JHEP02(2017)012 Springer February 3, 2017 January 13, 2017 December 8, 2016 : : : September 14, 2016 : 5 S Revised 10.1007/JHEP02(2017)012 × Published Accepted Received 5 doi: Published for SISSA by , a gauge theory description of a small 3 S × 1 [email protected] d,a R , = 4 SYM on N is proposed. The change of the number of dynamical degrees of . and Jonathan Maltz ) is derived. A heuristic gauge theory argument supporting this 5 3 7 S T × N a,b,c 5 , 1608.03276 10 The Authors. G c Black Holes in String Theory, Gauge-gravity correspondence, M(atrix) The- (

Based on 4d / , 1 [email protected] ∼ E E-mail: Yukawa Institute for Theoretical Physics,Kitashirakawa Kyoto Oiwakecho, University, Sakyo-ku, Kyoto, 606-8502The Japan Hakubi Center forYoshida Advanced Ushinomiyacho, Research, Kyoto Sakyo-ku, University, Kyoto, 606-8501Berkeley Japan Center for Theoretical Physics,University Department of of California Physics, at Berkeley, 366 LeConte Hall, Berkeley, CA, 94720 U.S.A. Stanford Institute for Theoretical Physics,382 Department Via of Pueblo, Physics, Stanford, Stanford CA, University, 94305 U.S.A. b c d a Open Access Article funded by SCOAP ories ArXiv ePrint: reproduces the relation forconsequences the of our eleven-dimensional proposal Schwarzschild isseen black that from hole. the the small gauge One and theory of large point black the of holes view. areKeywords: very similar when freedom associated with thein emission this of description. the Bylong scalar analyzing strings fields’ the eigenvalues at microcanonical plays low ensemble,duality a for the energy crucial a Hagedorn is role large behavior obtained. blackhole of hole, the Modulo energy an ofassumption assumption the small is based ten-dimensional on also Schwarzschild black the given. AdS/CFT The same argument applied to the ABJM theory correctly Abstract: black hole in AdS Masanori Hanada A proposal of thesmall gauge Schwarzschild theory black description hole of in the AdS JHEP02(2017)012 N , M 10 X (1.1) limit, G 12 , N  direction, ) and 5 N , where T 4 + (fermion) 2 M X 1 2 − 2 ] 0 10 5 13 M 1) 1) 6 ]): 3 ,X = 4 super Yang-Mills (SYM) theory  1) at high temperature  M λ 4 λ  N 6 X N T ( [ , 1 ( λ 1 4 10 11 AdS /λ G . Here the gauge group is SU( + R 1 5 2 S ) . – 1 – ∼ 15 /N < 13 × M 1 5 E BH X /N µ  is set to 1. If the conjecture is correct, then the dual N BH 3 D λ (  N , whose action is given by 1 2 3 6 /λ + Hermitian matrices. We consider ’t Hooft large- 16 2 8 µν ] conjectured that 4d is large enough, a large AdS-BH, which fills the S N F 2 1 4 14 ] is believed to be a key idea in resolving the black hole infor- × E 1 13  N Tr x 1 4 15 d 16 are the ten-dimensional Newton constant and the AdS radius, respectively. 6) are Z , AdS , and the radius of the S 1 1 2 YM R ··· − g , 3.3.1 The phase transition from the big black hole to the small black hole 3.2.1 Gauge fixing N = When the energy is formed. Theand energy scales as Note that the large AdS-BH has a positive specific heat. ∝ = 1 S • 3.7 Other topologies 4.1 ABJM theory4.2 and 11d black More hole generic theories 3.3 10d Schwarzschild at 1 3.4 The Hagedorn3.5 growth at The case3.6 of weak coupling SO(6) breaking 2.1 Emission of D-branes and black hole evaporation 3.1 The large3.2 black hole (large Emission energy of and eigenvalues 2 YM M ensemble at strong coupling (see e.g. section 3.4.1 of [ can describe a black( hole (BH) ing AdS gravity description suggests the following behavior in phase diagram of the microcanonical Gauge/gravity duality [ mation paradox. Wittencompactified [ on a three-sphere S 5 Discussions 1 Introduction 4 Other cases 3 Analysis of the microcanonical ensemble Contents 1 Introduction 2 Stringy interpretation of the field theory degrees of freedom JHEP02(2017)012 ], 4 ]. 9 – 6 ] and references at strong coupling 14 3 . . S × 1 1 R side; see [ , the BH localizes along the 2 When the Schwarzschild radius AdS R 1 . We will call this localized BH, the 7 T N 1 , ] between the type IIA black zero-brane 10 is the length of the strings. G 10 L ∼ – 2 – E ], which is analogous to the large AdS-BH in 4d , the BH should behave like the ten-dimensional 13 – phase diagram of 4d SYM on correspondence, the supergravity limit and the uni- AdS 11 2 T R -vs- E /CFT 3 is the entropy and are for the duality [ by assuming the validity of the AdS/CFT duality 2 S ], 5 , where L 1), obtained ∝  S . The microcanonical and can be regarded as a ten-dimensional BH. This transition is of first order [ N ∝ 5 2 YM strings. As the small BHof shrinks long towards strings the becomeE string better. scale, the The description system of shows itFinally, when the as the a famous energy bunch Hagedorn is very behavior, small, the system is well described as a gas of short S and the BH becomesbecomes hotter much after smaller the than localization. Schwarzschild BH in flat spacetime, small BH. Note that the small BH has a negative specific heat. The large AdS-BH shrinks as theWhen energy the is decreased Schwarzschild and radius the becomes temperature goes of down. order g Although the gauge/gravity duality conjecture has not been proven, there is accumu- The authors would likeAnalytic to approaches thank for Jorge deriving Santos nontrivial for temperature the dependence clarification. have been discussed in [ • • • = 4 SYM. (For the AdS = 1 2 λ versal logarithmic correction havetherein.) been Hence studied we from needuse to CFT the test duality gauge/gravity duality to for understand a the small quantum BH. gravitational Furthermore, aspects if of we black holes, we have that 4d SYM has thediagram same directly phase from diagram. thebeen It gauge poorly is theory crucial tested for however atcurrently to several finite understand understood reasons. temperature. this [ phase First Asand of far D0-brane all, as quantum the mechanics we [ N know, duality the has only quantitative tests The relation between the energy and temperature is shownlating in evidence figure that it is most likely correct. Hence the majority of string theorists believe Figure 1 ( JHEP02(2017)012 ). ), 7 T BH N ]. eigen- N , 15 10 G ( , we derive < N / , we suggest 1 4 BH ∼ N E , is the main part of this paper. 3 for the large black hole , we remind the readers how the 2 of the small black hole – 3 – 3 ] is explained. Section is smaller. ). We would like to thank him for stimulating discussions 7 17 , T BH 16 N , N 10 matrix entries are excited. G 2 ( / N 1 ∼ While this work has been in progress, we have learned that Leonard E . All M X ]). In this paper, we propose a simple gauge theory description of the small BH. By hole is smaller when The large black holefields is described by a bound stateSuppose of some all of the the eigenvaluesvalues eigenvalues of form are emitted, a scalar after bound which state. and Such only matrices describe the small black hole. The black 3 This paper is organized as follows. In section Previously, the gauge theory description of the large AdS-BH, the Hagedorn parameter The idea that the size of the matrix blocks changes with the energy has also been an important ingredient • • 3 holes. On theorder gauge to theory derive side,and mathematically, collaboration this toward is the end exactly of what the we project. have done in of a proposal for a description of the Schwarzschild black hole in the Matrix Model of M-theory [ Susskind had essentiallyhole the is same described idea by(AdS independently. a space) small smaller He sub-matrix, in and conjectureddescribing order considered the the to a small small remove possibility BH. the black Then of In degrees making he of terms the assumed freedom of ‘box’ the which gauge are ‘corresponding theory, not this principle’ needed means which for a relates truncation the to large U( and small black that the same pictureto can superstring/M-theory. hold We for study aenergy-temperature the rather relation ABJM generic of theory the class as 11d an of Schwarzschild example theories black and holographically hole. derive dual theNote right added. and closed strings —(eigenvalues) from are the introduced, BH and [ We propose the a meaning gauge theory ofbetween description the the of energy, the emission entropy small and of blackmodulo temperature hole, the a and expected technical D-branes obtain assumption from explained the the at relationship conjectured the end gravity of dual the section. In section microscopic, stringy degrees ofSYM. Two freedom seemingly different, can but actually be equivalent, pictures read — ‘open off strings+D-branes’ from the fields (matrices) in 4d assuming the validity ofthe the relation AdS/CFT between correspondence the energyat and temperature strong coupling. We alsoonly give on a gauge heuristic theory. explanation Inthe supporting addition, Hagedorn this we behavior. will assumption show based In that short, the same picture correctly reproduces the gravitational description toanswer explain the gravity, we question. would be just assuming the answerregion, to and string gase.g. parameter [ region have been understood at least qualitatively (see to solve gauge theory. If we assumed the validity of the dual gravity description and used JHEP02(2017)012 5 ], ,ll 3 11 M with 9) on X , ,ji 4 ,kl ). Hence , and the 2 ··· 9 M 4 , M R X 2 X , ,lj ], the general- -th D0-branes. 3 ,ik j 1 13 M = 1 – M X X 11 M ,kl ( 2 = 0, and the physical M t M ]. -th and ] in the usual sense, that X A i X 19 , ,ik 22 -th D0-brane in , 1 ), while i 18 M 18 3 X ; i is large, a lot of strings are excited Vac | ]. Here we explain how they are related ) 4 = 4 SYM is straightforward. We work in 19 M,ij M , X N X . This upper bound appears because single 18 3 2 M N – 4 – X 2 M X 1 ). Here we use the adjective “nonlocal” because all pairs M 2 ]. The gauge field is set to zero, X describe open strings connecting ]. are regarded as the position of 20 21 M,ii M,ij is a closed loop made of three links and one site (figure X X i, k, l When we follow the usual D-brane effective theory point of view [ -th D0-branes. This is picture (1). 4 j is a closed loop made of four links (figure ] and (2) long, winding strings [ . Interpretation of matrix components as sites and links of a nonlocal lattice. 11 -th and i, j, k, l i with different Figure 2 ,li 4 When the number of spatial dimension is nonzero, the gauge fields form other links; In order to go to picture (2), let us regard the D0-branes and open strings as sites and Firstly let us consider the D0-brane quantum mechanics picture [ More precisely speaking, the adjoint fermions existThis as idea well. is not The new. gauge-singlet See condition e.g. follows [ from M 4 5 X the picture should be clear if one imaginesthe a Gauss-law constraint. lattice discretization of space-time. Then maximum possible number of stringa bits single (open trace strings) operator, istrace the which operators maximum beyond is possible this of length boundthis of order can is be slightly different expressed from bythe the using black ‘correspondence shorter hole principle’ operators. [ is Note always that described by strings in this picture [ of sites can be directly connectedFor by links. example, Gauge-invariant if states are we madedifferent consider of closed Tr( loops. · the black hole, which is aa bound long, state of winding D-branes string and in open strings, the is lattice naturally description. regarded as (More precisely, a few long strings.) The off-diagonal components Note that these stringsbetween are oriented. When links of a non-local lattice (figure ization to generic gauge theoriesthe including Hamiltonian 4d formulation [ Hilbert space is obtainedthe by vacuum acting state. traces ofthe products diagonal components of scalars In this section, wegauge explain theory. how There the are two stringyand seemingly strings micro-states different [ of pictures, (1) a theto black bound each hole states other. are of D-branes encoded into 2 Stringy interpretation of the field theory degrees of freedom JHEP02(2017)012 (2.1) , though 3 S × 1 R . matrices turn . i i N When one of the Vac Vac × | | 6 ) ) 4 ]. 4 N M M 17 X X , 3 3 = 4 SYM on M M 16 N X X 2 2 M M -th block carries the energy and X X i 1 1 M M ]. X X 19 , in Tr( ! in Tr( M ,ji ,li 0 . Each block is regarded as long strings with 4 4 x M M – 5 – ··· X X , M BH 0 2 ,ll ,lj X 3 3 M ,N M

1 X X N ,kl ,kl 2 2 M M X X , respectively. Typically, the ,ik ,ik 1 1 ··· M M , D-branes. Then strings inside each bunch are short, light and can 2 2 X X . ]. Suppose the D-branes form well separated and localized bunches . ··· ,N . , 2 1 12 2 i 2 N N ,N 1 N Figure 4 Figure 3 Next let us consider the emission of eigenvalues, following [ In the theories with flat directions, the emission is inevitable. In 4d 6 there are no flat directions, emission still plays an important role, as we will see shortly. entropy of order D-branes is emitted, openheavy and strings decouple from between the theto dynamics. block emitted Hence diagonal fully D-brane forms, noncommutative and the others become consisting of be very excited, whileexcited strings connecting much. different bunches Indiagonal are terms matrices long, of with heavy block and matrices, sizes maximum cannot such length be a configuration is expressed by almost block- 2.1 Emission of D-branesThe and dynamics black of hole D-branes play evaporation of an important the role black in holes the [ matrix (gauge theory) description the link variables alsoloops become (QCD strings; generic strings) long withcondensation string of scalar states long insertions. are strings. expressed For Here more by the details, Wilson see black [ brane can be regarded as a JHEP02(2017)012 . 8 6 . min 5 . The N E < E BH should )-components. describe the 5 ; see figure and , and hence the M min 5 N,N x ] may provide us T min 26 , . 4 is the minimum energy 3 S E > E × min 1 . temperature) increases. As E R ] along the S ' 25 3.2.1 completely, the AdS 1) 1) matrices and + 1, while the energy is conserved 5 − 2  eigenvalues, it may still be possible to 1) corresponding to the instability might λ N ( − N × min N does not have a counterpart in 4d SYM; it is ( E 1) , and the solutions at ∼ ; they should be of the same order but it is hard − – 6 – min min to N min E 2 eigenvalues (D3-branes); as is the large black hole. T N described by 4d SYM on E < E N 5 BH black hole can exist only when the energy is large ∼ to be S 5 × This is the Higgs mechanism. The number of light physical min 5 black holes, respectively. (Hence T 7 5 BH to refer to a small AdS black hole which still fills the S = 5 T ]. BH solution at 24 , 5 ]. 23 black hole is obtained by rolling up a black three-brane with charge 17 , 5 are fully noncommutative ( 16 . As we declared in the introduction, we assume the AdS/CFT duality between 5 ]. M BH 21 X Strictly speaking, the ‘minimum energy’ The small AdS When the black hole on the gravity side fills the S In the next section, we show that the emission of D-branes can explain the properties The speculation that a subset of D-branes describes the small black hole existed for quite some time; Here we implicitly took a gauge in which the emitted D-brane is described by the ( 7 8 see e.g. [ 3.2 Emission ofThe eigenvalues above argument, however,Even does if the not energy take is not into large account enough the toTechnical bind emission details all of about this eigenvalues. gauge choice will be explained in section be slightly different from the energyto at determine the orderwith one a factor. concrete A number. dualare In gravity sensitive the analysis to following such this we as will ambiguity. [ not consider order one coefficients which 10d BH becomes thecan appropriate be description. understood as Fromblock) the follows. to point In be of order formed, for view (almost)a a of all bound lot gauge of state of theory, the of it off-diagonal energy.enough. eigenvalues elements (non-commutative must Hence be the excited, AdS which costs to be ‘large’ and ‘small’ AdS of the ‘large’ BH.) Toblack avoid confusion hole we and will small use AdS small BH to refer tounstable a with 10d respect Schwarzschild to the Gregory-Laflamme instability [ in practice), the dualitystudies, can see be e.g. [ tested by Monte Carlo simulation.be used For in recent the numerical dualLet gravity us calculation. call It the has energy the at minimum temperature 3.1 The largeThe black large AdS hole (large energyblack and brane is aSee bound figure state of all SYM and the large BH is correct. In principle (and within a ten-year span, probably details, see [ of the small black hole in AdS 3 Analysis of the microcanonical ensemble position of emitted D-brane. degrees of freedom decreaseduring from emission. Hence thethe emission energy continues per more degree eigenvaluesand can of be hotter. freedom emitted The ( and negative the specific black hole heat becomes of hotter the black hole follows from this simple fact. For where JHEP02(2017)012 3.2.1 (3.1) particles, 6) should , gas ··· N , . Our proposal 2 3 , S ]. Also note that, × 1 12 = 1 . R 0 9 N M , (almost) all off-diagonal ( 2 . Such a correction is negligible radius. Hence the emitted M 7 N ), there is some ‘fuzziness’ which 3 X = 4 on ∼ 3.1 , N E         = 0 can be realized. gas M N gas x N . is small but of order , . . becomes smaller. The number of degrees of BH /N M 2 N BH BH – 7 – x N N = eigenvalues and a gas consisting of M 1 x N . Note that, because the space is compactified, there BH M BH N < N X matrix.         BH N BH N , can be described by matrices of the form . AdS black hole is obtained by rolling up black 3-brane. × gas BH N N + Figure 5 BH is an N eigenvalues, with = BH BH X N freedom decreases dynamically, and the temperature of the system goes up. At some (almost) all off-diagonal elements must be excited,If which the costs total a energy lot ofelements of the can energy. system be is excited. big enough, Hence say If the energy is not that big, then In order for a bound state of eigenvalues (non-commutative block) to be formed, N Intuitively, We consider a state consisting of BH and gas in 4d Here we implicitly took a gauge in which the ‘black hole’ comes to the upper-left corner. In section • • • 9 we show how thisnegligible. gauge choice Strictly can speaking, bedescribes in achieved, short open and addition strings that to stretchedin the between the the nearby gauge elements situations D-branes; fixing shown see we and figure consider in Faddeev-Popov below, ( terms where are both where particles do not rollagain to by the infinity; black they hole. form At a some point, finite the density emissionis gas and that the and a absorption that rates blackwhere can can hole balance. be consisting absorbed of bind is no superselection ofbe vacua; determined the dynamically, like valueunlike in of the Matrix the the Model Matrix scalar ofthe Model field M-theory, scalars this of have theory M-theory a does [ mass not possess proportional flat to directions, the because inverse of the S JHEP02(2017)012 completely. 5 ) gauge theory. ) can be excited. can be obtained. BH 2 BH 7 N N ∼ becomes too big, it ends up ) and figure BH -BH which is fills the S N 5 becomes smaller so that the string can become small enough so that 3.1 ]. This gauge condition is based on 27 BH BH N N . Here we are assuming that most of them 7 – 8 – ), there is some ‘fuzziness’ which describes short open 3.1 phase diagram of AdS (number of degrees of freedom T BH . -vs- X 5 E does not have a counterpart in 4d SYM; it is unstable with respect to the Gregory- . The microcanonical . A more precise representation of the shape of the matrices describing a small black hole. min can take more complicated shapes. off-diagonal elements in In the closedpossible string with picture: the theentropy. available string amount When wants the of to energy energy.with is a become long not It as but big does not long enough, very this wound and i.e. string. in winding Then order as to gain more point, (if the total energy is not too small) Strictly speaking, there are strings connecting different gas-branes, and also the ones In our calculation, we neglect the interaction between the emitted eigenvalues and the • small BH, and treat it similarly as3.2.1 the ‘large BH’ in Gauge the fixing Here truncated we U( explain how theLet (almost) us block-diagonal first form introduce ( the maximally-diagonal gauge [ connecting the BH and gas-branes.are long, See heavy figure and do not play an important role. strings stretched between nearby D-branes.if The the light matrix blue were lines blockblue indicate diagonal. implying where The the a color blocks larger refers would magnitude. to be the magnitude of the matrix elements, darker Figure 7 In addition to the elements shown in ( Laflamme instability along the S Figure 6 E < E JHEP02(2017)012 N max size ). (3.4) (3.2) (3.3) BH 3.1 N 0’s. We can be a projector n . As long as we n − 0 P are parametrically N N the truncated action BH ) to the leading order t t ( N . X 2 | . and 1’s and ij 2 ) ) 1 n is of order N − BH D-branes), any gauge-equivalent N Ω (Ω). Unless there are accidental . P ii /N 2 transforms as M and simultaneous permutations of | BH R . Note that this ‘truncation’ makes M X ij BH N 1 X 0) with − R =1 N BH M,ij (Ω N i , | , ˜ N BH X X 1 | P defined by N − =1 6 ··· are ‘as close to simultaneously diagonal as diagonal elements are spread out. We can P to , X Ω diagonal elements in the maximally diagonal ij =1 6 = M 0 X − R 1 M M , x X BH R 3 0 − max BH – 9 – x X M . Then, the small black hole is stable, namely the , d 3 N Ω N 0 1 Ω d X , M Z − N → Z X ≡ Tr ( N ··· , ≡ M 1 max = M X , ij X (Ω) Ω c R ij gas ≡ R = N is unique up to U(1) M → ext ˜ X S ij = diag(1 max which maximizes Tr R n P is small but of order max ) is the same as the original action written by t ( form the small black hole. We assume both /N block, BH . BH BH ˜ n X does not fluctuate much. The ) parts of the actions agree), as long as N N /N 2 × BH N n ( N O Another way to fix the gauge is to introduce an ‘external field’. Let In our arguments throughout the paper, intuitively, we are ‘truncating’ (or freezing Now let us apply this gauge choice to the situation under consideration. Among the to the introduce an ‘external field’ which pushes BH to the upper-left corner, for example as the interaction with emitted particles.written This by is because, at(i.e. each time consider physics of the small blackprofiles hole give (the simply bunch of theorder same in path-integral 1 weight as the truncated theory, to the leading size is fixed andFaddeev-Popov temperature terms increases associated in with the theaccount. truncated maximally-diagonal But theory. gauge the The should maximally-diagonal gauge be gaugeis fixing taken condition time-dependent!) and into is the rather and nontrivialPopov (note it terms. that is In Ω difficult the to situation we write have down in the mind, however, gauge these fixing terms and are the negligible up Faddeev- to bunch in the upper-left corner of the matrices.out Then the the matrices gas takes degrees thesense of form only freedom ( in to theextra reduce) microcanonical energy ensemble, and is for added, a the fixed bunch value of size the increases energy; in when the the original theory, while the bunch large, and value gauge should then form aare bunch not (implying that small), the while(partly) off-diagonal other fix elements the connecting ambiguity them of the permutation of rows and columns by putting the degeneracies, such arows Ω and columns. The possible’. D-branes, Under the gauge transformation We choose Ω = Ω off-diagonal elements are small.Hence, we impose However the this conditionas is that possible’. actually ‘the For matrices a that are purpose gauge-dependent as we statement. close introduce to simultaneously diagonal the implicit assumption of the ‘D-brane+open string’ picture — diagonal elements are large, JHEP02(2017)012 . 2 (3.7) (3.8) (3.5) (3.6) α 11 2 . Then, 1, the two N 4 / 1 =  − BH 4 λ / which is small 2 to expect that 1 BH , assuming the − c 4 N ) 10 / is varied. When 1 BH /λ 1, we obtain , ∼ . = 0. λ ) ) BH ∼ ). Let us firstly give 2 YM BH . In order to measure M almost completely. In λ N N g ) 4 ( 2 YM N / 5 X , 2 YM 1 O 2 is the radial coordinate of ) , g 2 YM ≡ ; hence the eigenvalues of 0 N /λ BH > N, g X ) = when ( 4 ∼ 1) N, g R N / BH ( 4 ( ) 1 BH / λ min − 1  N min λ , α 2 λ min YM 4 T λ − / , ≡ 1 < E ) symmetry. Then the BH is pushed , where 0 − N N, g 1 ( 0 gas ( ( N 1. It would be natural α > X BH E ,S ∼ R ∼ E 4 min  + / scale as is almost touching the D-branes. When the . This is nothing but the usual prescription 4 E 7 c / ' gas 3 α 1 BH /N < and X BH 2 which occurs at positive value − -term, which should be true at strong coupling. λ E 3 ) E 2 – 10 – N ] S BH 0 ext /λ = × = N S M 1 ,E 4 BH ) R / 2 ,X λ 1  total ( − N M E = 1 makes the situation subtle. When ( , the radius of the bunch scales as ) ( , we turn off for given ∼ 3 X /λ O S ) /λ BH = 1 are clearly separated. Then energy scale should be dominated by R BH Tr[ N = 3 BH S 2 YM → ∞ λ N N λ R is made of disappears in terms of ( 2 , g BH N 5 , and we assumed λ N 2 BH λ BH and N ∼ N 4 . / ( BH N 1 ∼ N gas − BH ) ) min . Intuitively, the boundary S E E = T 6 /λ 2 YM R BH BH , g λ which is the size of the eigenvalue distribution. N BH ). After taking 4 / 0 2 1 YM N ( g − N ) ( = A better argument inspired by string theory goes as follows. In the dual gravity picture, We have to determine The energy is /λ min O are of order 1, and the eigenvalues of The existence of another scale We would like to thank Juan Maldacena and Kostas Skenderis for pointing out several miscalculations E 0 BH 10 11 BH λ the energy of this bunch, we imagine aenergy scales sphere ( right( outside of the bunch, andin consider the first version of this paper, including this part. It helped us debug a few wrong points in the argument. the ‘smallest large black hole’the D-brane is picture, a AdS bunch ofthe eigenvalues transverse filling AdS bunch shrinks, the radius becomes smaller by a factor of ( and combined with the ’t Hooft counting and interactions with the emitted branesλ do not affect the sizethe of typical the energy bunch scale significantly.combined is Here with set the by fact the that inverse the of large the BH eigenvalue should distribution, have a heuristic gauge theoryaction argument is which dominated does by the notNote rely that on the the coupling gravityX dual. Suppose the the bunch size decreases to and roughly speaking, Hence we ignore We will now makeorigin this of scenario AdS more corresponds precise. to a The bound AdS-Schwarzschild state BH of sitting eigenvalues around at the to the upper-left corner bybut minimizing used for detecting spontaneous symmetry breaking. 3.3 10d Schwarzschild at 1 Note that this term manifestly breaks the full SU( JHEP02(2017)012 ). If 1, 14 to N , 5 (3.9)  N (3.11) (3.12) (3.10) G N 2 YM (2 ) and will ; here we g naturally / 4 2 YM 3 + / 3 = g 3.13 1 S r ) 2 is the area of λ R = )–( π /λ λ 3.9 = 5 AdS BH R λ 3 π when = ( 3 13 , AdS-BH S , because we are using the dual S R 2 ) more quantitatively, let us This scaling of /N 2 3.8 . λ 12 ) ∼ 2 + . , r N , + 10 2 r AdS 2 AdS 2 AdS . G √ R R πR 2 AdS N black hole in order to derive ( 5 AdS + , + R 2 10 + 4 + πR 5 r ) = r 2 G π 2 ( large 2 ) gauge theory with 3 + ), where the denominator – 11 – 2 YM r = N , BH 3 AdS 5 AdS 10 = N R N, g 4 R G ( S 3 π 8 = 1, , and hence the entropy is 3 π ( AdS-BH min 3 + · T r T = 2 2 AdS N , π R E 5 ). Note that the ’t Hooft coupling changes from G = black hole in the following. black hole is given by 2 + 3.8 = 5 r N , small 10 , we obtain G with the surface right outside the bunch. ; this does not affect the result because the quantities of interest do not T dS 3 , by BH = 2 N is the 5d Newton constant, which is related to the 10d Newton constant /N dE N 2 1 YM , 5 g ∼ . Hence G 5 = In order to calculate the energy, entropy, and justify ( N ) can change as the energy grows, and hence the identification with the ‘smallest large black hole’ fails. , As mentioned above, if this rescaling is notThis performed, then is the the number only of assumption D-branes which in relies the on bunch (i.e. the dual gravityThis description. is different Note from that usual we value assumed in the here Einstein frame, 13 14 12 10 BH BH N the validity of thederive dual the gravity energy description of for the the frame. By using The area ofHere the horizon isG 2 the S which is minimized at 2 appeal to the AdS/CFT duality forwe large assume BH the from dual here gravity on, calculationthe of and temperature the go large of to BH the AdS to gravity be picture. correct identified the S suggests the scalings in ( λ explicitly depend on the ’t Hooft coupling. is not included.the From sense of the usual dualis open gravity very string/closed theory string problematic. point duality),gravity a of side As naive view is we truncation (‘closed something nearside. will string the different; Mapping see horizon picture’ back the shortly, to in curvaturethe the the radius fully gauge counterpart noncommutative changes theory, phase of the as of this energy U( well of restriction on this on bunch the should the gravity be described by decrease the energy insideup this the restricted energy, region, the the bunchnot becomes bunch have becomes hotter. to smaller. go (Without up thisbunch If because restriction, is we the the ‘the pump temperature bunch smallest can does possiblethe become large gauge larger.) black theory, Hence, hole’. or with ‘open Note this string’, that restriction, this point the restriction of is view, natural in from which the gravitational back-reaction only the interactions between D-branes and open strings inside this sphere. If we further JHEP02(2017)012 ), 3.8 (3.13) (3.14) (3.15) (3.16) associated ) into ( 4 / 1 ) cannot change increases and as 3.15 /N ]. BH BH N . 22 BH , . ) N N N 18 , , one would not have an 2 YM decreases, if one naively 3 8 AdS 10 = ( S , g ; namely the bunch size is R G = 1 and we obtain R BH BH 2 YM ∼ 0 AdS N g N 2 AdS ) R ( R 15 . 2 YM min cancel with each other. Hence we ∼ . ). By substituting ( S 4 7 BH 2 + / N, g T r 1 BH = 2 ( YM 1 N − , N ], the transition from the large black hole α , g 10 BH min 4 → ∼ G BH N ∼ N ) theory. When ( BH BH – 12 – 2 ,S ,S BH 7 min ) dS BH dE N N T and N , T = 2 YM 7 AdS 10 ∼ ) theory without rescaling R G , g 0 AdS BH BH R BH T ∼ BH N ) are given by using E ) = 1 should be replaced by N → ( 2 YM 2 YM ) does not explicitly depend on ) via the U( AdS min AdS R E 3.8 . The Newton constant remains unchanged, because the rescaling N, g 3.8 N, g R 16 3 ( ( S = R min min BH S E E ) again. ], and then becomes unstable.) 3.8 ) and 28 . The result ( 2 YM 2 YM g N, g dependence. This treatment is wrong because the truncation and the variation of the ( In our gauge theory argument, the order of the transition is not clear. However, if we Before closing this section, let us give a comment on a confusing point associated We identify the energy and entropy of the small black hole with these values: Because the curvature radiusThis changes, kind this of is matching different has from a the flavor ‘truncation’ similar of to the the geometry. correspondence principle [ 15 16 BH min assume the transition is ofHiggsing. first order, then the small black hole must be hotter, due to the 3.3.1 The phaseAccording transition to from the calculation the in bigto the black the gravity side hole small [ to blackblack the hole hole small is at black of the hole firstblack same order, hole energy. and [ (More the precisely small speaking, black hole the is large hotter BH than becomes the a large lumpy and the temperature hasis provided to at go the up.However, we beginning do of The not this find argumentvaried a section, on problem may the there, seem because scaling we todetermined of did solely suffer not eigenvalues, by from change which the the the energy. energy, same rather we subtlety. with the evaluation of‘truncated’ ( the theory to theN U( energy do not commute. Whena energy is result added, the in temperature the original can theory go down, while in the truncated theory we obtain Then, Note that this might be different from factors associated with should have ( When the bunch shrinks, with the rescaling of E JHEP02(2017)012 1, we ]. We 29 . BH , and hence a λ ), which is the N . The constant- N , 2 10 4 rather the shape /λ / G 1 7 s . This is exactly the l ( ∼ reaches E / 1. When 2 ]. 1 3 BH ∼  N E/N E 1) 1. The difference from the one at BH 1, ) are proportional to each other, N   ∼ E -independent constant. The growth λ ( λ 2 YM S ( g BH BH increases with λ T . This region is rather different from the ≡ /λ 2 1 8 BH , at N BH . λ /λ 1 1 – 13 – , which is of same order as the deconfining and /N ∼  min BH , the energy becomes T λ . Hence the SO(6) rotational symmetry should be /N and the entropy 5 N /λ ) and goes to a 1 0 BH E has been known for quite some time, see e.g. [ N N ). The difference is not just a factor 3 ∼ 8 1 /N BH N ] which breaks SO(6) and the eigenvalue distribution in gauge theory . In other words, when energy is added, it is used for exciting more 28 2 BH N ) is not just an overall factor. ∼ 1 BH . Phase diagram of the weakly coupled 4d SYM, S ∼ 1 (figure This phase diagram figure Note also that, at behavior below this point looks like the Hagedorn behavior; actually, in the closed  BH 3.6 SO(6) breaking The 10d BHbroken. is A localized natural on possibilitylike would the a be lumpy S that BH the [ side ‘smallest also possible breaks large SO(6). BH’ More is analysis something is desirable concerning this point. strong coupling region (figure is different. have just rephrased the known result, in order to show the consistency of our proposal. 3.5 The case ofThe weak same coupling idea of4d emission SYM. of In eigenvalues this can regionnegative be the specific Hagedorn heat applied growth is to continues not the until expected; weakly see coupled figure region of Hagedorn behavior! The energy E matrix degrees of freedom, rather than increasing the energyendpoint per of degree the of Hagedorn freedom. growth expected on the gravity side [ In the previouscan section, use we perturbation have theory.hagedorn assumed There, temperatures, is of O( of the temperature stopsT when string picture, the length of the long string Figure 8 λ 3.4 The Hagedorn growth at JHEP02(2017)012 1 3 N S Z × (3.17) (3.18) 2 and S can be made 3 The important There is a BS is unbroken, the E 18 -brane in the T- , p 3 N , 19 . 17 Z L . 1 BS N ⊂ S N × p N , and increase the energy 1 , due to the Eguchi-Kawai ≡ Z L 1 S goes to infinity. Hence they × At larger 1 1 should behave in the same way 20 , α BS = S 4 E / 3 1 . − α 6 , due to the dimensional reduction along BS ∼ ,p T and T ] and its higher-dimensional analogues; for is changed, the Eguchi-Kawai equivalence 1 . N 1 4 . We take the radius of the two-sphere to 1 , 1 / R 1 25 S L 1 10 S − × G ) – 14 – × 2 , which we denote by 2 /λ ∼ 1 first. The crucial difference from S BS BS is unbroken, translational invariant quantities such as λ 3 should be related to the temperature as E ( 1 ∼ we used before; we just emphasized that this is related /N is fixed and ) + 1)-d SYM on BS BH as the circumference of the S 2 BS YM p N N E , g R = is broken and the Eguchi-Kawai equivalence along the directions is not intrinsically curved. For this reason, the circumference of the is sufficiently large, the center symmetry is unbroken. When the BS × α 1 N N 7 E ( Z min ]. -independence of the system. In gravity side, the topology of the small T 34 L ]. – 30 31 is the same as is unbroken. The gravity side is described by the black N BS Z N Next let us consider 4d SYM on S Let us consider 4d SYM on T We thank our anonymous refereeWe impose for a suggesting periodic toOutside study boundary of these condition examples. along the S ’t Hooft limit, it is possibleIn to the take T-dual picture, the we limit find D2-branes describing smeared the along the M-theory T-dual circle, regime which of also can the be interpreted does not necessarily introduce a ‘scale’ to the setup under consideration; indeed, as long from zero. When 18 19 20 17 1 Matrix Model of M-theory,limit where in the this 11d paper. Schwarzschild black hole may emerge.as a We black do string. not consider this Here, to the black string. Combined with the ’t Hooft counting, the dimensional analysis and This can be derived fromnot gauge change, theory and as hence follows. Firstly, the scaling of the eigenvalues do guarantees the black hole becomes S behave as a blacksmaller string without (BS), breaking rather the than centeras a symmetry. a black There 9d hole. black hole, details, see [ be 1. LetE us fix the circumferenceenergy of density S per length theory is approximated by ( the directions where where dual picture. In thethe gravity black side, hole/black the string center-breaking phase phase transition [ transitions are interpreted as equivalence [ compactifications is that D3-branes wrapping onTherefore, three-torus have Ramond-Ramond this charge. theorycenter does symmetry, which not can describe be broken a at Schwarzschild small black volume. hole. When Let us consider other topologiesdifference such here as is that S S S as the center symmetry alongthe the energy S density is independent of the circumference of the S 3.7 Other topologies JHEP02(2017)012 , φ )), 0 can N (4.1) (4.2) (4.3) ( (3.19) (3.20) where O 4.2 9 P l = . Namely = 4 SYM, 2 ∼ S N N/k × . The gravity N , 1 N 11 R . The same behavior G = 6 / 1 ) ] on . . and , . 3 2 N/k 36 / 4 4 / P 4 / / N/k . The fact that this scales in BH P 3 l BH 6 7 l 4 = / 6 N α α 1 N / = ( 2 2 λ ]. λ 1 6 ∼ ∼ / N N 37 N 1 λ = = . In the ’t Hooft limit ( BH BH = , 6 4 4 / P / / 1 P 2 1 l /l N 6 − − 6 / ) ) / . Then the argument of section 1 1 2 ) which we have discussed in the previous ,E ,S /λ /λ − 2 BH 3 5 / / ]) suggests that the expansion with respect 3 N P BS BS 4 kN l ( λ λ 35 , the analytic treatment is difficult; however, N N ( ( ∼ c ∼ ∼ E 2 2 BS BS ∼ – 15 – BH BH N N N 21 = , it leads to , N , 2 -suppressed corrections [ AdS = 1, ’t Hooft coupling , the scaling should be dS dE 9 AdS 11 ∼ ∼ E 8 AdS 11 R N k R /N G /λ ) ) R G = 6 ∼ ∼ 2 2 ∼ YM YM ) Nφ BH N , g , g T . In this case, the microscopic picture on the gravity side is , N 2 , YM 4 7 corresponds to the number of M2-branes. However it would 2 AdS 4 AdS BS BS S G R G R N N N × N, g ( ( ( ∼ 4 ∼ min min min E S that the small BH is described in the same way as in 4d min min S is important. Here, we need to use a similar relation: the bifundamental scalars S M2-branes, and the calculation goes through in the same manner. , E 2 AdS BH N R N ∼ are the eleven-dimensional Newton constant and Planck scale, respectively, assume ) = 4 SYM, we used the fact that the eigenvalues scales as ) naturally follows. 9 P l N 2 YM works well. has a potential of the form 3.17 c φ and N, g ( With this assumption, by using When the energy is small, then the center symmetry breaks. On the gravity side, N , For 4d E/E 21 11 min the same manner which describes the moduli spacebecause of M2-branes, should scale as is expected in the M-theory region, up to 1 Therefore, the energy and entropy are estimated as and not clear, other thanbe that natural to by a bunch of G 4.1 ABJM theoryThe and argument 11d shown black above hole canexample, be let applied us consider to the otherwe M-theory quantum consider region field the of theories the Chern-Simons as ABJMdual level well. theory is [ As M-theory on an AdS be applied. Near theprevious transition studies point in analogous situationsto (e.g. [ 4 Other cases in the T-dual picture,the the smeared center D2-branes breaking (BS) can torather different the be from localized interpreted the D2-branes as localizationsections; (BH). a along the Note the topology that topology S change this change from sufficiently transition takes small, is place 4d only SYM in reduces to the 3d T-dual SYM picture. on S When the energy is Hence ( E JHEP02(2017)012 (4.5) (4.6) (4.7) (4.8) (4.4) , and hence BH N = 11 for theories D . 2 , − 2 D BH − = 10 and . N T 2 D 1 N ) N D − D, D, D G D, Rα G R ( G region. Seen from gauge theory, it is . ∼ ∼ ∼ , T ) 8 BH T min BH BH . Let us also assume that the eigenvalues 1 Rα does not set in. N , ( / 11 AdS 1 3.4 G R – 16 – ∼ ,S ,S ,S ∼ 3 3 3 N − − − BH N may depend on the theory. This gives the expected D, D D D BH BH T ) D, p G T R E 1 N G Rα ∼ ( D, G compact dimensions (where ∼ min d ∼ E BH − E BH X (where X is a compact manifold), as long as the notion of large = 1) the system is strongly coupled even for small D E × . The power k p ) /N in the dual geometry, like BH R N = ( scale in the same manner as a function of the coupling constant. Then, it is natural α non-compact and R d A rather striking consequencelarge of BH our from proposalBH’, the is which point is that of continuously the connectedsimply view small to a of BH the thermal the high- state, isprovides gauge but essentially us with theory; like with different it a important ‘matrix lessons is size’. on the Hence the the ‘smallest small study BH. possible of Another large the lesson large is BH the importance of 5 Discussions and where scaling, By going to the dualblack field hole, theory description we and obtain applying the same argument for the small to expect thatanalysis the as minimum energy and temperature are obtained by the dimensional with respectively string and M-theorybe duals). a The product space-time like doesand AdS not small necessarily black have to holeslength still scale makes sense.and Furthermore, suppose there is only one typical 4.2 More genericThe theories same powertheories, counting if will they possess hold a similarof for matrix description. other We theories assume that as the geometry well, consists including non-conformal This correctly reproduces the propertyIn of this the case eleven-dimensional (ABJM, Schwarzschild blackthe hole. Hagedorn behavior discussed in section yielding finally JHEP02(2017)012 black ]. We 5 26 , 4 ’ describe es- min E ] which integrates out the 39 ]. Since the agreement should 38 – 17 – (30) email exchanges with Daniel Harlow within two which can be found in [ O 5 ], it has somehow almost been forgotten after the Maldacena ]. There is also an attempt for studying the small AdS 12 26 , 4 ]. Due to this, a detailed study of eigenvalue dynamics should help lead us to 17 , 16 In this paper we truncated large matrices to small matrices. More refined treatments, More tests would be desirable to establish the proposal more rigidly. One interesting part by the Grant-in-Aid ofSports the and Japanese Ministry Culture of (MEXT) Education,supported for Sciences by Scientific and the Research Technology, California (No. Alliance 25287046). fellowship (NSF The grant work 32540). of J.M. is Furthermore we thank Ahmed Almheiri,Dyer, Tom Blaise Banks, Evan Gout´eraux,Guy Berkowitz, Gur-Ari,Jorge Oscar Daniel Dias, Santos, Ethan Harlow, Andreas Sch¨afer,Stephen Joaosions Shenker, Penedones, and David comments. Enrico Tong In Rinaldi, anddays particular, Ying certainly Zhao helped for us discus- improving the presentation. The work of M.H. is supported in Acknowledgments We would like toproject. thank As Leonard explained Susskindwould in also for the like introduction, the to thank he collaborationlations Juan in had toward the Maldacena essentially first and the the version Kostas of end same Skenderis this idea of for paper, which pointing as the helped out ours. us miscalcu- correcting We a few wrong arguments. eigenvalues. It would be nice if we could makefor progress example in something near like the future. emitted matrix eigenvalues, renormalization would group allow [ us to extract more information about black holes. have been performed in [ hole, which is not localized alongbecome S better when 10don black the hole very is hard smaller, numericalhave such calculations not tests needed might yet to be include understood the doable how finite-size without the effects relying breakdown [ of SO(6) symmetry can be seen in terms of and doable direction wouldour be proposal a is consistency correct, check thesentially based small the same black on dual hole dual gauge and theory, gravitywith the up calculations. the large to emission the black If of rescaling hole of D-branes. at the ‘ Recently, ’t dual Hooft gravity coupling calculations associated for the small black hole an understanding of the bulkthe geometry, large including BH the can horizonsimulation be of rather studied the straightforward black by with hole. using the Matsubaraa the Note formalism. first-principle canonical that study ensemble, It would of which provide makes thegauge us the geometric theory. with numerical structure Such of a the studyblack Schwarzschild hole should black information be hole puzzle. important based for on various problems associated with the of eigenvalue dynamics has been widelyMatrix appreciated Theory in conjecture the [ 20thconjecture century, for (AdS/CFT). example in It the is,evaporation, even however, in an the important contextand of piece refs. the [ for Maldacena conjecture, understanding as black emphasized hole in this paper eigenvalue dynamics in the gauge/gravity duality a la Maldacena. 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