (4.7)
, 4.6  +1 } . 2 h ) , δ 1) 6) , ) + − − )) 2 . 2 α h h ( ( h δ , 2 2 2 +( δ } + 1) 1)

+ − h 3 −

( δ +1 2 h δ , 2 δ 2 αδh αδ h ) 4 δ − +2( , − 2 + 60

) + . δ 2 1 δ 2 α
12 1) + 1) 7 + 18 ( − (1 + 4 +1) + 2 αδ h − h h δ h δ k α h 1) 1) 1) 6) + 51 αδ
1)( − − +( − 6 − 4 ) + −
h + 38 ) h − δ h h − δ δ − (( ( h : 2 (( h 2 2 +6 +
αδ 1 2 α − α 2( h α + 2( 2 − 2 +6 2 α δ
36 { − 2 α = 2 +4 h . h ) 2 ) + 4 ) 2 − h 4 δ 1) + 24 2 h + 1) δ δ +12 2 2 δ 6 − + 1) 2 δ )) (2( h − 1) } 6 + 1) ( α ( 2 ) δ 7 h 5 + 12 2 − h 2 1+6 2 δ ) − − δ ) δ 1) h 2 1 δ − δ − α 2 h ( 2 − + α 3 + ( 2 ) − = 0. The solution is obtained from eq. ( − 2 – 10 – α δ α h
, , − + ( 5( +6)+1) 8 − α 25( h αδ 1 + 10 2 h 2 6 α )) +1) ( + 2 α + 2 +3( δ k , h h k − kδ
h ( 4 ( < k < , 2 2
+ 1) + 1) + + 1) ( δ + δ + 22 2 ) αδ +1 2 + 1) ( + 4 h h 0. One possible range where these conditions is met is 3 δ 2 ) + kδ 4 α δ 2 ( h ) 2 2 + 1
( 2 ) h δ − 1 + 6( (2 2 h δ − h 2 δ 2 α − δ k > 2 ( )
+ ) − δ h 2 δh 2 +
+ 8 − δ − + 228 , 2 ( 0. An acceptable solution with vanishing horizon area then ( t α δ αδ + δ δ 2 2 ) + 2 k 2 +1 + α δ ) 2( α kαδ A δ δ 2 δ δ 2 kδ 2 + 2 , 1+4 2 h 1) + − + 2( kαδ (1 + 2 1) ))( + 2 αδ γ > 10 λ
− 3)+2) + 1) + 2 α δ 1) + 2 − − δ − 2 k α 2 8 2 k 2 − ) − + 14 + 6 )) ) h − α , α 2 − − ) α ), δ − δ 2( h − 2 kδ δ ( δ δ h − δ 2 δ 2 z α 4) ( 3 + 80 2 h α ))( 2 ) + 2 β 2 + 2) (2 α + 18 − δ
= 0 − k 5 10( δ δ − 2) ( + 6 − h +2)+3)+16 independent, but the rest of the formulae do depend on α h 2 + 2 + 2 + 6 ( + 2 ) α α α + ( − h k h δ h 6 12 1 k δ − + ( + 1 h h k 1 α α 2 (( y α α ( (3 k h αk 12 2 + ( 2 α α β + 2) h α + 6 < δ < h − is k (( 6 α αδ − ( 12 0 and also 1 + − − k − + 1) ( { 2 4 2( ( + 2 + 2 t + δk ( (2 4 δ { k − αδ δk 2 δ , 0 2 > ( 20( k k 4 A 2 k 1 + ×{ − x h (4  6 ( k = (2 5 α − β = 0: √ + +12 ( 4 ( k k ( × × z = (1 = = = = h β t 2 z y ≤ = (1+ = = = = λ Λ = β β A t m 2 z y λ This messy formula gives, for example, with Λ = β β A m < α Note that 4.2 Type IIIThis ( is a limitingsetting case of type VI, with We take Λ requires ( 0 JHEP03(2013)126 , , 2 3 λ , Λ, ) by q ), (5.1) (5.2) (4.9) (4.8) t z A β 4.6
, 4 + λ δ y < α < β ), z + β + 36 . x 3 , + β .  2 αδ 2 y , δ = 4. Again we will
δ β ))) 3 6 12 d δ ω , 1+6 where these conditions +


− + 3 − x 2 )) λ δ α β 2 δ . δ, k ( δ
+ δ ,
2 2 ) with + 3 2 2 α α
δ δ − 2 < k < ω 3.15 4 2 attractors , αδ + 6 (1 + 2 δ δ as free parameters. The ansatz for 2  2 + 3 } − k 2 0 − − 2 αδ

α yz + Λ α 4 2 β αδ m . 2 ) δ 2 δ ) (1 + 4 k δ + 8 − − r 2 3 , δ )) 2 k ) 1+4 2 δ 2 ) + δ + − − αδ δ α, δ, , α 2 − 2 δ 2 + − ) 2 4 2 α α ) ) + δ α δ − δ α α 2
+ 6 − dx 2 α 2 2( x α − 2 = 1. The solution is obtained from eq. ( 4( – 11 – 1 + 2 + α β − ( ( )) (4 − α 2 + 2 kδ k h α r α δ k 3 ))( α < δ < 2 , δ δ 4( + + 6 ) − 2 , 0 ) (2( 3 + 2 − . ) + δ δ 2 3 )(1 + 2 δ 2 dt α δ + 12 δ kδ 2 2 + 6( δ αδ ( γ 2 +
6 − q + 6 k δ 2 kδ , . k 2 2 r 2 1+6 k k α α − − + α 1 + ω kδ − − θ 2(
+ α = 1 + − ))( 2 r ( (1 + 4 ( 2 γ δ α δ z k − 2 + 6 )) 2 k < α < ( 6 + 12 β − r dr δ − + δ A δ 2 a 2 2 ) = 2 = = = α − ) < k < ) + C p ( α δ k t δ + 6 2 y α 2 λ k 2 Λ = 2 δ ( β A = = δ = 1 k 2 m + 1 k 2 +4 − 2 − h − α A 12 1 + − α α αδ kδ , , so that the solutions will have ds 2 − δ ( 4( (2( ×{ − − 1+4 + 0. We can find some allowable range of values for = = (1 = = , = 1: α t 2 α 2 y λ Λ = h β A = γ > m  symmetry using the same theory as before, i.e eq. ( 0. These conditions can be met, e.g., one possible range of parameters is 0 0 , Λ, < δ < t have the various fields is chosen as 5 Hyperscaling violation inIn 5d this Bianchi section VII weVII will obtain a hyperscaling violating solutions in 5d with Bianchi type Again, an acceptable solutionA with vanishing horizon area arisesare met. when ( For example: 0 This is a limitingsetting case of type VI, with An acceptable solution withγ vanishing > horizon area0 arises if ( 4.3 Type V ( JHEP03(2013)126 , 6 0, 1 √ (5.7) (5.6) > (5.3) (5.4) (5.5) , . , , , , , , yz
in terms β )) λ 2 = 0 = 0 ) λ A

+ 2 < δ < λ )) = 0 7 λ , Λ x λ 2 δ , the solutions

− 6 2 λ 2Λ)) = 0 2Λ)) = 0

β 2 1 2 a λ 2 √ m δ dz . m C

δ − − + 2Λ)) = 0 , ) , 0, 2 + 2Λ))) = 0 − δ (1 + Λ) x λ λ λ δ k + 8 λ λ , + 2 + 4 , > 4 2 (2 + kα . α 24+(24 δ 2 λ 2 a + 4 α 2 2 + 2 + 2 + 2 yz α − − 2 2 C k − 2 ( β − − − (2 +

) A λ A α ( ( ( 2 λ ) λ A λ + cos( + + 2 δ λ λ λ 2 λ yz 0, 2 A λ − β yz m 1 + , Λ, dy 3 + )) 1 + 4 − 2 ) a > (13+ δ 2 + geometry: kβ = ) + 2 ) + 2 ) − 2 2 2 1 + x 2 C 2 − λ λ λ a k δ 2 + α 0 + k 2 2 2 2 2 − C + 2 )( α ( k A + m m m ( λ sin( ) + k 2 ) + 2( k 5 δ δ + A ) λ ))+12 − λ δ 2 + − 1 + yz 2 + δ , θ λ yz ) + (2 + (2 + β ) − m β 2 2 = 2 2 yz , λ ( yz + + 2 k A A β λ 2 β 3 )
+ 8 α A λ 2 + λ , δ 1 + (2 + ) 90+19 ) + − δ k = − )) δ ( (2 + (2 + 2 + 6 (2 + − ( 2 + )(1 + 2 + 2 2 2 + 6 λ a a ( k + 2 λ − − A λ − δ C C yz
+ 4 ( ( ( α yz β ) + 4 ( 2 2 2 + + a a a
) 2 yz 2 – 12 – yz C λ C C λ yz ) + 4 αβ − β 10 + 3 5 ) 2 m β + λ + − yz 2 2 αβ 2 a (117+ − A ) kδ ,  2 kα − β ( A 1 + λ C A ) λ − ) λ A 2 5 ( − + ) (11 + 12 λ 2 + (2( − ( k 2 A 3 λ 2 − dz ω ) − δ λ ) ( yz ) 2 ) λ λ (2 + , ) 1 + k β x λ = 1 + λ ) + 2
− yz + (12 + 2 1+ ( λ 1 + 6 β yz 2 ) + 3 + . − 1 + − yz δ − β ( ) − + yz ) ) 4 (4 + 4 2 − ( λ (4 + (4 + ) + 2 ) β ( δ 2 ( r + sin( 2 ) λ 2 yz λ A kαβ yz ) kα 2 δ β 2 2 β 3 + − A δ kδ , γ dy A ( + : log( ) 1 + 11 + 12 1 + − + . 2 − α δ x ) parameter space, where the resulting expressions are more compact. − k − yz )(
( 4 + λ − yz ( 22+144 2 2 2 − = 2
+ 2 − , δ − δ δ (1 + 6 = δ 2 (11 + 12 2 x α, δ 3 2 kαβ α 2 + kαβ α = ) 2 (8 + (12 + δ β 2 φ a 2 2 − λ − δ = cos( A C 8 + ( yz A A 1 + 6 2 α 6 0 are satisfied. These conditions are indeed met when 2 (1 + 6 β 2 2 − − 1+ − − are two of the invariant one-forms of Type VII ( 22 + 24 1 + 6 2 ω 2 − − i 1 + yz 4 + 2 − ( 2 2 − − A 4 + 4 ω β 2 − . Although we were able to solve above equations in terms of 2 λ > 2 2 k = = = = = 2 λ 2 2 2 a k , 2 Λ = − 0, yz δ A C m β A solution with vanishing horizon area would arise if All the equations of motion become algebraic if we choose , > α Note that all of the 7 equations are written in vierbein coordinates. Six of the equations are Λ cial values in the ( Solution for independent, and can be usedof to solve for the six parameters are very complicated and not very illuminating. Instead we present below solutions at spe- The Einstein equations, gauge field, and scalar equations respectively give: Here, the JHEP03(2013)126  (5.9) (5.8) 1. 6= 1, ))) , , , ] λ    λ , ))) ))) ))) , ] λ λ ] λ  954949  38 + 39 √ ))) ))) − , 4469 ))) λ ( λ ] 0 and λ } λ 14 + 9  38 + 39 38 + 39 − ))) ( − 327+4 − , ))) . ( λ λ ( , 14 + 9 , λ λ λ 14 + 9  38 + 39  λ > − 25 39 − ( − , 89725 ( ))) ))) ( ))) λ . 1219 + 39 λ λ λ λ λ 14 + 9 − , = , − 14 + 9 )))]  − ( (113 + 9 ( λ − λ λ λ ( λ ))) ))) )))  1219 + 39 1219 + 39 λ λ λ λ 2 − −  38 + 39 (113 + 9 38 + 39  38 + 39 ))) ( )) ( (113 + 9 λ  λ λ − − λ ))) − λ λ 1219 + 39 ))) ( ( ))) 112 + (  38 + 39 λ (380 + λ λ λ λ − λ ))) 14 + 9 − 14 + 9 14 + 9 λ − ( (113 + 9 ( λ ))) ( − λ − − (113 + 9 λ λ λ λ ( ( ( 112 + λ 0 except 112 + λ λ λ (380 + 38 + 39 (380 + 14 + 23 − 14 + 9 λ − ( 14 + 9 λ + 217)) + 44235) − − ( 900 + − 38 + 39 λ ( 14 + 9 ( − ( λ 2 λ 112 + ( λ 144 + − λ 14 + 9 p λ (380 + − λ > 1219 + 39 1219 + 39 1219 + 39 ( 112 + )) λ ) ( − 2 λ ) − λ p − ( − λ − λ (34 λ − ( ( ( 900 + λ ( )) 900 + ( 1219 + 39 λ (113 + 9 (113 + 9 (113 + 9 λ )) λ )) λ λ 144 + λ λ λ λ λ λ 30 + − 144 + λ p λ p ( p )(27 + − 327 λ p 900 + λ  (113 + 9 34 (113 + 9 1219 + 39 − 14 + 23 144 + λ p ) + 1991 λ (113 + 9 37 + 144 + 112 + 112 + − 112 + − − (380 + (380 + )(11 + 39 λ (380 + 1219 + 39 14 + 23 p ( 1 + λ  – 13 – )(11 + 39 ( (113 + 9 ) + 477 14 + 23 14 + 23 λ − λ λ − λ − 7 p λ − λ λ λ ( − : λ ( − ( ( 12125 − − (380 + ( ) + 754 λ λ λ 1 463 ( − ( 13 λ λ λ 112 + λ − λ λ [ 112 + 49( − 2425 − [ (130567 − can be obtained from dimensional reduction, but we − 2 112 + ) + 3379 − ( [26 − 900 + 900 + λ ) 900 + ) ( [ λ )) 112 + ( λ − δ 25 + 39 λ )(27 + (380 + λ = λ 25 + 39 )) ( λ 144 + λ p p 144 + 144 + p − )) λ λ [ − 900 + λ λ − 8 ( − (380 + )(27 + δ ( λ p ( p p 7(27 + λ 7(27 + λ λ p 166 + 81 11109 + 9589 − 8 ) + : = 1 + 2075 + 3081 ) + 9 ) 144 + − − λ 1 79 + 9 144 + − ( ) + 6 + 17 ( ) + λ λ − − 3395) 900 + 1 + α ( 6 + 17 p 144 + λ λ − λ 2 λ − (217 + 34 − 6 + 41 p and ( )(11 + 39 λ − 2425) − ( 144 + λ − 900 + p λ 49( λ p λ − . ( λ ( 3 1 = −
p λ 6) p 1 938 49( λ 5 327 7 + 9 19 + 39 − δ + 26 6) − 1 + 4978 1 − − − 21 + 37 83 + 273 82 + 117 654 − 1 λ ( ( − λ 247 − − 11965 + 12 + = = ( − − − λ λ − ( 2 ( ( 11 + 13 28 λ − − √ 1661 + 104 357 + 52 10000 + 39 20395 + 5 ) (41 11 + 13 λ λ λ 2793 α α  2065 + 13 −    λ − − λ (  − 8 +  2 (31 {− 2 λ (  λ 10 + 39  81 λ λ λ 130567 λ λ 6 λ λ 3395 + 109 2 44235 220485 −  10 1 4 − 30 + 138 5 + − + + + + λ − (109 −    185 + 2 ( λ 39 + 4 380 + 3  λ λ λ λ = = = = = 1 − 2 10 40 (25 5 + + +(109 + +39 2 2 2 k a Λ = yz 1 A C = = = = = m β 221 2 2 2 k a Λ = 2. This specific case yz A 6= C m β An acceptable solution with vanishing horizon area arises if This solution is physically acceptable for all λ Solution for Solution for • • < λ < can get more generic solutions, which do not allow an uplift to higher dimensions, as follows. 1 JHEP03(2013)126 ), ]. 2.1 (6.5) (6.6) (6.7) (6.8) (6.9) (6.1) (6.2) (6.3) (6.4) case. 36 (6.10) (6.11) – 0 , , 2  31 ) 2 3 ) 3 dx 3 B 0 c dx . + 2 + 2 s 2 2 , dx λ 2  λ )( , we re-write the metric , so it is clear that this s dx + 2 2 2 1 ) s 2 − µ x x 146 2 c ( 2 λ − dx , we can also obtain simple λ ) + ) = 2 2 x + 11 1 c ( = sin ( 2 µ x s ) ( − βr B 3 = 4. The lower dimensional scalar 144 2 2 , e d λ + ) , 5 and 3 s dx + 4 , dz 1 2 + λ ( dx + ) x ) 3 ) 6 c , f 3 2 2 11 + x λ ( B ], which takes the following ansatz βr 2 φ − dx − 1 e 2 50 λ + 2 2 c dx 2 )( α 3 + 3 , = cos ( 2 ˜ λ 2 1 λ A  4 4 e 5 c – 14 – x − 11) λ ( λ dx − ) and setting q βr ( 2 + f 2 2 11 − 10 + χ 2 e = 25 βr λ r 2 2 2 − − 2.17 solution [ + , ds + λ λ e , ) 2) 2 + 2 3 2 3 0 2 x ) (5 λ + ( λ 2 ) through 3 3 2 − 5 2 1 φ − − dx λ . This is the solution of the 5d action given by eq. ( ) 1 λ 1 2 1 − 1 2 2 α , χ x  dx λ 95 x 2 2 s dx λ 3 e  2 dx r λ 2 cs 2 ( s + 2 − + ( ) 2 = χB using eq. ( 1 2 2 +( 2 2 50 √ 2(11 + 2 √ λ 2 λ 2 = sin 3 − ˆ s dt 1 √ + = = = = r dt d s s − t c dx r 2 t ( β t 2 2 − β βr c Λ = 2 β (1 β ˜ e A 2 e βr m = 2 e e and ˜ = 2 1 − A 1 − ˜ 3 α A 2 2. 2 x q 2 B √ dr q dr  − = = = = = λ < = cos 2 ˆ 2 2 ˆ A ˆ A s ˆ s c α . ≤ d d δ This metric can be written in the form In order to dimensionally reduce this 5d solution to 4d along Let’s consider the 5d type VII These two classes of solutions are quite messy, but are presented (in a somewhat Pyrrhic , α where field depends non-trivially on can give geometries dual to striped phases. where and gauge field as and 1 where where analytical examples of stripedSee phases. for For examples that of purpose, the we other start constructions with of 5d “striped” type phases VII in the literature [ victory) to illustrate that physically sensibleof solutions of this sort do exist for several values 6 Striped phases byBy Kaluza-Klein using reduction the of dimensional 5d reduction type we VII reviewed in section JHEP03(2013)126 , and (6.18) (6.12) (6.14) (6.17) (6.19) (6.13) (6.15) (6.16) 7.1 2 ) , ), ∂χ  ( , 2 φ ) 1  2.18 3 α 2 4 ) dx 3 e ( 1 2 δ dx 6 r − )( )˜ , 1 2 1
x x F ) ( c , ( 1 φ δ f f 1 2 x ˜ α r ( . βr 2 + 2 g 2 ˜ − e β 2 ]. The holographic dual of a four- A . e ) 1 1 ), the field configuration simplifies + + 1 4 β  q r . t φ 2 dx 1 + log ) β − = ( α 1 δ 2 2 3 7.2 log(˜ 2 βr = e dx r H β δ dx φ 3 + 1 3 2 1 + ˜ , t + ( + Λ α β 3 √ 2 B 6 , χ 2 2 − − ) dt χ dx = = dt e 1 γ 3 φ – 15 – r , 2 = r x 1 4 1 t ˜ r ( B β α β g 2 6 β − e − e = + 2 ). 2 + 2 ) γ t − t ˜ r 2 m ,B β 3 log 2 ˜ r β d , φ 1 4 3 ∂φ 2 6.10 ( 2 √ 2 dr ) s 1 2 − =  2 β − sdx ,B ] was to classify homogeneous, anisotropic extremal horizons 2 ) λ γ − ) r 1 1 3 1 ] yields 12 different classes of four-dimensional real Lie alge- βr + 2 x e t + R ( 2 44 χB β  2 log ˜ g sdx ˜ ( c A δ | δ − 2 βr g  | ˜ r e p q 3 ) 2 A 1 ( √ ˜ p = = A 2 x 2 x ( ) are given in ( ) = 2 4 m 1 q g − A 1 d x 1 4 x ds ( ( = = = − Z f g 2 φ , A = χ ds , S 3 B The classification of [ Here, we try to extend the classification proposed in [ If we perform the coordinate change The 4d action (after appropriate Weyl rescaling etc) is given by eq. ( bras (including some indexedembeddings of by fixed continuous three parameters), sub-algebrasHere, (corresponding with we to give a the the field variety datadescribe theory of space) corresponding the inequivalent into classification to each. of the subalgebras four-dimensional in algebras section in section as a tractable starting point for a more generaldimensional quantum classification. field theory has four-dimensionaldirection), spatial and slices (including therefore the a “radial” classificationmore based natural on for four-dimensional static realmore metrics. Lie non-trivially algebras This intertwined seems allows with for the the “field possibility theory that spatial” dimensions. the radial direction is 7 Classification of fourWe dimensional have seen homogeneous so spaces farsible that by to various exhibit simple modifications analyticThe of striped more the ambitious phases Bianchi goal horizons, and of it [ hyperscaling is violation pos- in anisotropic phases. The “striped” structure ofin the the solution metric. is evident and is seen in both the scalar field and where where slightly to: The 4d solution is given by, JHEP03(2013)126 i 4 e ∂ 4 + (7.1) (7.2) (7.3) ∂ 3 4 ∂ + ∂ 3 2 x ∂ + directions 3 + 2 j x 2 ∂ e ∂ 3 + ) x 3 1 4 x ∂ + 2 1 + x dx 4 ∂ 2 1 x = = x dx 4 4 +( e = = ω 1 4 4 ∂ e 1 4 ω along all the 3 ax dx i 3 ω ∂ dx = = 4 4 = = e 3 ω 3 3 3 e ω ∂ dx j 3 . = = ω k dx 3 3 3 . These satisfy an algebra 4 ω e ⊗ k ω x e i H dx ] in naming the algebras — we call them 3 − ∧ k 4 ij ω ∂ e j x 44 ij C = = ω η − 3 3 i jk e 3 2 = 1 ω + ] = 2 C has four linearly independent Killing vectors j – 16 – dx 3 ∂ dx 34 1 2 ) 4 ··· , e 3 H = = x i = 2 2 e ,C = dx [ − e i ω 4 2 2 x 2 dω = 1 ∂ dx ds − 3 2 2 34 = = . The Lie derivatives of the i 2 2 dx dx ( e 2 4 ω ω 4 ,C x x 1 2 2 − ∂ e = 1 = 1 = 1 + = = 2 1 2 2 2 24 2 2 34 34 e ω dx 12. We list only the non-vanishing structure constants (up to obvious dx 4 , 4 ,C ,C x x ’s can be normalized to satisfy the relation 1 a, C i 1 − − ··· ω dx ∂ e 1 = 1 = = 1 , 4 1 4, which generate the isometries of = = ∂ dx 1 1 1 ax 14 14 24 1 1 1 = 1 − e C C C = = ··· ω e ∂ , 1 1 k : : : independent of the relevant four dimensions) will be invariant under the isometry the structure constants of the related four-dimensional real Lie algebra. It also = = e 3 2 1 ω , , , 1 1 4 a 4 4 k ij e = 1 ω ··· A A A C i with In the following, we adopt the notation of [ • • • ,k 4 permutation of indices). (with group. A Here, we will list theas structure well constants as of the convenientparticularly 12 choices useful inequivalent because for four-dimensional a algebras, the metric Killing written in vectors terms and of one-forms. them The one-forms are with has four invariant one-forms vanish, and the A four-dimensional homogeneous space with 7.1 Real four-dimensional Lie algebras JHEP03(2013)126 4 3 ∂ 4 dx 4 1 ∂ = = ∂ 3 dx 2 + 4 e 3 x ω = ∂ 1 2 4 ] 3 3 + 3 ω 3 bx 1 dx ∂ dx ∂ ) + 4 2 4 = x 2 x 1 x 3 3 ∂ − ∂ e 1 2 2 ∂ e 2 + = = sin( x 3 ax 3 3 3 2 1 − ∂ e + ω ) 2 dx 1 + 3 = 4 4 3 ∂ x 3 dx ) 1 ) ∂ e dx e 3 4 4 x dx = = x x dx = = 3 3 1 4 1 − e 4 4 ∂ ω x ∂ 2 e 3 3 ω b [cos( − x x 4 dx 2 1 2 ( 1 2 = 4 bx 4 1 dx 4 x 3 − 2 − − = 1 = 1 ∂ ( 3 2 34 dx ∂ e − 2 4 3 2 ∂ e dx ∂ dx x 1 3 23 34 x = = = 4 b 2 − = = = 2 2 4 = 1 2 dx ,C ∂ e e 2 2 4 bx 4 2 ω ω = e ,C ,C x e 3 − ω ω − = = 2 ∂ e − 2 2 = 1 3 34 4 e ∂ e ω ∂ = = = 1 = 1 2 34 = = 3 3 + ,C – 17 – e 2 2 3 2 4 34 34  ω 3 e 2 ∂ ∂ ω 4 ,C ) ∂ ) 1 + 1 2 = 1 dx ,C ,C 2 3 3 + x 2 b − − ∂ 3 x  2 24 3 2 dx − ∂ 3 1 2 dx ) = = = 1 = 1 3 = x 3 4 4 dx x − dx + x bx 3 3 2 2 3 34 3 34 34 24 24 2 4 ax 3 2 b, C + x x 2 ∂ − 2 +( ) dx ∂ e 1 2 1 2 ∂ 2 3 ,C ,C ,C 2 4 ) ∂ x b, C +sin( a, C x + = = 3 − ∂ ) 3 2 x 1 2 2 2 3 3 + + = = 1 + = 1 = 1 = = 1 e 2 x ω + 3 dx + dx x dx 2 ) 4 1 + ∂ 2 1 2 2 2 1 24 2 14 24 24 24 24 4 x 3 2 x x x +( dx 3 x 1 − 1 1 2 bx +(  1 ,C ,C ,C ,C ,C 1 ∂ − 1 dx 4 ) dx a, C + 2 4 ∂ x +( 2 4 dx dx (cos( 1 2 x 1 ∂ 1 4 x x 4 = 1 = 1 = 2 = 1 = = 1 2 1 1 x  − − ∂ 1 − 4 ∂ e e + 1 ax bx x dx ∂ 1 ∂ e x 1 1 1 1 1 1 1 14 23 23 14 14 14 1 − − ∂ = = =2 = 1 − = = = x ∂ ax e e = = C C C C C C 1 4 1 3 e ∂ 1 4 1 = 1 1 e e = = = = ω ω e e : : : : : : 1 ω = =( = e 4 1 3 1 ω 9 8 7 6 5 4 e , , , , , , 1 4 1 e e ω ω b 4 4 4 a,b 4 a,b 4 4 e e ω A A A A A A • • • • • • JHEP03(2013)126 4 2 ∂ 4 + dx dx ) ) 3 1 4 ∂ = x ) 4 2 dx ω ) ) x 4 3 − x +sin( 3 dx 3 1 ) 3 ∂ 4 ax 2 dx x dx ) x 4 4 + sin( +( 1 2 2 x bx 2 2 ∂ a sin( + − ∂ 1 3 ) e dx x − = 3 ∂ ) 2 = reflects the intertwining 4 x − = =cos( 3 3 34 x 3 3 4 + dx 1 ω e ) 2 ω ∂ A 4 2 ,C x ax x 1 (cos( 3 in 3 ∂ 2 + 4 = 1 x 3 3 +( dx ) 2 4 x 1 − (cos( A dx 4 2 ∂ ∂ e dx 34 ∂ 1 2 4 4 x 1 x = = = = 4 ax − − 2 4 2 4 ,C 2 e ax − sin( e e 1 e dx ∂ ω ω = − − 2 =2 = = = 2 1 ω 4 2 4 2 ∂ = 1 = e e 3 dx ω ω ) x ) 1 24 4 3 24 2 1 2  x 2 – 18 –  − dx ,C 2 ) 2 ) dx 1 2 ∂ 4 a, C 3 dx 4 − x x = =cos( = 1 3 dx ∂ = 1 2 ) 2 2 x = 4 + e 2 ω 34 1 2 − 2 24 3 sin( x 3 2 ∂ 14 − 3 − 2 ,C 3 2 dx 1 1 2 bx ∂ ,C a, C dx x 2 dx 3 − +sin( + dx 2 1 2 x x 2 3 ) 4 x ∂ 1 2 = 2 = = 1 4 + ∂ + 2 1 2 dx 1 x − x + 1 ) 2 1 3 23 14 24 3 2 + 4 ∂ dx + ∂ 1 1 x 1 3  dx 1 x ∂ ,C ,C ,C 4 2 (cos( x ∂ dx 1 x 3 x 2 + x + 3 x  1 2 x ) 3 (cos( 4 1 3 = 1 = 1 = 1 − b +1) 1 2 ∂ 4 b ∂ ∂ e dx + 2 ( ax 1 4 1 1 1 ] , where the full four-dimensional algebra was semi-simple and contained a 13 23 23 2 ax + x − = = = = 1 1 1 3 e − − C C C 1 3 1 3 ∂ − dx dx ∂ ∂ e e e e : : : =(1+ = ω ω = = = = . The associated Killing vectors will generate isometries of the spatial dimensions 4 1 = = = = 4 1 4 1 12 11 10 e ,k , , , ω 3 1 3 1 e e ω ω 4 4 a 4 4 e e ω ω A A A A • • • ⊂ 3 A in the dual fieldof theory; the the “spatial” non-trivial andsolutions “radial” embedding of directions of [ along thetrivial factor flow, corresponding which to allows scale us transformations. to generalize the 7.2 Three-dimensional subalgebras We would like thefield bulk theory. geometry to For this reflect to homogeneity of happen, the we spatial wish slices to in embed the a dual three-dimensional real Lie algebra JHEP03(2013)126 ) ,k 4 A 1 > | a/b | into a given b/a, 3 = A z 1; 3 2 3 ) 1) 2 < , e , e , e φ | 2 1 1 , e ) , e , e , e a > φ 1 4 4 sin( a/b ( | e e e 3 e /a sin( 3 3 3 3 3 1) a/b, e , e , e , e , e = 1 ) + − 2 2 = 2 1 φ z z , e , e , e , e = 3 3 ) + 1 4 1 4 φ 3 a e e e e , e , e 1), cos( ( 2 2 2 , e 2 , e , e 2 cos( , e 1 4 a < 1 1 , e e e , e ( 4 1 a , e – 19 – , e e 4 4 , e ] for three-algebras with the more standard Bianchi nomen- e e = 4 e 5 5 5 1 44 z , , , a 3 b 3 z 3 2 , A 3 5 3 5 1 1 , , , , e 2 3 A A A 3 a 3 3 a 3 ; A A 1 3 2 , e 3 3 A A A A 1 1 1 , e , e , e 3 2 4 , e , e xe 2 2 4 3 4 , e , e e xe 6=0): + 1 3 , e , e , e , e 3 1 2 2 3 3 2 e e + 6= 0): 6= 0): ]. The results are as follows. We name the subalgebras following 2 4 5 1 , ab 4 , e , e , e , e , e , e , , , We also list the generators after each subalgebra. xe 1 3 3 z 3 . 4 1 1 2 2 1 e 44 , a A , a e e e 5 1 3 + 1 A A A B , e , e , e ]. 4 1 4 4 1 e e e e 44 A 1 1 1: a < a < , a | > b | b b 1 + 1+ | | , , b b 1 b 1+ 1+ = 1 2 = z z − 1; 1 1; = < , e b < 3 | | 3 b b , e , e b 2 1 e 1+ | 1 + | , e , 4 b b, e b 1+ = ) z = 1 + φ 3 – 20 – z 3 3 6= 0 , e 2 sin( 1 3 , e , e x 3 , e 2 2 6= 0 e 1 3 2 2 , e 1 ye e 4 3 , e , e 1 1 x , e ; , e , e e , e 1 4 + 3 4 1 1 4 , e 2 2 2 e e , e , e ) + 2 2 3 e 3 3 3 φ , e , e , e , e , e , e , e 2 1 4 3 1 1 1 , e , e 1 1 1 , e xe , e 7 , e 1 e e e 3 e e , 2 2 2 1 5 4 5 2 b 3 , , , , ; ; , e , e , e , e xe e A e e + cos( 3 z 3 3 z 3 2 3 3 , e 3 4 3 1 xe 3 1 A 2 1 2 4 1 e e A A A A + e , e , e , e e , e xe + 1 4 4 4 2 , e , e , e , e 3 , 1 3 1 1 e e e e 4 3 + e , e 0): A , e , e , e , e 4 2 1): 4 2 4 4 e e 1 1 e e e e < A A 1 3 2 1 ): | , , , , , b > | 3 6 7 1 3 3 3 3 b , , , , ⊕ ⊕ x 5 1 1 5 3 | 1 3 3 | 3 , 3 , , , , A A A A 2 1 1 2 3 2 2 3 3 3 3 < a A A A A A 6= 0 < 3 A A A A A A A a (0 : : (0 ( : : : : : 12 10 11 9 9 9 8 6 7 1 5 , , , , , , , , , , , 4 4 4 0 4 1 4 b 4 4 4 4 1 4 a,b A A A A A A A A A A • • • • • • • • • • JHEP03(2013)126 i s (8.4) (8.5) (8.2) (8.3) ) then ). The i 8.2 (no sum for i . Equation ( ω i i σ λ , t = σ 4. Since the NEC amounts to i , i σ 3 , , X 2 is identified with one of the exact i 0 , , s r and 2 ≥ ) 4 . =1 t j = 1 ij i X 0 (8.1) s σ δ 0 i i ( , i s + i λ ≥ ≥ t s ij − ν ν 2 = X M ) , where we will only consider a simple scaling 1 2 N i N ij µ µ σ dt ≡  λ ( ) – 21 – 2 i N ν N r s ( 4 =1 Via Einstein’s equations this translates into a con- µν N i µν X tt 4 µ T . =1 g i G X µ = N p  2 N µν = = ds , and analyzing the constraints obtained by requiring that G t ij ~ σ . The radial coordinate N r M t β 2 to be diagonal: are the invariant one-forms listed before. For simplicity we shall e j ij ω λ ) = such that the other 3 one-forms form a sub-algebra. In this basis an r i ( must be non-negative. σ tt g ij = are dual vectors to the invariant one-forms , where M i j dr i ω X 4) in the form: λ , ij , 3 t λ , X 2 ≡ , i In the following we will study the NEC for each of the 12 types of space-times (in a We will study natural multiparameter families of metrics realizing a given symmetry We will work in an abstract orthonormal basis, in which the metric takes the form σ = 1 i case, we conclude that the related type is notnatural realized metric parametrization physically. realizing the associatedically four-algebra), the by eigenvalues writing of out the specif- all of these eigenvalues be non-negative. of the matrix structure. We conclude thatsupported if by a physically given reasonablethose set matter parameters of as fields. parameters capturing a violates“natural” In physically the set unrealizable this of NEC, geometry. case, possibilities In it we some for cannot cases realizing view be the a the entire given metric four-algebra with can be eliminated — in this becomes a bilinear function for arbitrary choices of theimposing four real positive parameters definiteness of this bilinear function, it then requires that the eigenvalues ( where only take the matrix time-like one-form is given by red-shift factor: one-forms arbitrary future-directed null vector can easily be written by using 4 real parameters for for all future directed nullstraint vectors on the geometry, namely that the Einstein tensor has to satisfy We have proposed a classification of extremalfour-algebras. near-horizon metrics The for first 5d check black braneswhether of using they validity can for be space-times supported whichtensor by realize reasonable that matter these supports content, symmetries i.e. the is whether space-time the satisfies stress the energy Null Energy Condition (NEC): 8 Null energy condition for 5d space-times with four-algebras JHEP03(2013)126 (8.8) (8.9) (8.6) (8.7) (8.10) (8.11) , are 2 2 . λ r M + = .
4
). One of the x 2 2 λ = 0, and one could + 2 8.5 t 4, it is easy to see i + , k β 2 4 3 2 gives λ ) , t 2 3 4 2 β λ , + 2) signifies a scaling of the ω cases, where the natural 2 , and the limit where the 4 i a r − λ t k ,k β 4 4 = 1 4 2 + ( λ = i = e A + 2 4 3 2 4 . . λ i a a ir , σ 0 2 ( 2 ). The eigenvalues of C 0 ) is 2 3 ) 2 3 defined by eq. ( r i = r < λ λ ( ω < = 2 . ( (1) for i tt ir 4 4 , all conditions are satisfied. dr M 2 i 4 2 g 0 4 4 O 2 4 − C λ λ λ x 4 3 λ λ < 2 3 < a λ = 3 = + 4 =1 =1 2 2 i λ X i 2 X i 2 2 3 2 2 λ t λ λ − λ β + 2 1 + 2) q 2 : 2 λ a ( ± – 22 – ,M dt − t r with ) , at least in some cases. ) t β t t = β t ≤ β β 2 β 1 e s Law 0 + M − becomes a 2 + should have the same qualitative scaling behavior as the = )( − t 2 4 M 2 r 2 4 β i a λ . The red-shift factor k i λ ds in the IR limit corresponds to a ground state entropy density − Nernst − σ as a radial coordinate ( as a radial coordinate; =1 r 3 i i 4 4 k P + 2 x x + 1)( − =1 a 3 i t e ( β P ( − ∼ − approaches zero corresponds to the IR in the dual theory. A field theory e = = tt g 6 ∼ 1 4 , 3 M M for the one-form : We identify : We identify near-horizon geometries, which violate this criterion, are ubiquitous in simple approximations r 2 1 i d , , . In general, a structure coefficient of the form k i 4 a 4 R − parameter space; for example:that at with large all negative values of Nernst’s law forces It can be shown that the NEC and Nernst’s law are satisfied in several open sets of which is negative. Therefore this geometry isA ruled out. A It is straightforward toeigenvalues of calculate the the matrix matrix e , β × t We will now study the NEC separately for each of the Before listing the results of applying the NEC to each type, let’s discuss the constraint 2 As already mentioned in some of the discussion above this may in fact be too strong. For instance, • • β 6 AdS to string theory, andcontemplate softening show the fascinating condition physical above to properties. However, these have We will impose this, as“Nernst’s well law”. as the NEC, as a criterionmetric of ansatz reasonableness, we which will we’ll consider call is field theory entropy densitythe goes spatial to volume zero inred-shift the factor, in IR. the On sense the that gravity side, this means that on form red-shift factor spatial volume of the field theory. In order to satisfy the laws of thermodynamics, we require that the JHEP03(2013)126 , we (8.16) (8.19) (8.20) (8.12) (8.14) (8.15) (8.21) (8.17) (8.18) (8.13) are are are are . .
) 4 3 M M M M → −∞ b λ , t β + . + a 4 2
2 2 λ + 1) λ t β (1 + + t − 2 3 β 2 4 λ b λ + 2) + 2 4 2 + λ 2 + 2 1) ) b t a , − β )( + . , 2 t t t t − 2 β β β β − . ( a ). The eigenvalues of ). The eigenvalues of ). The eigenvalues of ). The eigenvalues of b . ) r r r r + 2 2 2 3 + 2 t 4 4 4 4 0 λ λ a 2 2 2 3 β λ 2 4 λ + 2 = = = = 4 4 ( 1 + ( λ λ 4 3 0. Therefore the NEC can be easily 4 3 λ < + λ 4 4 4 4 a 2 4 λ O λ 2 − − 4 3 ) x x x x 2 3 2 2 4 2 + λ + 6 + 6 2 2 (( < b λ λ λ 2 t λ + λ = = 2 2 2 3 2 3 2 t 0. 2 β λ λ + λ 2 2 4 t 2 4 β 2 q 2 2 λ β a ) + + λ t − , ± b > 2 1 2 2 β ) ) 2 4 2 4 → 2 λ λ = t t – 23 – +
λ λ are roots of a higher order polynomial, which is + ,M 4 β ,M (1 + β 2 , 2 3 4 3 2 − t a b 3 4 λ , λ , eigenvalues behave as β ( − − t 2 2 2 3 b = , b β b λ 2 1 + 1) M + 1) M ,M 1) t t − M + 4 M 2 4 + 2 2 4 β + 2 β 1 t 4 3 − λ λ a a λ 2 4 t − − − with 1 + aβ 0. The NEC can be easily satisfied at large negative values )( β λ )( b b 2 t t t t 4 4 4 2 2 4 β 2( β < β β λ λ λ + + as a radial coordinate ( as a radial coordinate ( as a radial coordinate ( as a radial coordinate ( + 2 4 2 t + a + 2 a λ 4 4 4 4 β = = a b a x x x x )( )( 4 1 2 ( q ( t t , 3 β β − − ± − M M + = = = b (1 + ( 2 4 1 , 3 − − M M M = = 3 1 : We identify : We identify : We identify : We identify . t M M 6 5 4 3 β , , , , a,b 4 4 a,b 4 4 A Nernst’s law forces We see that therelaw is for an large negative open parameter set satisfying both the NEC and Nernst’s A On the other hand,satisfied in Nernst’s the law regime forces with large negative values of The other three eigenvalues too cumbersome to writecan down. easily check However that in the the rest large negative limit Nernst’s law forces of A A • • • • JHEP03(2013)126 1 and (8.27) (8.29) (8.31) (8.32) (8.22) (8.23) (8.26) (8.28) (8.30) (8.24) (8.25) − are are are . M M M
b > 2 2 λ + , hence this type 2 3 2 4 2 λ λ , 2 − , 4 4 4) λ 2 3 = 4 3 − λ + 12 2 1 1 , , λ , 2 2 t 2 4 λ 2 2 2 3 4 2 β λ 2 3 2 3 2 3 λ λ M λ ( λ 2 λ λ λ 2 2 1 2 1 2 1 2 2 2 2 2 2 2 + λ λ 4 4 λ + λ . λ . λ t λ 2 2 2 β 4 3 ). The eigenvalues of ). The eigenvalues of ). The eigenvalues of 1) 8 λ r r r − + − + 1) + 1) − 10 2 ± ) 2 ) 2 4 2 4 t . t = = = t t t . t λ λ λ β 0 β = β β β β 4 4 4 0 ( q 2 x x x t = < . + + < β ) 0 2 ± b b b 2 4 2 4 2 3 + < λ λ λ 0. 2 4 ,M 2 4 + 1) + 2 1 2 3 t + 1) 2 2 + 2 λ t λ λ 2 3 λ β + 2)(1 + + 2)( + 2)( 2 1 b λ ,M 2 2 β a λ ( – 24 – t t t 2 2 1 λ ( t 2 4 b > 2 2 λ t β β β 2 λ + β λ 2 λ β − b 2 − − − − − + 2 ( b b b b + a = = 2( (2 (2 (2 + 1) 8 has one negative eigenvalue, 1 4 , t − − − − 3 − β M 2 4 with M t = = = = M λ . β 2 4 t 4)( t 4 1 2 3 2 β λ β as a radial coordinate ( as a radial coordinate ( as a radial coordinate ( − M M M M − 4 4 4 t 2 t x x x β β ( . t β = = 1 4 , 3 M M : We identify : We identify : We identify 9 8 7 , , , b 4 4 4 Nernst’s law forces Therefore, the NEC and Nernst’slarge law negative are satisfied on an open set where A We find that the matrix of metric is ruled out by the NEC. law at large negative A Nernst’s law gives It is obvious that there is an open parameter range satisfying the NEC and Nernst’s We see that therelaw is for an large negative open parameter setA satisfying both the NEC and Nernst’s Nernst’s law forces • • • JHEP03(2013)126 , 3 λ = (8.37) (8.38) (8.39) (8.40) (8.34) (8.36) (8.41) (8.33) where 2 are are t . λ , β
2 M M 4 3 . λ − ]. Therefore,
2 3 2 2 . However, in 1 + 4 2 ) (8.35) λ λ 3 = 1 and the rest . . 4 2 2 λ λ ) 3 λ
+ 1 32 3 4 3 + 2 2 2 3 + = C ω dx λ λ λ 2 3 4 1 2 4 − 2 λ λ + = + λ λ 2 2 + ( t = 2 4 2 3 λ + 2 λ )
1 aβ 2 2 23 2 λ C 2 1 ω +2 +8 , ω λ 2 (( 2 t 4 2 2
2 2 a β λ 0 and large negative with λ 3 2 + 2) + dx 12 k ). The eigenvalues of ). The eigenvalues of λ 2 + 2 + r r 2 ω 2 2 ) , = −
2 t 2 a λ a > ∧ 2 3 2 3 2 ) = = 1 2 β 4 2 = j 1 . λ λ λ 4 4 4 2 2 2 ω ω 2 − 2 x x λ ( λ 4 4 − 2 i t 4 4 . jk 2 2 , ω
λ β λ ) C 0 − 2 2 4 t / 2 2 4 3 , 2 4 3 1 1 2 4 3 2 4 λ 2 1 β (( λ λ λ λ 2 3 < λ λ ,M λ 2 3 4 2 4 2 dx are = 2 1 λ t 2 1 − = 3 λ λ 2 3 λ 2 2 is ruled out by the NEC. In the case i 4 4 2 2 λ + ( 2 x 2 1 2 λ 2 λ 1 2 λ aβ ) + 3 λ M 2 2 2 λ 1 2 2 dω – 25 – t 4 2 > λ λ are obvious candidates for the radial coordinate. λ λ of the sort studied in section 4.1 of [ β λ 2 2 t 2 dr 4 3 − 4 + 7
2 4 6= β λ + q x 3 2 3 2 4 ,M 2 2 4 t λ + 2 λ a λ λ β q 2 ± λ 2 2 dx 2 ) gives t 2 3 +
2 4 )( 2 4 4 λ β λ 2 2 t 2 and = ± λ x 2 2 λ λ β 2 2 8.36 − 2 3 1 3 1 2 + 2 3 λ dt λ 2 r λ 2 t x with 4 3 λ − − 2 1 t 2 2 + λ β 2 1 λ , we have the metric a λ 2 aβ as a radial coordinate ( as a radial coordinate ( 2 4 10 − 1 3 2 λ , 2 1 e 2 4 4 λ (4 4 λ ,M 4 4 2 2 4 t λ 2 3 λ dx − x x A 2 λ β − − 4 2 λ − =
2 4 2 2 2 λ 2 4 = = 2 3 , the eigenvalues of 2 λ a λ = = is negative and hence violates the NEC unless λ 4 λ 1 2 2 3 λ 8 2 q 4 3 x 1 4 1 λ ω 4 , λ − − ds 2 2 ± − 3 x 4 2 = M M 2 2 λ λ M 2 λ r = = 2 = t β 1 4 r , 2 − 3 M : in this class both : We identify : We identify = = M 4 2 , 12 11 10 , , , 3 4 a 4 4 M M A Case 1: If we pick Therefore, there is anboth open the parameter NEC range and for Nernst’s law are satisfied. Nernst’s law forces A This is a type IIthe Bianchi geometry generic case of the NEC imposes the constraint Notice that the case where where A = 0. One can easily check that basis ( • • • 7 i jk C JHEP03(2013)126 . t can and β r r (8.42) (8.43) (8.44) (8.45) t β 2 e , , one can also take 2 12 ,
2 4 4 2 2 λ A λ 2 2 λ , − 2 1 3 2 1 λ 2 λ , but generically we can take more EAdS − i 2 choice. The rest of the classes x is negative and hence the NEC ~ t x r β d 1 2 3 . 2) = M λ 2 3 2 4 4 2 4 λ are then λ − λ + Generically, the radial direction x 2 2 t . 2 λ ) “by hand” to have the form 8 β − r M 2 1 . ( r 4 1 dr ( λ 0 2 4 λ 2 tt = . . λ g t 2 3 < 2 β ± metric with unit length-scale. In this case the t + /λ β with the 3 2 2 4 – 26 – λ dt 12 2 , ,M r + 1) t 4 AdS t β > A β 2 . It is quite possible that more elaborate choices than 2 3 e 2 t ’s. While this freedom does not enter in any of the data r λ i 2 3 + 1) β − 2)( x t λ β − as radial coordinates, = . , and simply identified with one of the t 2 2( 4 2 r 10 β λ x , ( − 4 ds = A case, we see that in this case = = and 1 , , the geometry becomes a product of the form 8 λ 3 2 4 1 , , instead, the eigenvalues of 10 x 4 , λ 3 is the Euclidean 4 M 3 A 3 M x = A , 3 1 1 x , = λ 4 r = EAdS A 2 r ~ x d Therefore, both the NEC and Nernst’s law are easily satisfied at large negative Nernst’s law forces where NEC then imposes just that Case 2: If we pick is violated, unless In the case Similar to the In summary, we have seen that the types of space-times that are eliminated byThe the reader should note that we have made the “obvious” choice of the holographic “Obvious” since we have taken 8 realizable from reasonable matterples content. explicitly, in In a particular, system we with will a work similar outcomplicated combinations effective a of action few themany to as exam- linear radial one combinations coordinate. of we’ve For discussed example, in before the type those considered here could resurrect some of the algebraic structures we have left9 for dead. 5d space-time avatars ofIn this four-algebras section we will demonstrate that some of the afore-mentioned geometries are indeed radial coordinate in each case inbe the a more analysis complicated above. function of involving the spatial 4-slices,is it because is we relevant have in selected theits the contributions construction redshift in of factor the our NECcoordinate 5d calculations we space-time. clearly choose depend to This consider upon, as among other things, which NEC are types all contain at leastare one satisfied, open around set large negative of values parameters of where both the NEC and “Nernst’s law” JHEP03(2013)126 t (9.4) (9.6) (9.1) σ αφ 2 ∧ e r  αφ i ri ) 2 σ . 2 ) e C  )] r r B i ( ) ( ) 2 mpl B r f = 2 r 0 t  rt ( ( ). The structure ). The equations M f )) θ C Λ = 0 (9.5) t r rt ikl , β 0 ) r (  + ( r C δφ o ) f 2 ( ) 2 f r t r ) i rj ( A αφ there ( r δe A δ 2 2 t = 0 (9.3) A C k ( rp e αφ 0 A 2 C i φ M = 2 e αφ ) → ( ) + )] ) 2 r r ) + 2 r r e ( ( ) + 2 i βφ rj ( ( 2 r here f j t ri C B pql ( rt e ) j rm ) + 4(  2 r f B 1 4 C , δ C r C ( ) ) k ( ) 0 t i ikl r r − M σ r )  t ( ) (9.2) rt ( A ) ( 0 t r j r there k j 2 pq C + ∧ ( B ( α A ) t A j 2 A C + [ F h r j rm A σ i ( i A + = 2 ) C σ f 2 i A jk ) ) + 2 βφ r ) ) A ) + θr ) + their field strengths. r r 2 ( r r ∧ r C e r F ( ( M e ( ( dr ( ) j ri ( here ( 0 r ( B j ) generalized to include two gauge fields, up to the f f r 2 f A f ) φ σ C ( ) 2 A αφ 0 i βφ ) r , α ) = ,F i r 2 = M 2 ( r o 3.15 ) ( r A 0 i – 27 – e A e ( r ( + 4 r j 1 2 0 m 1 4 F A ) + ( 2 = β σ αφ f there r 2 A j A ri φ − ( ijk + e − 1 2 + [  C f Λ i ) nh ) ) ) ) ) + r . . = r r 2 δφ B αφ r ( r ( ( 2 2 ( ( B B j F m e e ri 0 m ) = t rt f A here r ) C + + A C ( φ r 2 h ) mkl 2 ( A ) m  r 0 i ) + ( mjk F + φ A t r A  ( ( jpq ) h ∇ A  βφ r f αφ 2 ( ) 2( j 2 kl e r f e C ( − ) ) + 2 A 0 i 2 α r r 2 i pq R ( ( A 0 M C h f −  φ ) ) g 2 r r ( ( − i 0 t ) = r A √ A ( 9 5 1 4 A two abelian gauge fields, and nh )): F dx − Z with a similar equation for B. The gauge curvature is given by with a similar equation for Scalar field: Magnetic field: with a similar equation for Electric field: 3.15 A, B However, we are going to relax the requirement that the spacetime be scale-invariant, = In fact the above action is the same as eq. ( • • • 9 S following simple field redefinitions: for a hyperscaling violating metric (withconstants involving hyperscaling-violation the exponent radial index willof need motion to change will via: be modified in the orthonormal basis to be: four-algebras. In the orthonormal basisthe notation, this radial means one-form that to we are be going to generalize where with and look for a possibly hyperscaling violating generalization of the space times based on (eq. ( JHEP03(2013)126 (9.7) (9.8) (9.11) (9.10) (9.13) (9.12) ) , 2 2 ) )) = 0 )) = 0 ) kr . r ω )) = 0 ) = 0 ω ( ) 2 ) kω kω 2 f t ) kω 1 β β ) = − 3 β r t )) kω ) ( + 2 2 2 B β ) ω ( 2 β − )(1 + 3 (2 + 2 1 t ) = t 2 B − 3 kβ ω r β β β , φ ( β B + ( 1 + + 3 2 − 3 3 )) 34 ω + ( +2 σ a − 2 2 , 2 1 B ) 2 C ( 2 A ) + β ) r β ) 2 1 kβ − ( + ) = 0 (9.9) 2 +4Λ( ω M β 2 t 2 1 2 2 kω ( ω − 2 ωφ ) = β 2 t ( + 2 − β kω (1+ B − r − 2 a 2 1 2 ( 2 ( 1 − ( A e β β 2 1 β + − k β 3 ( 2 2 2 A 24 kβ B − (2 + t k B + 2 2 kβ + β )) M kβ (1 + +2 M ( 4 2 t 2 ( = B t +8 ( 2 + 1 2 ) 1 2 B 2 t β 2 2 2 1 β + − 1 β 2 ,C β + A 2 ) + B ω ω M ) a σ t ) ) ( ββ r )+ r β + 3 2 2 t − ( 2 , θ − kβ − ( 4 + ( k β 3 f B 2 2 B θ k B 1 1 2 − ωφ ω 2 β is the first such case. A hyperscaling violating 1 − − 2 1 2 − + 1 M +8 β B a 2+ = ( – 28 – 2 2 aβ 2 e ( , β 2 2 2 β 1 2 2 1 β 2 3 a 4 ) − 2 3 β β 3 β ( r dr k B B 2 A 1 kβ B ( ) B ) = ) 2 β 2 f 3 , δ 2 kβ r a 3 + 2 )+2 ( B B θ B ) + k + 2 ) = 2 B + t 2 1 B + 2(2 + 14 2 r 2 β β + 2 ( k − 2 A B M 2 2 8(2+ dt ( ) + 4 + β r 2 2 M 1 B (1+ t − (8 + 2 ,C +4( β 2 β 2 B = )+4 β t ,B ) 2( ω kβ 2 2 1 2 r B t 1 2 β e B α − ( β 8 + σ + 2 + − ) + ) f M kβ a 2 r − 1 2 2 B 3 . We will turn on the two massive vector fields and the dilaton as: ( +2 k 2 = 2 β kω − B θr 2 +2 2 β M B ωφ 2 e ( β 1 − − +4 geometry takes the following metric: 2 3 M ds e 2 3 − β ( t 2 B ( , (1 + : 2 3 2 B B ) = a 4 2 2 ) = A (12+ φ ( r B r β 2 β 1 2 A 1 ( M ( 2 2 1 1 β 2 β 2 f β β 2 34 2 ) = 4Λ + 2 3 B r C +2 − 4Λ+ ( − B B 2 − , 2 2 A a 4 − A Einstein’s Equations: EOM for EOM for B: EOM for A: The structure coefficients in the orthnomal basis becomes: • • • • If we take then the equations of motion becomes algebraic: where again Many of the space-timesspatial in 3-fold the belonging classification beforewith to in are one previous simply of sections. scaling We theactions geometries will on 3d therefore with the focus Bianchi a spatial on types. geometries,version the of and geometries the with They non-trivial have radial already been dealt 9.1 JHEP03(2013)126 ). = 5 2 )) ) t (9.19) ) 2 (9.16) ,B AdS 2 β ) ) 5 5). Also , + (9.14) √ kω √ ) kω 2 )=0 (9.15) ) − ))  )) = 0 2 kβ , − t ) (9.18) t − t β +5 β ω 24 + kω )) = 0 β 2 β ( 1 (6 +  ( 1 +  , 2 1 (3 9 2 p (9.17) 2 1 )) ) )) β kω − 2 β kβ  2 2 β 2 2 kβ ( ) − )) = 0 (2 + 2  12 β  β )) + a 2 3 2 1 2 = 2 + 2 4 = + 5 (2+ − kω 1+ 2 β β B 2 2 )) = 0 3 2  t A + 5 + 2 2 − 2 B − β 1 + A  ( (1 + B − ) 26 + β β M − 2 kω 1 (3 a ,A 2 +2( t M ) 6 ( ( (2 + ,M 4 6 ( β (1 + β (3 2 2 t β q 2 B 1 )+ 2 3 2 t 9 2 2 2 1 will yield real solutions with − − β kω A 12 β B A M 18 ) 2 − kω − (2 + ( 2 t ) + − − 2 3 3 = t − 2 β − = B ) β,  + 2 , β ) + B ) 2(1 + B 2 (2 + 2 = 2 t k 2 2 B ) (2+ A 2 M 2 26 + 2 B β )+ β ) 2 2 ( r 2 t A kω B  M B 2 2  1 + r ) + M β B ( ,M − ( 2 +5 2 B 2 ) ) + − + 2 2 − = 2 β ( kω t  t B 1+ = 0, which corresponds to the conformal − + (2 + 2 B (3 2) + 8 β β kβ kω ,M 2  2 β − 2 kβ 4 4 + ) ( − + 27 we have β − B 12 ) t 2 − 2 + 1 (2 +  2 − t  , β β 2 2 kβ + 2) 2 Λ = 108 ,B β 2 + β 2 β 2 2 1 β  , 3 +5 2 − ( 1 k a 2 B β + β – 29 – 3 2 β 2 2 a  ( β ) 3 B  3 2 β ) +  26 + = 1 3 2 B (3 2 kβ 3 + 2 √ 162 k ) B B 2 − β ) 2 B  2 2 B 2 B  − q + 4( + 4 + 2 β 2 2 / B 12 2 4 β a t B kω B 2 2 , 2 k kω 3 β B +4( / 2 B β 4 1 2 β + 5 3 − t 4 3 54 + B +2 4 − + − β ) 2 − − (2+ β + 60 12 + − t a + t 5) − 2 2 4 − 1 2 + 2 − 6 26 + = 2. The following solution is obtained: β ) B 2 a β − a 2 β ) β ( √ 2 1 2 − ( ( 2 a = 2 β 2 k 2 1 2 2 is not realizable unless it has non-zero hyperscaling violation 1 1 (2 q 2 2 2 1 2 β β + β  β β 2 ω β β Λ = 36 , β (4+4 − , β 2 a 2 1 − 2 a 4 ω  2 + 2 B − (4 + 4 2 2 + 2 1 + 2 A − 2 +3 4 + 4 (1 + , ) + 2 kβ (8 + 4 a  2 ω 9  2 B = (1 + 36 + 2 M 2 2 − B A  2 2 6 2 1 ω A kβ a 2 = . We see that a large open set of ( ( kω 2 3 1 M 2 − 3 2 t 1 β r M M − B − ( M kβ ( ( β +2 2 B ( 3 (4(3 + 2 kβ ,B 2 2 t 3 , β − 2 2 t 2 = = = = 1 (4 + 4 (1+ 2 2 B 3 2 (4+4 +2 A B 2 2 1 − A β / . ω t 3 2 1 − k 2 B β β 2 θ − − − = β A − 2 5) B 1 = 4Λ + − M 1 β = 0. This fact remains true for other values of the parameters, indicating √ +2  2 − 4Λ 2 +2 4Λ , β θ B − 2 (1 + = 2 − notice that the solution iscase not valid when that in this system exponent where a positive value ofFor example, Λq taking (so that it is, quite plausibly, ultimately gluable to a A general solution iswe impractical will for such show a thatwhich system a gives of solution algebraic a equations. exists thermodynamically for Instead, reasonable some space-time. particular values In of the particular, parameters, we pick JHEP03(2013)126 ) r ( (9.20) (9.21) f 1 λβ − , )= , ) r ) 2  ( ) . = 2, the solutions  3 24 +2 2 kr 2 2  β 2 ) + (  3  β , β 2 β 9 ) 2 ω ) ) = ,C  ( can yield real solutions  r − 2 ) ( 9(3  r λ . + ( + 2 = 2 ) − 10 +  f β + t r ) = 0 (9.22) β 1 ( ( 2 , φ = q λ 3( and β , β ) f β 2 kω 2 1 2 3 2 A 9 σ β β ω ) − )= r = ( t r ( 2(1 + 1 β + ( ) = ωφ ( β ,M 10 + 2 2 34 r 2 1 r − ) will reduce the equations to algebraic ) ( )  β e 1 r ) ω ) 2 1  14 2 ω = − B ,C +2 2 kβ − 2  β β ) + ( ( ( , r − ,C β ( 11 2 k 1 ) = ) 9 b ) r ) f β  – 30 –  r ( 2 1 2 ,B = 2 will suffice to be general. Therefore we will turn ( )  2 + 2 +2 f 2 r dr bβ 3   , θ 1 ( β 81 +  10 + a ( 2 θ k β f , σ β ) ,B 2 )= 2 9 +  q t + β with the structure coefficients r β = + σ ( σ β + 2( + 2 9 2 ) 2 , 3 3 r 34 , δ a 4 2 ( 1 + A β − dt ) = θ k C + 18 10 + A r r − ωφ t 3( M ( 2 ( β − q − 2 β 2 )= 2 34  e Λ = β e 9 r t C (  − A = 2 2 24 (1 + , 9 +  9  = α  2 3 2 r 2 ) = 2 r B − − ( ,C ds ) A = = = = r ( t 3 k 2 B f A B 1 M aβ 6 3 , , : a,b 4 4 t )= r A A A ( As before, taking Since the radial action rotates the 2-3 plane of the geometry, choosing one of the vector Similarly to the previous case, a large open set of It can be shown that with the same matter content as the previous example, this class 1 • 14 C equations: fields to be alignedon with the either following of fields: with structure coefficients: 9.3 This geometry has therotation interesting on feature the that spatial the 3-manifold. radialthe The action following hyperscaling metric: involves both violating version scaling of and the geometry has with positive Λ; andviolation. this system can only support geometries with non-zero hyperscaling take the form: This is a simplified version of of geometries is also realizable. Again, if we fix that 9.2 JHEP03(2013)126 2 ω (9.27) (9.28) (9.29) (9.25) ) (9.26) t + 34 ) = 0 (9.30) t = 0 (9.31) β = 0 = 0 ) β 4  ω = 0  ω ω ) ω βω ) = 0  4) = 0 2 − ) 2 ))  1) 4)) ) + λ 2 2 ) b  − ) ) )) ) ω β (9.23) (9.24) 2 β − − 2 t 2 2 (8 )) 2 kω 2 t k ))) 2 λ 2 2 ) 2 − a kω ω + 1) A 4 + 16 − ω ) A ( β 2 t λ ω − )) B b 2 = 0 2 + 2 t + ω 2 ) t p kω − − A + 4 )) − λ ( b b 2 − + β ω A 2 )) = 0 2 2 β ( ( ) + ( 2 ( (4 + b k − ω = 2 ( bk a 2 + b 2 2 ( 4 B ω b kλω t λ k kω λ k k 2 2 + ( k 1 kλ ( (4 + + 4Λ b 2 − ab (4( 2 β k + + 8 + + + t + + λ 3 + a k 2 ab t 2 β ω b + b b ω 2 + ( ω ) ( β bλ (1 + λ + 2 2 2 + t ) ( 2 (4( 2 − 2 2 2 t (2 b a t (( λ bλ ) + β 1) + 2 λ β + ) a β (1 + 2 2 2 8 1 B B β ) t a ω 2 A 2 t ,A 2 2 ( 2) t + − β β B 2) 2 1 β − 2 − β t 2) B 2 − M 2 t β k 15 β 2 2 ) + − ( − 2 t − λ − ) + 4 2 t 2 − 2 + 4( 2 A λ λ t b √ 2 t 1 A β )( ) − 1 + (2 + 4 λ 2 t 2 + 4( B β − 1 + t + 2 2 2 t A + + − kβ + 2 ) = β A  − − − 2  2 2 A 2 t 1 kβ  a 2 )+( b λ 2 + 2 1 − − k + ( β 2 β 1 b kω − + ( ) β 2 1 + ( )) b − β ) + ( + t 2 t + b ω 3 + ( − kβ b β − kω kβ β + 4 – 31 – a + a 2 + 2 A 4 ( t  ( (  + 2 2 + (2 λ + 2 + 2 2 1 a β 2 1 + t − b 2 2 1 ( ( − λ (2( 2 β ( a kω β a kβ 2 t λ ) 2 4 + 2 t k λ ) kβ − kβ 2 ) kβ B A λ 1) (4 − 2 A (2( 2 B 2 + 2 2 2 + 2 B ( + 12 +2 k (1 + M +2 + − + λ 2 2 M 2 2 2 2 1 2 2 bB M b − 2 2 λ 2 2 λ a λ β 2 2 − 2 ) 1) + ) B λ λ 2 − B 2 ( b B 2 B +4 B 2 − − a 1) + − + 2 − λ )+3 M 2 t + a M 2 t A 2 + 4 2 2 ) − 2 2 2 2 + A 2 β − A = 2, the following solution is obtained: λ ) + bβ 2 − B ( M B 2 ) ( 2 8 B M 2 + 12 2 2 t 4) 1 M 2 λ 2 t kλω + , λ 2 t − + − A λ − 2 M 2 A ( kβ 2 2 t A A 2 A + 2 kλω − 8 − − − ,B A − λ M − M 2 2 + − bλ 1 2 t ω = 2 t + 4Λ +( B t β (( A A − 4Λ bλ (4Λ 2 − 2 2 B ( , β − (( − B = ( 2 2 M (2 + + 2 k 2 B b = 1 4Λ (4Λ + B : − b ( − − : 2 ( − Einstein’s Equations: φ B Again, we are notat going to give a general solution, but instead simply observe that • • • JHEP03(2013)126 0, > (9.32) 1, we can  is apparently ε ω = β → for some ε = 0 θ ∼ ω , 4 − β ω . = 2 − 1 2 β 3 β − − = = 0, hence is not thermodynamically allowed 2 B = 0. We leave this to further work. All the ex- b < – 32 – The conformal case θ . , a 5 + 2 which appear thermodynamically stable; and Λ ) a 2 , 0 ω AdS a > + 121 2 ω βω , we have discussed 5d space-time. In this section, we discuss 4d 8 9 ,M 2 1 β (16 + 64 2 2 1 β − = and section 2 A Λ = 8 M which allows it to be gluedvalid, to except that it(by will “Nernst’s make law” of section 8). It is easy toobtain see real that solutions with by making for example 10.1 Bianchi three-algebras forWe radial first and discuss two-spatial 4dthree-algebras. directions space-times Here, where we the giveizable. radial some direction examples is of nontrivially this involved sort, in which the should be easily general- dinates are intertwined in a non-trivialond way by part the of Bianchi this three-algebras. section,metry Then, we algebras in investigate of the the sec- horizons examples in wherenates 4d, can four-algebras where all can the be arise intertwined. radial as and Weconditions restrict sym- field our that theory attention the to spatial NEC only and the places time static on coordi- metrics, them. and discuss We will see that some of the types are excluded. geneous horizons in 5d canspatial be coordinates governed by and a the four-algebra radialcan involving direction. the arise three as Quite “field symmetry analogously, theory” the algebrasalgebra Bianchi of mixes three-algebras general the horizons two in fieldpart 4d. theory of spatial In this such coordinates section, cases, and we the the investigate Bianchi examples radial three- where direction. the In radial the and first field theory spatial coor- 10 4d metrics governed byIn three section and four-algebras space-time where the radial direction isand nontrivially four-algebras involved in we’ve realizations been of discussing. the real We three have seen that the fairly general static homo- directions in which theeven field more theory lives. novel possibilities Thegroup where gravity which description the suggests does bulk there geometry nottheory could directions is have be alone. homogeneous such with An aalso a exploration left three-dimensional symmetry of for sub-algebra such the that geometries future. and acts their on field the theory duals field is In summary, in this section,scaling we have metrics found governed (hyperscaling-violating“Nernst’s analogues by of) law.” several extremal of Itwithout is the hyperscaling interesting violationi, four-algebras that i.e. consistent with amples we with we have have been the considered have unable, NEC a as three-dimensional and sub-algebra yet, which to acts find on the such spatial metrics 9.4 Comment JHEP03(2013)126 } 3 and , e 2 t (10.6) (10.3) (10.4) (10.5) (10.2) (10.1) e { t∂ t , β } 2 and adding + t , e 2 . 1 ∂ t∂ e . 2 t i { x . β 2  ) of type III satisfy 2 2 i + + ) i 2 e r r ) 2 dx 3 ∂ ∂ ( ∂φ 1 dx x . 2( dx 1 = 0. The Killing vectors = = 2 3 1 x − − 3 2 − e + ( i j,k e e e dx dx 2 2 C + A ) . = = = 2 to 2 to 2 1 2 3 ) m 3 2 1 dx ω ω ω e e 1 4 ( r dx 2 ) = 0 dt , ( − t ), we consider the ansatz for the gauge r − h t 2 β e 2 2 β . F e L + t 10.1 3 1 4 + 2 2 x A + 2 1 − i dr L p φ 2 2 + – 33 – = = dr m + Λ 2 ( t φ + A dt R A r 2  t β p dt 2 = 1 and the rest of the g r e t 2 − β ∂ 1 31 − 2 2 √ h e C x 4 2 − − L h + dx 2 1 2 3 1 = = ∂ ∂ ∂ L Z 2 1 13 = = = = to correspond to symmetries along the field theory spatial directions to correspond to symmetries along the field theory spatial directions = C ds 1 2 3 2 } } to the set of Killing vectors. S e e e 3 2 t ds ∂ , e , e to be the radial direction. Then the metric can be written as to be the radial direction. Then the metric can be written as 1 1 e e = where r r { { 4 k to the set of Killing vectors. e = = e t . We consider two possible embeddings of the two-algebras in the three-algebra. 3 1 k ij ∂ } x x 3 C = , e 4 1 The scalar field equation is identically satisfied. The vector field equation gives Along with the metric ansatz givenfield by and eq. ( scalar field: Case 1 Case 1 can bemassless obtained scalar from field, the Einstein action coupled to a massive vector and a Time is added ase an extra direction, modifying adding Case 2: Consider and Consider and Time is added as an extra direction, modifying Case 1: e ] = j { We will illustrate that both case 1 and case 2 can be obtained as solutions of Einstein This three-algebra has three two-dimensional sub-algebras given by We will illustrate the type III examples. The Killing vectors • • • , e i e gravity coupled to a (either massive or massless) gauge field with/without scalar fields. and [ and invariant one forms are given as JHEP03(2013)126 0 > t (10.7) (10.8) (10.9) A (10.11) (10.12) (10.14) (10.15) (10.16) (10.10) . = 0 = 0 = 0 = 0 0. With this, 2 2 2 2 > L L L L 2 1 . φ 4Λ  − 2 . , ) ) + 4Λ ) + 4Λ ) + 4Λ (10.13) F 2 1 2 1 2 1 2 1  0 and , 1 4 φ φ φ φ , . ) Λ = 0 = 0 2 t t > . 2 2 − − + 1  β β ) 2 1 2 1 2 t L 2 ), we consider the gauge field ansatz t 1 + t 1 dt . β L L β + β m r − − t , + Λ t + Λ 8( β t 1 − 2Λ 2 t β 10.2 (1 + + 8(1 + 2 2 e R β 2 β + 8( 1 t −
L − + 2Λ 2  − (1 L
2 A t
− 2 1 2 t 2 2 1 2 2 1 + − t 2 t β g − t β β p L L β  β 4 − 2 = = 2 = 2 = 2 1  – 34 – = 1 2 + 2 t L √ 2 2 2 1 − − + 2 + 4 4 2 L φ A L L A 2 2 t + 8( 2 L 2 2 t 2 1 Λ β dx L 2 = 2 =
2 1 L β t 2 L m 2 t 2 t t L 2 m t 0, and therefore A Z β m A m β A t < 2 = t 2 t 2 + 2 t A L A S L β A 2 L L 2 m t 2 1. 0. In order to satisfy Nernst’s law (i.e. that the horizon area vanishes A L 1. > > − 2 2 L < t β The solution is given by The Maxwell equation is identicallytwo satisfied. independent equations: Then the Einstein equations give only Along with the metric ansatzto given be by eq. ( So we have Λ at the horizon), we need implies Case 2 Case 2 can be obtained as a solution of the Einstein-Maxwell action, The equations can be solved to get The Einstein equations are given by 0 implies Λ • > 2 t β JHEP03(2013)126 , 5 10 , a,b 4 A ) and , (10.19) (10.17) (10.18) 3 , 4 A 10.17 , 2 , a 4 6 respectively. , , A 2 5 , dy 3 0. Since the static y , η > + = 2 i 2 η k ) . So we should not choose 4 rdy x with , − type given by eq. ( ,k in the 3d Bianchi’s classification 4 b dx ,k ( A 4 as the radial directions, the metric x ) and “Nernst law” constraint as in A 4 Bianchi η 2 x D ) + 8.2 i 2 3 . Since this involves the field theory time 2 ω ,k ds ( i 4 dr η and we require all r + ]) for the A η 2 4 ,k 44 =1 only runs over three values. i 4 X + dt – 35 – i r A 2 t = , which acts on the field theory spatial coordinates β  2 2 G e in 4d Bianchi attractors dy ds 2 − as the time and ,k r 4 1 = 1 2 A x 2 + ), but here ds 3), then the diagonal metric ansatz yields a stationary metric. 8.4 , rdx 2 in the notation of [ , − 7 , dt b 3 = 1  A i , ( − 5 i , ]. a 3 x = 1 A 2 , 1 ds , 3 are the invariant one-forms of : A as time if we wish to obtain a static metric. Furthermore, if we choose time as 1 , , i 4 2 4 , ω 3 and this generically induceswithout closed further time-like exploration. curves (CTC). So we discard this case one of the For example, if we choose ansatz becomes like; A The invariant one-forms arex manifestly dependent on We will investigate the metric ansatz for each A suitable static metric is as follows. For simplicity, we consider the diagonal metric A . The three dimensional subgroup Once we allow the stationary cases, we have to worry about the possible presence of closed time-like • 6 , 10 a,b 4 curves, as seen in [ NEC constraints as in eq. ( and radial direction, turns out(or, to be type IV, II, VI, VII also the corresponding nullsection energy 8 condition for eq. each ( class of static metrics. We choose our null vectors in investigating the we will see later that thisA form can be obtained only from the four-algebras ansatz is where property restricts the metric to the form time-coordinate explicitly. Thisshould be leaves static, us or with itin should only 4d, be but two stationary. it Here possibilities; is we straightforward restrict either to our the generalize attention to the metric ansatz static analysis metrics and to include takes the the stationary “obvious” cases. choice for the coordinate identification. Then, the metric So far we investigatedinvestigate examples 4d realizations of of the the four-algebras three-algebrascoordinates realized as well, in generically 4d this space-time.rather induces a interested We time-dependent in now metric. time-independent Inground metrics, this states paper, which of we may doped are be field dual theories. to Therefore (time-independent) we seek metrics which do not involve the 10.2 Bianchi four-algebras JHEP03(2013)126 2 a , as 3 4 A x (10.25) (10.26) (10.27) (10.28) (10.20) (10.21) (10.22) (10.23) (10.24) 0 as time in ), the null y 1 ≥ 3 η , x s , 2 , √ 0 , , ) y , y 3 η  x > s x η 2 2 η as time and ( η y s 4 √ r dy η 1 f + 2) 2 η y , x + 2) 2 η a + 2) r a + 1 2 ). η a s 3 a + y √ . ( s 1)( η 2 , 2 → 1)( 2 ) , − s ) 2 − 4 a 3 3 ) − , i f f y a = s rdy ( y a η x ( η r + 34 η − 4 η ≥ 2 =1 + 2( 2 3 i ) 4 , f + 2( 2 − x f dx 0 s x 2 ( P η ( . η f x 2 0. In this case, the spatial part of the as time since then the metric will be − > y 4 q η + 1 f ,G η = 4 y  > 2 , then the metric becomes stationary at = a
η + x 2 1 r 0. a y y 2 4 = ( 2 2 x 4 η η a . ) − ). , = 2 2 1 µ e y s 0, → a > η B 4) ( 22 N + → + 1) 2 , 1 – 36 – , f > ) f 2 2 , f 2 a 3 y , x y f + 1) η dr η y ( + η = r a η x ) forms type IV of Bianchi’s 3d classification, or 1) 2 η = 1 r r 0, r η ,G a  + η µν η i − , + √ 4 4 , f y 2 y 2 > y G a 2 η η a 0 ν . Therefore we need , r y 2 + 4( 3 η x dt η 0 ( 4( a N s > r + 2) x µ ar + 4( η η y as the radial direction instead, but keep using ≥ − + 12 16 a 2 4 3 x i N ]. (See appendix − x x 3 ) allows at least solutions at large ( η 2( aη f e η η → x 4 x 44 = , and − − − η 4 − − 2 f s or → µν = = 2 = 10.27 = = = , T f 1 2 = 1 by a coordinate change. We can then show for large positive 1 1 3 2 ν f 4 x s f f 33 44 11 x ds N G G G η µ N , y η : 2 , a 4 Therefore, eq. ( If we choose order to obtain a static metric,alone. the warp And factor is if notmost, we a function but do of not not the static. radial choose direction We time postpone as further investigation of these geometries. Let us set and for arbitrary Also, to satisfy Nernst’s law, we need where Then, for theenergy arbitrary condition null gives: vector which is static. Wemetric have coordinatized by ( in the notation of [ The non-zero components of the Einstein tensor are given by Again in this case,manifestly we time-dependent. should Then, not thethe choose radial best direction, choice yielding is a to metric select ansatz A • JHEP03(2013)126 , 2 to x 3 , x 1 ). x (10.34) (10.32) (10.33) (10.29) (10.30) (10.31) B as the radial 4 x (appendix . . 1 2 , 2 3 as the radial direction, dy , since it admits warp dy A y 4 4 , η ar x x 2 r x , + η − η 2 e 0 2 ) . − + < 0 2 y = rdy x < 0), the Null Energy Condition is η η , dx 4 34 − y r only, consistent with the algebraic is the way in which radial warping 0 2 η − x , r 2 − r as time. If we choose , η dx r η e a 4 case above. So again to obtain a static ( = η 1 2 1 x A case. Choosing 2 √ + x , η − 22 , a 4 1 ,G 2 , 4 A + ) = dr y A 2 r η . η = (1 µν dr y – 37 – y r η G as the radial direction, then the warp factors are + ,G µ η + 4 η ν 4 r r 2 y 4 x N η η η + x N η , but any choice always admits at most a stationary − x 4 dt 4 r 3 µ 2 η η x (3 br x 4 η 2 N dt x , r − η 2 − − 2 e x − − = = = e , 1 − = 11 33 44 x 2 G G G = 2 ds ) form Bianchi’s 3d classification type II, or ds r , y , x . All give equivalently good static metric ansatzes. For example, setting 3 : : : 5 4 3 x , , , a,b 4 4 4 structure. (If we donot not functions choose of radius alone).or There are threebe equally time, good we choice have for a time: static metric metric, but not a static one. A In this case, thefactors simplest in choice the for metric a which radial are direction functions is of A This has a very similar structurethe to warp the factors aschoices a for time function from of r are the only difference. Again, there are three It follows that forviolated; a null vector Therefore we cannot obtain this space-time from physical matter systems. Let us first check whether the Null Energy ConditionThe is non-zero satisfied. components of the Einstein tensor are given by The only difference betweenappears this in case various components. and nated In by these ( cases, the spatial partNote of that the in metric this coordi- casespecial the care horizon must volume is be independent taken of for the analysis radial of coordinate. Nernst’s So law or “physical reasonableness”. This has a very similarmetric, structure the to simplest the approachdirection, is then to the choose metric ends up with following form: A • • • JHEP03(2013)126 ) = B µ N (10.43) (10.39) (10.40) (10.41) (10.42) (10.35) (10.36) (10.37) (10.38) . . 0), 0 ) , 2 1 ≥ ) , (appendix 0 , 5 rdx , a 3 + 1) A b , . , and on the other = (1 2 2 2 +sin + µ r R η a dy dy N b , rdy × br br 2 2 1)( 2 . + − − 0), − , e e a (cos 0 b y . , as the radial direction. Then, + 1 ( + + AdS + . η r 0 r 2 2 a 4 η 1 − 1 η + ba x √ 0. In order to satisfy the Nernst’s 2 + dx dx ≤ . , ) r = , b > 2 2 ar b 1 → a 2 0 − + 22 − rdy e ≤ tt ≤ = (1 e ≥ = a g 0 + = 1, the metric reduces to Bianchi’s type µ sin + ). It is pretty straightforward to calculate 44 2 ⇒ a 2 N − dr + 1) = 0, we have 0 ,G dr r , b , as time and b where r η b 2 10.35 0 at the horizon, we have a parameter regime rdx 1 1 η 1 b → – 38 – + + r . x ,G r = + ≤ ≤ 4 a η + ) → 2 b 2 as time, we obtain respectively, (cos a a a r → ∞ + dt ba η x 1)( yy 2 a dt + 1 r ( ≤ η g ar r AdS x r b ( 2 2 + − − 0 η e − − 2 br a + or 2 e e a ( ) gives 2 − 1 − − b − − e x ). 0 and = = + = = 2 , B 2 2 10.39 → ) are in type VI of Bianchi’s classification, or 0 dr 11 33 r ds ds r 1. For the special case G G , 0, ( ≥ xx η , = 1, we have y g 2 + b > , b 2 6= 0 x r = dt η − a (appendix a b ar 2 3 r , + − 3 η 1), respectively, we obtain e 2 , A a 0 = , : 2 6 0 − , , a,b 4 a ds A Again, the best choicewe is obtain to the select static metric ansatz where the Null Energy Condition is satisfied, Interestingly, on one boundary boundary Then, with Assuming that The black brane horizon islaw, at we need Therefore, for the choice(1 of null vectors Let’s consider the metric ansatzthe eq. Einstein ( tensor, and we obtain it in diagonal form as coordinates ( for generic V, or and Note that these are just generalized Lifshitz geometries. In all cases, the three spatial Similarly setting either • JHEP03(2013)126 , or b (10.44) (10.46) (10.47) (10.48) (10.49) (10.50) . b (10.45) 0 . 6 , ≥ a,b 4 2 A ) are given by: 3 i s f ( 4 , and f 5 , in Bianchi’s classification, + a,b 4 ), the allowed cases for b 3 A , . , has two unit eigenvalues and . s , 2 2 2 y y y 2 2 2 ab η η , s ) η a 4 3 r 3 n , 10.17 f η f ], of trying to classify a wide − + − A y ( y 1 b . 0. Here, the 2 2 2 x x x η η + η η η x x ≥ = 2 > η η ) 4 − − 3 34 2 y 2 f √ ) + ) s η ) 2 b b ( ), we have the NEC y ,C f 2 y η 0, 4 f 3 η s + 2 + 2 + √ ) + 1) > = 1 + b a a , , and furthermore decomposing them into a sym- x ). x 2 2 34 η , we can see that x ) )( )( η dc − , 2 c η 1 b b B as the generator of time-translations, and the spatial s − C a a s ,C √ 0, 1 4) ( ( − − )( ) form Bianchi’s 3d classification type VII , 1 e b , 1 abc abd b – 39 – r   a a r f 2 > − ) span a manifold of type VII 1 (2 ( ( η , , r s r + y y y ≡ y = √ , + 2 η = η η η y , , ab x x x 3 c 24 ab a = 1 , x 2 n η η η C µν ) x i (appendix i ≡ G s with b , C by ( 11 ν = 2 = 2 = 2( = 2 ] ab 3 0 ( = N s dc 1 2 3 4 =1 C 0). These are precisely the characteristics of Bianchi type VII i 3 44 µ f f f f , C 2 24 ≥ 0 P C i N b, f ]). We have chosen q = , and 43 = ( 2 s c µν = ( a , T we discuss the explicit parameter ranges where these conditions are 1 µ ν s C N N : By completely analogous reasoning to that appearing above, if we as- µ 12 , N 4 forms a real three-algebra where antisymmetric structure constants are given by A 4 - in the notation of [ , e 3 7 7 , , , e 4 3 b 2 e dependent warping factors in the metric. A sume a diagonal metricradial ansatz coordinates, and we can make obtain the at “obvious” most choice(s) stationary metrics, for but the not time static and metrics. In appendix satisfied. Other choices of coordinates yield either non-static metrics, or field-theory position- Therefore we need, for arbitrary A It is pretty straightforward tonull calculate vector, the Einstein tensor. Then, for the arbitrary The three spatial coordinate ( In summary, for the diagonal and static metric ansatz ( The fact that the three coordinates ( • 11 subset Defining the two-index constants metric and anti-symmetric partone as zero eigenvalue, and variety of less symmetric extremal near-horizon geometriestions. for black We branes, have in demonstrated several direc- thathyperscaling violation one in can the simply dual field modify theory. these We have geometries given to examples of incorporate suchcan hyperscaling be seen as follows (see [ which the NEC and Nernst law constraints can be11 met are, Discussion In this paper, we have extended the program initiated in [ JHEP03(2013)126 ]. In 51 , that can be 50 , 8 12 ] to classify 5d extremal 45 , 44 ]. It should be clear in each case 1 ] that some of these horizons can be 1 ]. It would be interesting to see if similar 55 , 54 – 40 – ]. The issue has also been addressed in isotropic 53 , 52 ]). Classifying 4d stationary space-times governed by the 58 ]. Such a study could include analysing both whether the solutions 1 ] (see also [ 1 ] for the study of interesting black brane solutions in gauged supergravity. 57 , 56 A simple, possibly interesting, extension of this work is to relax the condition of the As always, one can wonder to what extent the rich set of possible phases found here Another important question is to study the stability of the solutions found both in In addition, while we have given evidence in [ A number of issues remain to be clarified. In many cases, these near-horizon geometries In the direction of finding a more complete classification, we discussed the possible See also [ 12 geometry being static. Intime section naturally 10, we induce have metrics seenmust which that be are the careful not real with four-algebras static,as in such but illustrated 4d metrics, can in space- since [ be they stationary. can However, easily one contain closed time-like curves, to explore embeddings ofderived these from solutions consistent truncation into of gaugedbe IIA supergravity interesting and theories to IIB find supergravity, forto proposals instance. some for of It phases the would of also more matter exotic in symmetry real groups systems discussed which here. could give rise are perturbatively stable, i.e.,whether whether they they are stable have with modesof respect relevant which deformations to grow with changes with respect in time, to boundary and RG conditions, flow. also i.e., themanifest presence themselves in UV complete models derived from string theory. It would be useful in anything close todescribed. a comprehensive A careful way study for ofCFTs the which (perhaps classes full with of classes additional geometries of are currents truly anisotropic activated infrared on metrics phases the we’ve of boundary) doped wouldthis be paper worthwhile. and in [ singularity have been discussed inspace-times [ with hyperscaling violation inphysics [ arises for the anisotropic space-times described here. glued into asymptotically AdS space-time by suitable RG flows, this has not been discussed static near-horizon space-times with symmetries governed byseen the that four-algebras, some and of we have the types are forbidden due tomanifest the NEC, infrared but singularities others arethe similar likely Lifshitz to attainable. case, that various of physical the smoothings or Lifshitz more space-time subtle [ “resolutions” of the metric application of the largernear-horizon algebraic geometries structures in uncovered terms in ofwe [ found real that four-algebras several with of these preferredto possibilities 3d matter will subgroups. remain theories unrealized While (which in sensible satisfysimple gravity matter the coupled sectors Null coupled to Energy gravity, andin Condition), likely others strongly-coupled arise as can quantum duals be to field realized suitable infrared theory. with phases Similarly, we have classified the 4d extremal the radial direction included inanalytical the “striped” algebra. metrics We starting have from also thethat shown horizons in that of providing one examples, [ can we have easilysets barely obtain of scratched the solutions surface to of the what Einstein are equations likely or very appropriate rich low-energy limits of string theory. violation both in 4d and 5d bulk space-time metrics which realize the Bianchi types, with JHEP03(2013)126 ], this 61 – 59 – 41 – 13 The most ambitious possible extension would be to try and classify all extremal in- We thank Shiraz Minwalla, Mukund Rangamani and especially Sayantani Bhattacharyya for a discus- 13 Weizmann Institute of Sciencessupport where from part a J. of C.Govt. this Bose of work Fellowship given India. was by H.W done. the is Department supported of S.P.T. Science by acknowledges and a Technology, Stanfordsion Graduate of this Fellowship. issue. the Government of India, andin thank the string people theory. of India S.P.T.String for thanks generously Theory”, the supporting held ICTS research in andcontract organisers Bangalore. DE-AC02-76SF00515 of and the S.K. byPHY-0756174. workshop was the on P.N supported National is “Aspects in grateful Science of part for Foundation the by under support the grant of US no. Feinberg DOE Postdoctoral under Fellowship at the S.K. would like tothe thank the theory organizers groups of atthe the Imperial “Topology, “Holographic Entanglement, Way” College and conference and Strongat at Correlations Cambridge the Nordita, University in University, of and Condensed Illinoispleted. Matter” the for conference organizers providing N.K. stimulating of would environmentsheld like as in this to Puri, work thank India was the for com- organizers their hospitality. of N.K., the N.S. “Advanced and String S.P.T. School, acknowledge funding 2012” from and V. Tripathy for very helpfulJ. discussions, Maldacena and for E. fun Fradkin, conversations J.to after Gauntlett, thank S. various the Kivelson talks CERN and about COFUNDlow this at fellowship, material. CERN. much N.I. of N.I. would this and“Gauge/Gravity like work S.P.T. Duality” would was held like done to at while thank he YITP, the Kyoto, was organizers a for of fel- providing a long-term a workshop stimulating atmosphere. Acknowledgments We are grateful toganayagam, N. S. Bao, Minwalla, G. S. Mandal, Bhattacharyya, S. K. Nampuri, Damle, M. S. Rangamani, A. Harrison, Sen, A. T. Karch, Takayanagi R. Lo- mogeneous phases is a challengingsolving question algebraic since equations the now analysis and cannotential equations be instead in reduced requires their to us full merely glory. toan The inhomogeneous confront striped phase coupled phase and partial discussed our differ- in discussionthis section in direction. 6 that is Clearly section an can though, example be much of viewed more as effort a is small step needed in to make progress on this issue. be worth studying further. homogeneous, anisotropic black brane horizons.shown In in light holographic of the transportproblem interesting could in scaling be the features interestingas simplest for a “applied inhomogeneous question holography” in geometries in general [ addition relativity and to string its theory. intrinsic Needless interest to say, finding all such inho- duals. In fieldcurrents theory to systems, flow. if Thisgravity external would dual naturally sources of correspond the perturbature to system. the or a system, A an stationary concrete electric they metricdirections. example potential can in which is Gravity the induce duals is that putative of time of extremal independent a geometries and subject fluid varying to subjected slowly such to in potential a the gradients would spatial temper- four-algebras which do not contain any such pathological features, could lead to interesting JHEP03(2013)126 are µ . (A.4) (A.2) (A.3) (A.1) σ G , in keeping dependence: 2 ]; the factors t 1 dt ) r ( tt . Metrics which are , annihilated by the g i G ω = } ) 2 2 a ) t A s; in the most trivial case, σ 2 a ω m 1 4 j to be identified with one of the 2 ω ) r t + σ ⊗ . 2 a ( i F α ω − 1 4 σ ij ( 2 ], though we will allow for slightly more ) ∧ λ 1 r a ν X σ + σ 2 − µ να + ( dt , described by the Lagrangian: ) C 2 – 42 – a ) r 2 1 i ( + Λ A σ tt ( = R g { µ 3 − =1 g i X , such that the other 3 one-forms form a sub-algebra. − dσ = 4 with constant coefficients, and with trivial = 2 √ i 2 dx x = 1. We further demand that ( ω ds 5 ds 4 d = λ Z dr 4 = -invariant. Here we slightly modify the notation of [ λ G S = r which generate the 4-dimensional isometry group ). σ i will be homogeneous spaces as in [ A.1 ], the basic objects of interest are invariant one-forms } 1 3 ). Therefore, the orthnomal non-commuting basis satisfies the following commu- , x there, which incorporate the scaling of the coordinates under radial translations, 2 . In this formalism, we will in general lose the advantage that the one forms r i i A.2 , x ω β 1 We are interested in scale-invariant (or conformally scale-invariant, in the case of In order to exploit the underlying symmetries of the homogeneous geometries, we will As in [ In this section, we discuss some basic facts about the Einstein equations for the theory − = e i r, x exact one-forms In this appendix A,with we ( set tation relation: exact, and hence couldnon-commuting be basis. integrated into coordinates; instead they form anhyperscaling-violating orthonormal metrics) near-horizon geometries.ing In to order make for sense, the we generalised require scal- that the radial coordinate metric tensor takes the form: The vielbein elements areσ simple linear combinations of the will automatically be of are built into the forms now, and the radialwrite scaling down symmetry the is Einstein-Maxwell treated equations as in part an of orthnomal (vielbein) basis, in which the Killing vectors e written purely in terms of the A.1 Ricci curvature We will be interested in{ static solutions of Einsteingeneral gravity. possibilities The intertwining spatial the sliceslater spanned radial sections by of and the “field paper. theory” spatial coordinates in the with matter ( In many sections of thecoupled paper, to we a will reasonable want matter to sector,can exhibit to arise explicit show in that solutions physically the of sensible near-horizon Einstein systems.of geometries gravity a we We will propose set usually of choose massive our Abelian matter sector gauge to fields consist A Einstein gravity coupled to massive vector fields JHEP03(2013)126 (A.7) (A.8) (A.5) (A.6) ]. (A.10) . Then i 47 σ , ) k r 46 ( σ i ∧ A i j σ µτ t rt σ P C i jk )Γ ) + C i t τ αβ j rm mnl σ Γ A .  ) C 1 2 ) r ) are given by the data of − ikl ( r t  ( + µ } αβ j τ βα A t k rn C A , r σ C (Γ 3 ) − (A.9) , , the generalization to multiple ∧ ) = j − a ri 2 completely in terms of the struc- ) + r r , τ r σα ( C A σ = 0 1 ( ] j σ τβ ] for detail. The Riemann curvature C A ) A t m 0 Γ rt α µαν r 5 43 µσ ( A C ∈ { R mnl ? τ µα g )  A r 2 Γ ). With this set-up the connection form ( = τβ + ( r ikl t g m ( 0 −  ) ) + α, β A 1 2 tt µν r + . The Maxwell equations for a massive vector k σ τα ( mn j R − rm /g 0 i ν Γ ) C τ ) + σβ C – 43 – A σ ) r and = r ) ( C τ µβ ( j r ri ∧ 0 t 0 } tt ( F + ( µσ C j g 3 µ A 5 ) g , 1 2 + Γ σ A r 2 ( − , τα + [ mkl d ? j µν  g in this basis. See [ 1 i F ( ) + A σ µα,β = σ 1 2 r 1 2 jpq µν Γ (  ∧ t ∈ { tr η − = 0 m ) + j r kl − C i r A σ ( = C F ] 0 i i j pq ri A for σ µβ,α µ αβ ( The Maxwell’s equations are: C C Γ ) ) ) = ( . = 0 ; r r r i αβ ( ( ( = Γ i j C r µν m A A C A ) = 0 1 4 σ µαβ 2 r ( − R t ) + m r 1 2 A ( 0 i A = [ F are no longer given by the Christoffel symbols, but instead are given by the structure Assume that the vector potential takes the form The constants is equal to the Minkowski µ νβ µν A.2.1 Magnetic field In this case, field are given in differential form as: Since the metric tensorflat is case. now It Minkowskian, is the therefore Levi-Civita straight-forward tensor to obtain reduces the to following the Maxwell’s usual equations. the curvature is given by with components given by Here, we present thethere Maxwell are equations no for cross-couplingsvectors a between is single trivial. the massive vector Abelian fields gauge field. Since ture constants, which only dependalong on the r. time direction, Intherefore the the case algebraic. entire where space-time We we is will alsoalgebraic homogeneous equations, have see and scaling as that the symmetry in Ricci in the curvature this story is formalism, of the generalizedA.2 attractors Einstein discussed equations in reduce e.g. Maxwell’s to [ equations tensor is given in this basis by the connection forms via: Hence we can compute the Ricci curvature the four-algebra; Γ constants via: g JHEP03(2013)126 1, − = (A.12) (A.13) 1 32 C ] − = 43 ,  1 23 ρ 42 ) (A.11) ]. C A r ( ρ t 43 A , A 2 1. 2 4 42 2 m − m + = = = 0, Bianchi VII reduces to a i αβ ri a ] and [ a µν C F Bianchi I ) 44 Bianchi II Bianchi V T Bianchi IV Bianchi VI Bianchi IX Bianchi III ]. Bianchi VII ; or, in the case of scaling solutions, t αβ Bianchi VIII rt r a F ] with C 44 X ) 1 4 r 44 (  in [ t 6 µν in [ A , η 3 5 , 1 2 . By setting A a 3 + Λ) = ) + a Algebra in the notation of [ r A − R ( – 44 – . = 0 t ( ν ] A A µν µν 2 44 32 µ η η C A 1 2 + ( 2 0 − ) − m t 1) rt = 1 4 µν C < ) 0) 2 + R 23 1 | r ( λ a C ν A t | 1 2 3 8 9 1 , , , , , = 0 limit is called F a > 3 3 3 3 3 A A ⊕ . Classification of Real 3 dimensional Lie algebras. ( < a 3 A A A A A µλ 2 There is only one component of Maxwell’s equation: 7 , F (0 ) + A . a 3 1 2 r 5 ( A , 0 t a 3 = 1 and = , which we use many times in this paper, is the special limit of Bianchi . The A Table 1 A 0 0 ( ) = 0 2 31 ] is the special limit of the r µν ( T C i 44 above, we provide a dictionary relating two different common nomenclatures − A 1 Algebra in the notation of [ in [ = 4 , 2 3 13 Bianchi VII VII. More precisely, BianchiC VII has nonzero structureBianchi constant VII A fication Therefore we see that the Einstein equations • • B Rosetta stone relating different nomenclatures forIn the the Bianchi table classi- for the classification of 3d real Lie algebras, as given in [ are reduced to a set ofa ordinary differential set equations of in algebraic equations. where index contraction is done using Similar to before, intential a are scaling constants, solution, reducing Maxwell’s we equations expect to thatA.3 a the set components of of algebraic Einstein’s equations. the equations The vector stress po- energy tensor of a massive gauge field is given by: A.2.2 Electric field In this case, JHEP03(2013)126 . (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) → ∞ (C.10) r . 4 b + 3 b ) parameter space. We 2 , , b y −  η b 2 ) . 2 p 2 b ) , , . , , . 4 3 0. So the horizon is at 0 0 0 0 0 0 f + 2 − ( 2 b ≥ ≥ ≥ ≥ ≥ ≥ b ≥ a > 2 2 2 2 2 2 2 y y y y y y 4) gives 4 , η η η η η η , f − 0 2 = 1 by performing a coordinate re- 0. 2 b − − − − , f > 4 1 1 + 1 1 + x η y b > ) give respectively = 1 − − − − 4 + (2 + 2 1 + 2 i ) + 1 ) ) + 1 ) 6 b b b b , p C.7 , ≤ 4 0 ( – 45 – y + 4) , η + 2 + 2 A ) + 1) η ) + 1) > , 2 b b 0 a a b i 2 ≤ 4 , f )( )( )(1 + 2 )(1 + 2 − − ) and ( 4 b b b b − a b b > 0, ). We set (1 ( = 1 − − − − b b b C.6 + i > , we can always make a a 3 (2 (1 (1 (2 ( ( r r b y y y y y y η 10.49 2 η η η η η η 2 + 2 0 ( 2 2 2 2 2 2  − )–( ↔ − ≥ 1 2 b r i f p ≤ ) gives 2 10.46 y = 1. − η C.5 2 a b 2 = 1. Then the above 3 conditions become − a b = 1 and x . The darkest parameter region is the allowed region in the ( η 1 + 2 given by eq. ( i Let’s set f the first condition ( The second the third conditions ( In the regime where Furthermore, by flipping In order to satisfy the Nernst’s law, we require parameterization. Then, with C Null energy conditionWe for need 4d for Figure 1 have set JHEP03(2013)126 D JHEP Ann. (C.11) (C.13) (C.14) , , . A 42  . The limit 13 Phys. Rev. 1 Class. Quant. , , − b J. Phys. Adv. High Energy (2010) 027 At the frontier of 1) , , ]. . − 10 , in b  ) in figure ]. 2 ) 12( 2 SPIRE b JHEP 4 − IN C.12 , SPIRE ][ − , ) (C.12) )–( IN b 2 1 [ ]. g ]. ]. − C.9 + Holography of charged dilaton black ]. 1 g SPIRE + 9) + 41 ( SPIRE + 1) b IN SPIRE 1 4 b IN [ ( IN 4 + (2 + 2 + ]. SPIRE b ][ ≤ 2 ][ 4 IN p b y ]. ]. 4 ][ − η + arXiv:1108.1197 − 2 – 46 – SPIRE ≤ b 3 arXiv:0812.0530 (2006) 17 IN Doping a Mott insulator: physics of high-temperature ) b , 2 SPIRE SPIRE ][ Gravity duals of Lifshitz-like fixed points g + 4 78 IN IN (10 b gives additionally − ][ ][ b 2 (2012) 9 [ + 9) + 41 2 1 8 ) g The condensed matter physics of QCD − b 3 3 ( . p ]. 2 f arXiv:0911.3586 4 +  1 4 ( ]. arXiv:0909.0518 [ − 2 2 [  b ≥ arXiv:0903.3246 4 AdS [ 1 2 4 SPIRE f + 1) − SPIRE 2 IN ≥ b Holography of dyonic dilaton black branes 3 f IN b arXiv:0808.1725 ][ Rev. Mod. Phys. y (2 ][ [ η , (2010) 078 Bianchi attractors: a classification of extremal black brane geometries − (10 arXiv:0904.1975 arXiv:1201.4861 , volume 3, M. Shifman ed., World Scientific, Singapore (2000), pg. 2061 Lectures on holographic methods for condensed matter physics This article is distributed under the terms of the Creative Commons Holographic duality with a view toward many-body physics [ [ b r Lectures on holographic superfluidity and superconductivity 08 8 (2010) 723105 2 What can gauge-gravity duality teach us about condensed matter physics? Non-relativistic holography p √ (2009) 224002 = = JHEP 26 2010 1 2 , g g (2008) 106005 (2012) 193 = 1 corresponds to arXiv:1007.2490 hep-ph/0011333 b holes [ particle physics [ 78 Phys. Rev. Condensed Matter Phys. superconductivity Grav. (2009) 343001 07 We plot the allowed parameter ranges satisfying ( M. Taylor, K. Rajagopal and F. Wilczek, S. Kachru, X. Liu and M. Mulligan, S. Sachdev, P.A. Lee, N. Nagaosa and X.-G. Wen, S.A. Hartnoll, C.P. Herzog, J. McGreevy, N. Iizuka et al., K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, K. Goldstein et al., = [9] [7] [8] [5] [6] [2] [3] [4] [1] y [10] [11] References η Open Access. Attribution License which permits anyprovided use, the distribution original and author(s) reproduction and in source any are medium, credited. where Finally the condition 4 JHEP03(2013)126 , , , D 85 ]. 02 ]. D 86 Phys. 108 ]. , (2012) 065 SPIRE JHEP SPIRE 05 IN , (2011) 091 IN SPIRE ][ Phys. Rev. ][ IN , 12 ][ Phys. Rev. JHEP , , (2012) 165 JHEP Phys. Rev. Lett. (2010) 151 (2007) 141602 Effective holographic , ]. , ]. 06 11 99 ]. Aspects of holography for ]. SPIRE arXiv:1201.1905 SPIRE [ arXiv:1105.1162 ]. IN Cold nuclear matter in holographic JHEP JHEP IN [ , ]. SPIRE arXiv:1112.0573 ][ , ][ [ IN SPIRE Holographic Fermi and non-Fermi liquids IN ][ SPIRE ]. ][ IN SPIRE ][ IN (2012) 041 Phys. Rev. Lett. Spatially modulated instabilities of magnetic (2012) 094 ][ , SPIRE ]. ]. ]. 06 IN Holographic Fermi surfaces and entanglement 01 Gravity dual of spatially modulated phase – 47 – ][ ]. (2012) 035121 Hidden Fermi surfaces in compressible states of SPIRE SPIRE SPIRE arXiv:1011.4144 JHEP arXiv:1109.0471 [ IN IN IN , JHEP [ SPIRE , B 85 ][ ][ ][ arXiv:1007.3737 Baryon number-induced Chern-Simons couplings of vector IN [ arXiv:1111.1023 Generalized holographic quantum criticality at finite density Holographic helical superconductors Helical superconducting black holes Black holes dual to helical current phases ][ [ ]. ]. ]. ]. ]. arXiv:0708.1322 [ Holographic end-point of spatially modulated phase transition Spatially modulated phase in holographic quark-gluon plasma arXiv:0911.0679 [ (2012) 061 SPIRE SPIRE SPIRE SPIRE SPIRE arXiv:1107.2116 Phys. Rev. (2011) 061601 [ IN IN IN IN IN , 01 (2012) 125 ][ ][ ][ ][ ][ (2010) 126001 (2008) 053 106 Holographic entanglement entropy and Fermi surfaces 01 arXiv:1203.0533 arXiv:1204.1734 arXiv:1202.5935 Domain wall holography for finite temperature scaling solutions Hyperscaling violation from supergravity [ [ [ 01 JHEP On Lifshitz scaling and hyperscaling violation in string theory , arXiv:1006.2124 (2010) 044018 D 82 [ (2011) 036 JHEP , 12 JHEP D 81 , arXiv:1109.3866 arXiv:0704.1604 arXiv:1205.0242 arXiv:1112.2702 arXiv:1005.4690 [ (2012) 211601 (2012) 064010 Phys. Rev. Lett. black branes Rev. Phys. Rev. and axial-vector mesons in[ holographic QCD QCD (2011) 013 [ (2012) 106006 theories with hyperscaling violation [ with transitions in dilaton gravity entropy gauge-gravity duality theories for low-temperature condensed matter[ systems JHEP A. Donos and J.P. Gauntlett, A. Donos and J.P. Gauntlett, H. Ooguri and C.-S. Park, A. Donos, J.P. Gauntlett and C. Pantelidou, A. Donos and J.P. Gauntlett, S. Nakamura, H. Ooguri and C.-S. Park, H. Ooguri and C.-S. Park, S.K. Domokos and J.A. Harvey, M. Rozali, H.-H. Shieh, M. Van Raamsdonk and J. Wu, E. Perlmutter, K. Narayan, E. Shaghoulian, X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, E. Perlmutter, N. Ogawa, T. Takayanagi and T. Ugajin, L. Huijse, S. Sachdev and B. Swingle, B. Gouteraux and E. Kiritsis, N. Iizuka, N. Kundu, P. Narayan and S.P. Trivedi, C. Charmousis, B. Gouteraux, B. Kim, E. Kiritsis and R. Meyer, [29] [30] [26] [27] [28] [24] [25] [22] [23] [20] [21] [17] [18] [19] [15] [16] [13] [14] [12] JHEP03(2013)126 , , (in (2011) (2012) 03 , 01 (2012) 047 Phys. Rev. ]. , . 11 JHEP JHEP , SPIRE ]. , IN JHEP ][ , arXiv:1211.5600 ]. (2011) 140 , SPIRE , Princeton Series in Holography for IN 08 ]. ]. ][ SPIRE ]. , Pergamon Press, Oxford U.K. Towards a holographic IN [ SPIRE JHEP SPIRE , IN Black hole instability induced by a IN [ SPIRE Magnetic field induced lattice ground . ][ Competing orders in M-theory: IN arXiv:1210.6669 Baryonic popcorn ][ ]. ]. ]. Holographic stripes pg. 299 arXiv:1106.4551 (2012) [ SPIRE ]. [ SPIRE SPIRE A striped holographic superconductor Conductivity of strongly coupled striped IN Generalized attractor points in gauged supergravity IN IN arXiv:1212.0871 – 48 – Fluctuations and instabilities of a holographic metal ][ [ ][ , (1898) 267. SPIRE JHEP arXiv:1209.5915 IN arXiv:1206.3887 , 11 [ (2011) 94 [ ]. ]. arXiv:1104.2884 Generalized attractors in five-dimensional gauged The classical theory of fields [ Holographic striped phases http://faculty.physics.tamu.edu/pope/ihplec.pdf Homogeneous relativistic cosmologies , Subalgebras of real three-dimensional and four-dimensional Lie ]. ]. (1977) 1449 B 706 SPIRE SPIRE Pathologies in asymptotically Lifshitz spacetimes 18 IN IN (2012) 003 ][ ][ SPIRE SPIRE arXiv:1211.1381 , submitted to arXiv:1010.1775 On the classification of the real four-dimensional Lie algebras [ IN IN [ 09 ][ ][ , Springer, Germany (1999), (2011) 046003 arXiv:1208.2964 Phys. Lett. , , JHEP , Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti D 84 (2013) 007 Kaluza-Klein theory J. Math. Phys. ]. Soc. Ital. Sci. Mem. Mat. , 02 (2011) 064009 arXiv:1110.2320 arXiv:1011.3502 [ [ SPIRE arXiv:1201.1331 IN arXiv:1106.2004 Einstein-Maxwell-dilaton theories from generalized dimensional089 reduction 039 Phys. Rev. supergravity (1980) [ISBN:0-08-025072-6]. algebras On Einstein’s path Italian), Physics, Princeton University Press, Princeton (1975) [ISBN:0-691-0153-0]. states from holography magnetic field superfluids, stripes and metamagnetism [ realization of the Quarkyonic phase JHEP [ D 83 [ superconductor B. Gouteraux, J. Smolic, M. Smolic, K. Skenderis and M. Taylor, K. Copsey and R. Mann, S. Kachru, R. Kallosh and M. Shmakova, K. Inbasekar and P.K. Tripathy, C. Pope, J. Patera and P. Winternitz, M.A.H. MacCallum, L. Bianchi, M.P. Ryan and L.C. Shepley, L.D. Landau and E.M. Lifshitz, Y.-Y. Bu, J. Erdmenger, J.P. Shock and M. Strydom, M. Ammon, J. Erdmenger, P. Kerner and M. Strydom, V. Kaplunovsky, D. Melnikov and J. Sonnenschein, J. de Boer, B.D. Chowdhury, M.P. Heller and J. Jankowski, N. Jokela, M. Jarvinen and M. Lippert, M. Rozali, D. Smyth, E. Sorkin and J.B. Stang, A. Donos, J.P. Gauntlett, J. Sonner and B. Withers, A. Donos and J.P. Gauntlett, J.A. Hutasoit, G. Siopsis and J. Therrien, R. Flauger, E. Pajer and S. Papanikolaou, [49] [50] [46] [47] [48] [44] [45] [41] [42] [43] [39] [40] [37] [38] [34] [35] [36] [32] [33] [31] JHEP03(2013)126 ]. , JHEP , SPIRE IN ]. ][ ]. , SPIRE ]. IN Nernst branes in ]. ][ SPIRE IN (2012) 046008 SPIRE arXiv:1202.6635 ][ IN SPIRE , ][ Extremal black brane solutions IN arXiv:1211.0832 ]. D 85 [ ][ SPIRE IN arXiv:1207.0171 [ ][ Entangled dilaton dyons (2013) 103 ]. Phys. Rev. On the IR completion of geometries with The benefits of stress: resolution of the , arXiv:1208.4102 02 [ arXiv:1208.1752 SPIRE [ ]. arXiv:1108.0296 IN Further evidence for lattice-induced scaling Optical conductivity with holographic lattices [ – 49 – JHEP ][ , Resolving Lifshitz horizons (2012) 106008 SPIRE IN arXiv:1110.3840 ][ (2013) 147 (2012) 124046 [ D 86 (2011) 090 h ¨dlSh¨dnesaeieandThe stringy G¨odel-Schr¨odingerspacetime chronology 02 Universal linear in temperature resistivity from black hole ]. Lifshitz singularities 11 ]. D 86 JHEP SPIRE arXiv:1204.0519 (2012) 032 , SPIRE [ IN JHEP Phys. Rev. IN , [ , ][ 01 arXiv:1209.1098 Phys. Rev. [ , JHEP (2012) 168 , ]. 07 (2012) 102 SPIRE arXiv:1111.1243 IN superradiance 11 JHEP gauged supergravity in five-dimensional gauged supergravity protection hyperscaling violation arXiv:1208.2008 [ [ Lifshitz singularity G.T. Horowitz, J.E. Santos and D. Tong, G.T. Horowitz, J.E. Santos and D. Tong, A. Donos and S.A. Hartnoll, S. Barisch-Dick, G. Lopes Cardoso, M. Haack and S. Nampuri, C.M. Brown and O. DeWolfe, J. Bhattacharya, S. Cremonini and A. Sinkovics, N. Kundu, P. Narayan, N. Sircar and S.P. Trivedi, S. Barisch, G. Lopes Cardoso, M. Haack, S. Nampuri and N.A. Obers, S. Harrison, S. Kachru and H. Wang, N. Bao, X. Dong, S. Harrison and E. Silverstein, G.T. Horowitz and B. Way, [59] [60] [61] [57] [58] [54] [55] [56] [52] [53] [51]