Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
UUniversal Spectrum of 2d Conformal Field Theoryin the Large c Limit
Thomas Hartman, ∗ Christoph A. Keller, † and Bogdan Stoica ‡∗ Kavli Institute for Theoretical Physics, University of CaliforniaSanta Barbara, CA 93106-4030 USA † NHETC, Rutgers, The State University of New JerseyPiscataway, NJ 08854-8019 USA ‡ Walter Burke Institute for Theoretical Physics,California Institute of Technology, 452-48, Pasadena, CA 91125, USA [email protected], [email protected],[email protected]
Abstract
Two-dimensional conformal field theories exhibit a universal free energy inthe high temperature limit T → ∞ , and a universal spectrum in the Cardyregime, ∆ → ∞ . We show that a much stronger form of universality holds intheories with a large central charge c and a sparse light spectrum. In thesetheories, the free energy is universal at all values of the temperature, and themicroscopic spectrum matches the Cardy entropy for all ∆ ≥ c . The same is trueof three-dimensional quantum gravity; therefore our results provide simple nec-essary and sufficient criteria for 2d CFTs to behave holographically in terms ofthe leading spectrum and thermodynamics. We also discuss several applicationsto CFT and gravity, including operator dimension bounds derived from the mod-ular bootstrap, universality in symmetric orbifolds, and the role of non-universal‘enigma’ saddlepoints in the thermodynamics of 3d gravity.CALT 68-2889, RUNHETC-2014-07 a r X i v : . [ h e p - t h ] S e p ontents c partition function 7 A Density of states in the microcanonical ensemble 28B Mixed temperature calculations 30C Symmetric orbifold calculations 31
C.1 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31C.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
In quantum gravity different energy scales do not decouple in the same way as instandard effective field theory. Rather, as a consequence of diffeomorphism invariance,the theory in the UV is heavily constrained by the IR. The same effect must occurin conformal field theories (CFTs) with holographic duals. In this paper we explorethis connection in a class of 2d CFTs, where it is realized as invariance under largeconformal transformations of the theory on a torus, and provide a partial answer to1he question of what data in the UV is fixed by the IR. The results agree with knownuniversal features of 3d gravity. The calculations are entirely within CFT and do notassume holography.The UV/IR connection leads to universality. A famous example in gravity is blackhole entropy: to leading order, every UV theory governed by the Einstein action atlow energies has the same high energy density of states, dictated by the Bekenstein-Hawking entropy law S = Area / G N . This is an IR constraint on the UV completion.The area law has been derived in great detail for particular black holes in string theory[1]. Yet it is often mysterious in these calculations why the final answer is simple anduniversal, since the intermediate steps seem to rely on various UV details.In AdS gravity, the black hole entropy agrees with the Cardy formula [2] for theasymptotic density of states in any unitary, modular invariant 2d CFT [3]: S black hole ( E L , E R ) = S Cardy ( E L , E R ) ≡ π (cid:114) c E L + 2 π (cid:114) c E R . (1.1)The central charge takes the Brown-Henneaux value [4], c = 3 (cid:96) G N (cid:29) , (1.2)where (cid:96) is the AdS radius, G N is Newton’s constant, and E L,R are the left- and right-moving energies of the black hole (normalized so that the vacuum has E L = E R = − c ).This is a more universal derivation of the black hole entropy that does not rely on all ofthe microscopic details of the CFT. However, there is an important difference betweenthe black hole entropy and the Cardy formula. In general the Cardy formula only holdsin the Cardy limit c fixed , E L,R → ∞ , (1.3)whereas the Bekenstein-Hawking entropy should hold in a semiclassical limit, c → ∞ , E L,R ∼ c . (1.4)Having an extended range of validity of the Cardy formula is a key feature that distin-guishes holographic CFTs from the rest. Of course, in the explicit theories consideredin [1, 3], it is possible to check microscopically that the Cardy formula indeed appliesbeyond its usual range, but in other cases such as the Kerr/CFT correspondence the2ardy formula is applied without a clear justification [5].One aim of the present paper is to characterize the class of CFTs in which theCardy formula (1.1) extends to the regime (1.4). It is often stated that this should bethe case in a theory with a ‘large gap’ in operator dimensions above zero. We confirmthis intuition, give precise necessary and sufficient criteria, and identify the applicablerange of E L,R . The origin of the UV/IR connection in 2d CFT is modular invariance,so this is our starting point. In terms of the partition function at inverse temperature β , the modular S -transformation implies Z ( β ) = Z ( 4 π β ) . (1.5)The standard Cardy formula was derived by taking β → β limit at any value of c [2]. We will essentially repeat the analysis in thelimit c → ∞ with β held fixed. The result is the same formula for Z ( β ), but validin the large c limit at any value of β , under certain conditions on the light spectrumin addition to the usual assumptions of unitarity and modular invariance. This is thelimit that applies to 3d black holes.Constraints from modular invariance have been studied extensively in the simplifiedsettings of holomorphic CFT and rational CFT. In the holomorphic case, with onlyleft-movers, the partition function Z ( τ ) is a holomorphic function of the complexifiedtemperature τ . For a given central charge, the space of holomorphic partition functionsis finite dimensional, which yields powerful constraints. For example, the spectrum ofstates with E L > E L ≤
0, and there must be atleast one primary operator in the range − c < E L ≤ c + 1. Similar statements applyto other holomorphic objects such as BPS partition functions and elliptic genera insupersymmetric theories (see for example [6, 7, 8, 9]). Far less is known about modularinvariance in non-holomorphic theories. For some rational CFTs, the solutions of (1.5)can be classified explicitly [10]. For general non-rational partition functions, one ofthe only tools beyond the Cardy formula is the modular bootstrap [11], in which (1.5)is expanded order by order around the self-dual temperature β = 2 π . We use ourmethods to reproduce and clarify some results of the bootstrap in section 2.5. This Not to be confused with another common statement that it may apply when there is a ‘small gap’above the black hole threshold (discussed for example in [5]) suggesting a long string picture. We willnot address this latter criterion. c expansion may be a useful way to organize the constraints ofmodular invariance on non-holomorphic partition functions.This is similar in spirit to recent efforts to derive universal features of entanglemententropy [12, 13, 14, 15] and gravitational interactions [16] at large c . In fact, sincethe second Renyi entropy of two disjoint intervals can be conformally mapped to thetorus partition function at zero angular potential, the entanglement entropy is directlyrelated. Most of the entanglement calculations rely on a small interval expansion, butour results do not, so this rules out the possibility of missing saddlepoints in the secondRenyi entropy discussed in [12, 17]. Under what conditions universality holds for highergenus partition functions (or higher Renyi entropies) is an important open question. Operators in a unitary 2d CFT are labeled by their left and right conformal weights( h, ¯ h ) with h, ¯ h ≥ π , the operator-statecorrespondence associates to each operator a state with energies E L = h − c , E R = ¯ h − c
24 (1.6)and total energy E = E L + E R = ∆ − c . (1.7)In section 2 we study the partition function for zero angular potential, Z ( β ) = (cid:88) e − βE . (1.8)It is convenient to classify states as light , medium , or heavy :light : − c ≤ E ≤ (cid:15) , medium : (cid:15) < E < c , heavy : E ≥ c , (1.9)for some small positive number (cid:15) that is eventually taken to zero in the large c limit.We show that the free energy is fixed up to small corrections by the light spectrum.If in addition we also assume that the spectrum of light states is sparse, by which wemean that it is bounded as ρ ( E ) = exp[ S ( E )] (cid:46) exp (cid:104) π (cid:16) E + c (cid:17)(cid:105) , E ≤ (cid:15) (1.10)4 Π Π Π Π Π Π Π Π (cid:45) c c (cid:45) c c (a) β L β R Low temperature(gas) phaselog Z = c ( β L + β R )High temperature(black hole) phaselog Z = π c (cid:16) β L + β R (cid:17) (b) E R E L Universal S = S Cardy ( E L , E R ) S ≤ π (cid:112) ( E L + c )( E R + c )Light Enigma S boundedFigure 1: Universality in CFT with large c and a sparse light spectrum. (a) CanonicalEnsemble: The dashed line ( β L β R = 4 π ) separates high temperatures from low tem-peratures; in gravity, this would be the Hawking-Page phase transition. We show thatthe leading free energy is universal and equal to the Cardy value outside of the shadedsliver, and conjecture that this also holds in the sliver. (b) Spectrum: The density oflight states in the hatched region is bounded above by the sparseness assumption. Weshow that the density of states obeys the Cardy formula above the solid curve, andconjecture that this is true above the dashed curve ( E L E R = ( c/ ). In the enigmarange, the entropy is not universal, but satisfies an upper bound that prevents theenigma states from dominating the canonical ensemble.then at large c the free energy is universal to leading order :log Z ( β ) = c
12 max (cid:18) β, π β (cid:19) + O ( c ) . (1.11)There is a phase transition at β = 2 π . Furthermore the microscopic spectrum satisfiesthe Cardy formula for all heavy states, S ( E ) ∼ π (cid:114) c E ( E ≥ c
12 ) . (1.12)5he medium-energy regime does not have a universal entropy, but it is bounded by S ( E ) (cid:46) πc πE ( (cid:15) < E < c
12 ) . (1.13)The medium-energy states never dominate the canonical ensemble and therefore donot affect the leading free energy.The heavy states are holographically dual to stable black holes. The non-universalentropy at medium energies is related to the fact that in 3d gravity, black holes inthis range are thermodynamically unstable. In fact, the leading order spectrum of3d gravity plus matter (or gravity on AdS × X ) in this range is also non-universal,because in addition to the usual BTZ black holes there can be entropically dominant‘enigmatic’ black holes [18, 19]. These solutions, discussed in section 4, obey the bound(1.13).In section 3 we repeat the analysis for non-zero angular potential, which meanswe introduce β L and β R . The partition function at finite temperature and angularpotential is Z ( β L , β R ) = (cid:88) e − β L E L − β R E R . (1.14)The results are more intricate but qualitatively similar, and summarized in figure 1.In the quadrants β L , β R > π and β L , β R < π , the free energy is universal assuming asparse light spectrum (1.10). If we further restrict the mixed density of states as ρ ( E L , E R ) (cid:46) exp (cid:20) π (cid:114) ( E L + c
24 )( E R + c
24 ) (cid:21) ( E L < E R < , (1.15)then we can show that the universal behaviorlog Z ( β L , β R ) = c
24 max (cid:18) β L + β R , π β L + 4 π β R (cid:19) + O ( c ) (1.16)extends to the rest of the ( β L , β R ) plane outside of a small sliver near the line β L β R =4 π . The universal features of the free energy lead to corresponding universal featuresof the entropy S ( E L , E R ); it equals S Cardy ( E L , E R ) at high enough energies, and isbounded above in the intermediate range (see figure 1b). The derivation of the freeenergy is an iterative procedure that gradually eliminates larger portions of the ( β L , β R )plane. The sliver shown in the figure is what remains after three iterations, but weconjecture that more iterations would show that the free energy is universal for all6 L β R (cid:54) = 4 π . If so, then the Cardy entropy formula holds for all E L E R > (cid:0) c (cid:1) .The detailed comparison to 3d gravity is made in section 4. Finally in section 5we compare our results to symmetric orbifold CFTs, since certain symmetric orbifoldsare known to have holographic duals. We show that all symmetric orbifolds have freeenergy that satisfies (1.16) at all temperatures. We also show that the leading behaviorof the density of states is completely universal for all symmetric orbifold theories, andsaturates the bounds (1.10), (1.13) and (1.15). In this sense, symmetric orbifolds havethe maximally dense spectrum compatible with 3d gravity. c partition function We begin by analyzing the constraints of modular invariance on the partition functionat zero angular potential, β L = β R = β . Modular invariance requires Z ( β ) = Z ( β (cid:48) ) , β (cid:48) ≡ π β . (2.1)We denote the light states by L , and the medium and heavy states by H , L = { E ≤ (cid:15) } , H = { E > (cid:15) } , (2.2)and define the corresponding contributions to the partition function and its dual in theobvious way, Z [ L ] = (cid:88) L e − βE Z [ H ] = (cid:88) H e − βE (2.3) Z (cid:48) [ L ] = (cid:88) L e − β (cid:48) E Z (cid:48) [ H ] = (cid:88) H e − β (cid:48) E . Clearly the full partition function is Z ( β ) = Z [ L ] + Z [ H ] = Z (cid:48) [ L ] + Z (cid:48) [ H ] . (2.4)7 .2 Free energy Let us first discuss to what extent the light spectrum determines the free energy. Aspointed out in the introduction, in the holomorphic case, it is completely determinedby L . In the non-holomorphic case, clearly for very small temperature it is given bythe light states, or more precisely, by the vacuum. For very high temperature we knowfrom the usual Cardy formula that the behavior is again determined by the vacuumvia modular invariance. We want to investigate what we can say about intermediatetemperatures assuming that we know L completely.We can express modular invariance as Z [ L ] − Z (cid:48) [ L ] = Z (cid:48) [ H ] − Z [ H ] . (2.5)In a first step we want to bound Z [ H ]. Assume β > π . Then Z [ H ] = (cid:88) E>(cid:15) e ( β (cid:48) − β ) E e − β (cid:48) E ≤ e ( β (cid:48) − β ) (cid:15) Z (cid:48) [ H ] . (2.6)Therefore we have − Z (cid:48) [ H ](1 − e ( β (cid:48) − β ) (cid:15) ) ≥ Z [ H ] − Z (cid:48) [ H ] . (2.7)Using modular invariance, Z (cid:48) [ H ] ≤ (1 − e ( β (cid:48) − β ) (cid:15) ) − ( Z (cid:48) [ H ] − Z [ H ]) (2.8)= (1 − e ( β (cid:48) − β ) (cid:15) ) − ( Z [ L ] − Z (cid:48) [ L ]) ≤ (1 − e ( β (cid:48) − β ) (cid:15) ) − Z [ L ] , so in total we have Z [ H ] ≤ e ( β (cid:48) − β ) (cid:15) − e ( β (cid:48) − β ) (cid:15) Z [ L ] . (2.9)So for β > π we have for the free energylog Z [ L ] ≤ log Z ≤ log Z [ L ] − log(1 − e ( β (cid:48) − β ) (cid:15) ) . (2.10)By modular invariance we obtain an analogous expression for β < π .The two inequalities in (2.10) tell us that the free energy of a theory differs from thecontribution of the light states only within a universal range which does not dependon the theory. Crucially however this error is not bounded uniformly in β . The closer8he temperature is to the self-dual point (and the smaller we choose (cid:15) for that matter),the bigger an error we make. For β = β (cid:48) in particular we can only give a lower boundfor the free energy.Let us now consider families of CFTs depending on the central charge c , and in-vestigate the limit of large c . From (2.10) we can obtain the free energy of this familyas log Z ( β ) = log Z [ L ] + O (1) : β > π log Z (cid:48) [ L ] + O (1) : β < π (2.11)in the limit c → ∞ . We stress again that the error is not uniform in β : for large butfinite c , we can always find β close enough to 2 π so that the O (1) term is potentiallyof the same order as the light state contribution.This result is particularly powerful in a theory where the Z [ L ] is dominated by thevacuum state. In this caselog Z ( β ) = c β + O (1) : β > π π c β + O (1) : β < π . (2.12)It is straightforward to see that this holds if and only iflog (cid:88) < ∆ ≤ c/ (cid:15) e − β ∆ = O (1) , (2.13)for β > π . Allowing for o ( c ) corrections to the free energy, we can also choose to take (cid:15) → c limit (for example (cid:15) ∼ e − α √ c for some α > ρ ( E ) (cid:46) exp (cid:104) π ( E + c
12 ) (cid:105) ( E ≤ (cid:15) ) . (2.14) Let us now discuss what we can learn about the heavy spectrum of the theory from(2.12). Thermodynamically this means we are interested in the entropy S ( E ). This Approximation symbols are used with precise definitions: x ∼ y means lim x/y = 1, x ≈ y meanslim log x log y = 1, and depending on the context, inequalities x (cid:46) y mean lim x/y ≤ x = O ( c ) (forexample a free energy) or lim log x log y ≤
9e can obtain by performing the standard Legendre transform from F ( β ) to E ( S ).By the usual arguments, F ( β ) fixes E ( S ) completely, so naively we could expect that(2.12) gives the leading c behavior of S ( E ). It turns out that is not the case, and thatsubleading corrections to F can give large c corrections to S ( E ), so that we can onlyfix the leading order behavior of S ( E ) in a certain range of E .To see this more concretely, we compute the thermodynamic energy E ( β ) = − ∂ β log Z = − c + O (1) : β > π π c β + O (1) : β < π . (2.15)and thermodynamic entropy S ( β ) = (1 − β∂ β ) log Z = O (1) : β > π π c β + O (1) : β < π . (2.16)We see that at β = 2 π , E jumps from − c to c . For finite c of course E has to beregular. What this means is that a small change of order O (1) in log Z at β ∼ π willproduce a change of order c in E . This is the flip side of (2.10) which tells us that weshould only trust our approximations if β is far enough from the self-dual temperature.For the microcanonical density of states, this means that we should only trust ourapproximation if E is in the stable region > c . In that case we get the expectedCardy behavior S ( E ) ∼ π (cid:114) c E ( E > c
12 ) . (2.17)This entropy was obtained from thermodynamics, but it also holds for the microscopicdensity of states, ρ ( E ) ≈ e S ( E ) . (2.18)This is expected since c → ∞ behaves like a thermodynamic limit, but as usual itrequires some averaging to make precise. The details are relegated to appendix A.10 .4 Subleading saddles and the enigmatic range For reasons that will be clear when we compare to 3d gravity, we refer to the medium-energy states 0 < E < c
12 (2.19)as the ‘enigmatic’ range. The saddlepoint that dominates the partition function atlarge c never falls in this range, so S ( E ) is not universal. We can, however, easilyderive an upper bound. Setting β = 2 π in the expression Z ( β ) > ρ ( E ) e − βE gives S ( E ) (cid:46) πc πE . (2.20)This holds universally in theories obeying (2.13). We have not found a universal lowerbound — in particular, our results and the results in [11, 20] seem to be compatiblewith the possibility that there are no primary states within this range — but modularinvariance suggests a lower bound may hold in many theories. To see this, write thecontribution of heavy states to the partition function as Z [ H ] = Z (cid:48) [ L ] + (cid:0) Z (cid:48) [ H ] − Z [ L ] (cid:1) . (2.21)For β > π , the terms in parentheses dominate. Still, there is a contribution to thefirst term from the vacuum state, Z [ H ] = e c β (cid:48) + · · · . (2.22)If the heavy spectrum is precisely tuned so the dominant terms in parentheses cancelthis contribution, then Z [ H ] is completely unknown. If on the other hand we assumethis cancellation does not happen then we expect a corresponding contribution to thedensity of states, S ( E ) ∼ π (cid:112) c E + · · · . This suggests that in generic theories withoutfine tuning the entropy in the enigmatic range also satisfies a lower bound,2 π (cid:114) c E (cid:46) S ( E ) (cid:46) πc πE (0 < E < c
12 ) . (2.23)As we will see in section 5, there are theories which saturate the upper bound of(2.23). We can also construct leading order partition functions which saturate the lowerbound: Take for instance the partition function whose light spectrum only contains the11acuum representation, and whose heavy state contribution is given by Z [ H ] := Z (cid:48) [ L ]+subleading. We do not know of any examples which have fewer medium states thanthis. This certainly does not constitute a proof, and it may be possible to evade thelower bound if the heavy spectrum can be arranged to produce delicate cancellationswith the light spectrum. As mentioned in the introduction, the light spectrum of general CFTs can also beconstrained by the modular bootstrap. The idea of the modular bootstrap is to expandthe partition function around the self-dual temperature β = 2 π and then check (1.5)order by order. In [11], this technique was used to lowest order to prove that everyCFT has a state with scaling dimension ∆ = E L + E R + c ≤ c + 0 . . . . . Otherarguments such as extrapolating the result for holomorphic CFTs suggest that a tighterbound ∆ ∼ c may be possible. A more systematic numerical analysis of the modularbootstrap at relatively large values of c in [20] reproduces however the same asymptoticresult, ∆ (cid:46) c . (2.24)In our approach, this bound follows immediately from the fact that (2.17) is reliablemicroscopically. Here the reason that the bound is c and not c is that the stateswith c < ∆ < c never dominate the canonical ensemble. Our uncertainty about themedium-energy states (2.20) thus translates exactly into an uncertainty about the bestpossible bound.States above the lightest primary were incorporated into the modular bootstrapin [21]. Based on the pattern observed numerically, it was conjectured that there areactually an exponentially large number of primaries at or below ∆ ∼ c as c → ∞ ,specifically [21] log N primaries (∆ (cid:46) c (cid:38) πc . (2.25)For theories with a sparse light spectrum, the stronger boundlog N Cardyprimaries (∆ (cid:46) c ∼ πc ∼ c . However,by adding a large number of light states to a sparse light spectrum we can push up12he Cardy regime. Adding for example πc (1 + α ) light states at just below E = 0 with α >
0, the free energy is universal only for β < π (1 − α ). It then follows that (2.17)is valid only for E > c (1 − α ) − , so that it falls beyond the range of (2.25).Let us therefore drop our assumption on the light spectrum and see how this relaxesthe bound (2.26). We showed that Z [ H ] ≈ Z (cid:48) [ L ] ( β < π ) . (2.27)From this we would like to extract information about the microscopic density of statesat E (cid:46) c . The associated energy is E ( β ) ≡ − ∂ β log Z [ H ] ≈ π β ∂ β (cid:48) log Z (cid:48) [ L ] . (2.28)Since Z (cid:48) [ L ] has contributions only from − c ≤ E (cid:46) ∂ β (cid:48) log Z (cid:48) [ L ] ∈ [0 , c
12 ] . (2.29)It follows from (2.28) that as β → π , the energy E ( β ) must fall in the range [0 , c ]up to subleading corrections. Since Z [ H ] only has contributions from E >
0, it followsthat the dominating contribution E must satisfy0 (cid:46) E (cid:46) c , S ( E ) − πE ∼ log Z (cid:48) [ L ] (cid:38) πc , (2.30)where the lower bound in the last inequality is the contribution of the vacuum. Thelowest S ( E ) is achieved by assuming the dominant contribution comes from around E ∼
0, so S ( E ) (cid:38) πc . (2.31)The distinction between counting states and counting primaries does not matter toleading order in c , so this is a derivation of (2.25).13 Angular potential
Let us introduce the partition function with different left- and right-moving tempera-tures, Z ( β L , β R ) = Tr e − β L E L − β R E R . (3.1)We take β L and β R to be real, which corresponds to a real angular potential proportionalto β L − β R , and assume that the partition function is invariant under real modulartransformations, Z ( β L , β R ) = Z ( β (cid:48) L , β (cid:48) R ) , β (cid:48) L = 4 π β L , β (cid:48) R = 4 π β R . (3.2)This transformation at real temperatures is a consequence of modular invariance onthe Euclidean torus. Since we will rely on positivity, it is not straightforward to applyour argument directly to complex angular potential or to a chemical potential.The strategy to derive a universal free energy involves an iterative procedure, withresults summarized in figure 2. First, we use the results of section 2 to compute thefree energy in the quadrants β L,R > π and β L,R < π . This is then translated intonew constraints on the microsopic spectrum, and used to extend the universal freeenergy to a larger range of ( β L , β R ). This is iterated three times. The unknown range(the white sliver in figure 2) appears to shrink further with more iterations, so weconjecture that the universal behavior actually extends to the full phase diagram awayfrom β L β R = 4 π . We will first discuss the regime where both temperatures β L , β R are either high or low.This is the region labeled ‘first iteration’ in figure 2. It turns out that the constraintson the light states imposed in section 2 are enough to ensure universal behavior in thisregime. From eqs (2.20) we know that the large c density of states of such a theory is In Euclidean signature, the angular potential is imaginary, and Z ( τ, ¯ τ ) = Z ( − /τ, − / ¯ τ ) with τ = iβ L π complex and ¯ τ = τ ∗ . We may view Z ( τ, ¯ τ ) as a holomorphic function on a domain in C , with τ and ¯ τ independent complex numbers. The function f ( τ, ¯ τ ) = Z ( τ, ¯ τ ) − Z ( − /τ, − / ¯ τ )is also holomorphic, and vanishes for ¯ τ = τ ∗ . The Weierstrass preparation theorem implies thatthe vanishing locus of a holomorphic function must be specified (at least locally) by a holomorphicequation W ( τ, ¯ τ ) = 0. Since ¯ τ − τ ∗ = 0 is not holomorphic, it follows that f = 0. L Β R (cid:61) Π Third iterationSecond iterationFirst iteration0 2 Π Π Π Π Π Π Π Π Π Π Π Π β L β R Figure 2:
Derivation of universal free energy at finite angular potential.
We apply aniterative procedure to derive the universal free energy in larger and larger portions ofthe phase diagram. The shaded regions show the universal regions derived from thefirst three iterations. After three iterations the universal range encompasses all ( β L , β R )away from the white sliver.bounded by ρ ( E L , E R ) ≤ ρ ( E L + E R ) (cid:46) exp (cid:16) πc π ( E L + E R ) (cid:17) . (3.3)Therefore for β L,R > π , the total exponent in the partition function (cid:88) E L ,E R ρ ( E L , E R ) e − β L E L − β R E R (3.4)is bounded above by πc π ( E L + E R ) − β L E L − β R E R (cid:46) c
24 ( β L + β R ) . (3.5)15his implies that the vacuum exponentially dominates over other contributions to (3.1)at low temperatures, Z ( β L , β R ) ≈ exp (cid:104) c
24 ( β L + β R ) (cid:105) ( β L,R > π ) . (3.6)By modular invariance, we then immediately obtain at high temperatures Z ( β L , β R ) ≈ exp (cid:20) π c (cid:18) β L + 1 β R (cid:19)(cid:21) ( β L,R < π ) . (3.7) Just as in section 2, the free energies (3.6) and (3.7) lead to corresponding statementsabout the microscopic spectrum. The thermodynamic energies derived from this par-tition function are E L,R = − ∂ β L,R log Z ∼ π c β L,R β L,R < π β L,R > π (3.8)and the thermodynamic entropy is S = (1 − β L ∂ β L − β R ∂ β R ) log Z ∼ π c (cid:18) β L + 1 β R (cid:19) . (3.9)Legendre transforming to the microcanonical ensemble, this implies the Cardy behavior S ( E L , E R ) ∼ π (cid:114) c E L + 2 π (cid:114) c E R , ( E L,R > c
24 ) . (3.10)It is straightforward to prove using the method of appendix A that this Legendretransform is an accurate calculation of the microscopic density of states. For statesoutside the range (3.10), we can again only give an upper bound. The condition ρ ( E L , E R ) e − β L E L − β R E R ≤ Z ( β L , β R ) (3.11)gives the constraint: S ( E L , E R ) (cid:46) πc π ( E L + E R ) (all E L,R ) (3.12)16 ( E L , E R ) (cid:46) πc
12 + 2 πE L + 2 π (cid:114) c E R ( E R > c , all E L ) (3.13)and similarly for L ↔ R . Let us now turn to the regime where one temperature is high and the other is low.The situation here is more complicated, but we will derive universal behavior for partof this range. For this purpose however (2.14) is no longer good enough, and we needto replace it by something stronger. To this end it is useful to change the definition of‘light’ and ‘heavy’ states L : E L < E R < , H : E R > E L > . (3.14)The partition function is given by Z ( β L , β R ) = Z [ L ] + Z [ H ] (3.15)where the notation Z [ · · · ] means the contribution to Z ( β L , β R ) from the range specifiedin (3.14). Our strategy is then the same as in section 2: We first impose constraints onthe growth of the light states in such a way that their total contribution to leading orderis still given by the vacuum contribution, and then check if this is enough to ensurethat the full phase diagram is universal, or if the heavy states can make non-universalcontributions. For the first step we want to make sure that Z [ L ] ≈ exp (cid:104) c
24 ( β L + β R ) (cid:105) (3.16)for β L β R > π . This is the case if the growth of the light states is bounded by ρ ( E L , E R ) (cid:46) exp (cid:20) π (cid:114) ( E L + c
24 )( E R + c
24 ) (cid:21) ( E L < E R < . (3.17)To see this, we require ρ ( E L , E R ) ≤ e c ( β L + β R )+ β L E L + β R E R and then optimize over β L,R in the range β L β R > π . This guarantees that the light states give a universalcontribution to the free energy. Next we want to check if Z [ H ] is subleading in thisrange. For concreteness let us take β L > β (cid:48) R > π . The other case can be obtained by17xchanging L ↔ R . We then need to bound Z [ H ], and optimally we would hope tofind the analogue of (2.9), which would ensure that the heavy states never dominate inthis regime. Assuming only (3.17), we show in appendix B the slightly weaker result Z [ H ] (cid:46) exp (cid:104) πc
12 + c β (cid:48) R (cid:105) . (3.18)Unlike the case of zero angular potential, this is not enough to derive a universal freeenergy for all temperatures, as it is not dominated by (3.16) in the entire range weare considering. We do, however, find universal behavior in the range where Z [ H ] (cid:28) exp (cid:2) c ( β L + β R ) (cid:3) , i.e., for β L > π + β (cid:48) R − β R , in which case indeed Z ( β L , β R ) = Z [ L ] + Z [ H ] ≈ exp (cid:104) c
24 ( β L + β R ) (cid:105) . (3.19)In total we getlog Z ( β L , β R ) ∼ c
24 max( β L + β R , β (cid:48) L + β (cid:48) R ) ( β L , β R ) / ∈ S . (3.20)The sliver around β L β R = 4 π , S = { β L < π + β (cid:48) R − β R , β R < π } + L ↔ R + β L,R ↔ β (cid:48) L,R , (3.21)is the regime where the heavy states can contribute so that the free energy is not fixedso far. This extends the previous results to the region labeled ‘second iteration’ infigure 2.Turning to the microscopic spectrum, by the usual argument we obtain S ( E L , E R ) ∼ S Cardy ( E L , E R ) (0 < E R < c , E L > g ( E R )) (3.22) g ( E R ) ≡ E R − c
24 + c E R + c − E R (cid:112) E R /c . (3.23)We can also place an upper bound on a certain range where one energy is large andthe other is small. Let 0 < E R < c . In the inequality ρ ( E L , E R ) e − β L E L − β R E R < Z ,choose β R = π √ c √ E R , β (cid:48) L = 2 π + β R − β (cid:48) R (3.24)18hich falls in the regime where (3.20) is applicable. This implies S ( E L , E R ) (cid:46) g ( E R ) E L + g ( E R ) (0 < E R < c , E L >
0) (3.25)where g ( E R ) = 2 π √ cE R √ c − E R ) + √ cE R , g ( E R ) = πc
12 + πc (cid:112) E R /c + π (cid:114) c E R . (3.26)We can now perform another step in our iteration. Although the free energy is notuniversal inside the sliver S , (3.18) still imposes an upper bound, which we can use togive a stronger bound on the microscopic spectrum. The modular transform of (3.18)implies Z (cid:46) exp (cid:104) πc
12 + c β R (cid:105) (2 π < β R < β (cid:48) L < π + β R − β (cid:48) R ) . (3.27)Requiring ρ < Ze β R E R + β L E L and minimizing over β L , we find ρ ( E L , E R ) (cid:46) exp (cid:20) πc
12 + c β R + 4 π π + β R − β (cid:48) R E L + β R E R (cid:21) , (3.28)for any β R > π . The optimal bound is obtained by minimizing this expression over β R . This involves solving a quartic equation, so this step is performed numerically.However it is straightforward to see analytically that for E R = 0, this implies theasymptotic behavior ρ ( E L , (cid:46) exp (cid:20) π (cid:114) c E L (cid:21) ( E L → ∞ ) , (3.29)which is stronger than any of our previous bounds. When we apply this bound on thespectrum to the free energy, it reduces the size of the unknown range to a smaller sliver S , as shown in the ‘third iteration’ of figure 2 where S is the white region. The rangeof energies where the Cardy formula applies to the microsopic spectrum becomes veryclose to the line E L E R = ( c/ , as is shown in figure 1b.One can of course continue with this procedure iteratively. We conjecture that thesliver would collapse onto the line β L β R = 4 π . That is, we expect (but have notshown) that the leading free energy is universal everywhere away from the self-dualline, log Z ( β L , β R ) ∼ c
24 max( β L + β R , β (cid:48) L + β (cid:48) R ) ( β L β R (cid:54) = 4 π ) . (3.30)19n this case, using ρ ( E L , E R ) ≤ Ze β L E L + β R E R with (3.30) and optimizing the boundover β L,R implies S ( E L , E R ) (cid:46) π (cid:114)(cid:16) E L + c (cid:17) (cid:16) E R + c (cid:17) , (3.31)for all E L,R > − c . Moreover, repeating the arguments in section 3.2, we can transform(3.30) to the microcanonical ensemble to get S ( E L , E R ) ∼ S Cardy ( E L , E R ) , for E L E R > c . (3.32)The usual arguments (see appendix A) imply that this expression is accurate in themicrocanonical ensemble to leading order in 1 /c . Black holes provide UV data about quantum gravity, such as the approximate den-sity of states at high energy. Since their thermodynamics is determined by the lowenergy effective action, this means that any UV completion of quantum gravity sharesa number of universal features. In this section we will review some of the well knownuniversal features of 3d gravity, and show that they correspond exactly to the universalproperties of 2d CFT at large c derived above. Any theory of gravity+matter in AdS has (at least) two competing phases at finitetemperature: the BTZ black hole [22, 23] and a thermal gas. The black hole action is[24] log Z BH = π c (cid:18) β L + 1 β R (cid:19) , (4.1)where c = 3 (cid:96)/ G N , with (cid:96) the AdS radius and G N Newton’s constant. The thermalgas is the same classical solution as empty AdS but in a different quantum state. Itsclassical action is that of global AdS,log Z therm = c
24 ( β L + β R ) . (4.2)20oth of these classical solutions obey the same finite-temperature boundary condition,and in the canonical ensemble the partition function is a sum over such saddlepoints.Therefore, Z grav ( β ) ≈ e − I BTZ + e − I therm + · · · with I the Euclidean action, and we findlog Z grav ( β L , β R ) ≈ max (log Z BH , log Z therm ) . (4.3)There is a Hawking-Page phase transition at β L + β R = β (cid:48) L + β (cid:48) R [24, 25, 26].In principle, other saddlepoints should also be included. Even without matterfields, there is an infinite family of Euclidean solutions in pure gravity known as the SL (2 , Z ) black holes. These are obtained from the Lorentzian black hole by the analyticcontinuation to imaginary angular potential, τ = iβ L π , ¯ τ = − iβ R π , (4.4)followed by the SL (2 , Z ) transformation τ → aτ + bcτ + d . The resulting action islog Z = − iπc (cid:18) aτ + bcτ + d − a ¯ τ + bc ¯ τ + d (cid:19) . (4.5)Maximizing this expression over SL (2 , Z ) images leads to an intricate Euclidean phasediagram with an infinite number of phases tessellating the upper half τ -plane [6, 24, 27].However, in Lorentzian signature, β L,R are real and cosmic censorship imposes β L,R ≥ | Re τ | ≤ Im τ . (4.6)Within this range, the dominant phase is either Euclidean BTZ or thermal AdS. Inother words, when we compute the free energy for real angular potential, these arethe only two dominant phases in pure gravity. Allowing for matter fields could leadto new saddlepoints, but we do not know of any example where the new saddlepointsdominate the canonical ensemble.At zero angular potential, the gravity result (4.3) precisely agrees with our CFTresult (2.12) for all values of the temperature. At finite angular potential, the gravityformula was derived from CFT for all β L,R except within the sliver discussed in section3.3. This can be viewed as a prediction that in any theory of gravity+matter, BTZ orthermal AdS is indeed the dominant saddlepoint (at least outside the sliver).21 .2 BTZ black holes in the microcanonical ensemble
The known phases of 3d gravity in the microcanonical ensemble are much richer. Inaddition to BTZ black holes, there are other bulk solutions with O ( c ) entropy, includ-ing black holes localized on the internal manifold [18] and multicenter solutions [19].Within certain parameter ranges, these can have entropy greater than BTZ and thusdominate the microcanonical ensemble. Before turning to these more exotic solutionslet us compare the spectrum and entropy of the BTZ black hole to our CFT results.BTZ black holes have energies E L,R = π c β L,R , (4.7)and entropy given by the Cardy formula S BH ( E L , E R ) = S Cardy ( E L , E R ) . (4.8)They exist for all E L,R ≥ E L = E R = E/
2. The black holes exist and have Cardy entropy for E ≥
0, but in the CFT we onlyderived the Cardy entropy for
E > c (see section 2). In fact this is perfectly consistent:the black holes with 0 < E < c are unstable in the canonical ensemble. These unstableblack holes eventually tunnel into the gas phase. Therefore within this range the blackholes are subleading saddlepoints, much like the subleading saddles in CFT discussedin section 2.4. There we argued that, generically (assuming no delicate cancellations),the subleading saddle in CFT gives a reliable contribution to the microscopic densityof states; this contribution corresponds exactly to the unstable black holes.The situation at finite real angular potential is similar. In the regime where wefound a universal CFT entropy given by the Cardy formula, it agrees with the entropyof rotating BTZ (4.8). Outside the universal regime, we derived an upper bound onthe CFT density of states which is satisfied by (4.8). Subleading saddlepoints in theCFT with rotation were not discussed, but are easily seen to correspond to unstableblack holes with β L β R > π . 22 .3 Enigmatic phases in the microcanonical ensemble As mentioned above, there are known solutions in 3d gravity with entropy greater thanthat of BTZ at the same energies, S enigma ( E L , E R ) > S Cardy ( E L , E R ) . (4.9)The examples we will consider are the S -localized black holes in [18] and the moultingblack holes in [19]. These are similar to the enigmatic phases discussed in [28, 29] sowe adopt this terminology.We will see that the enigma saddlepoints fit nicely with our CFT results. They fallin the intermediate range 0 ≤ E L,R ≤ c , where we found that the CFT entropy is notuniversal but obeys S Cardy ( E L , E R ) ≤ S CF T ( E L , E R ) ≤ cπ π ( E L + E R ) . (4.10)The upper bound holds universally, while the lower bound holds provided we assumethat subleading saddlepoints are not cancelled. The upper bound is simply the state-ment that these states never dominate the canonical ensemble.The relevant solutions in [18] are BPS solutions of M-theory compactified on S × CY .In the decoupling limit, the 5d geometry is asymptotically an S fiber over AdS . Froma higher-dimensional perspective the twisting of the fiber is proportional to angularmomentum; from the 3d gravity or dual CFT point of view, twisting corresponds to SU (2) R charge. At high energies, the highest-entropy BPS solution with these asymp-totics is an uncharged extremal BTZ × S with energies ( E L ,
0) and entropy given bythe Cardy formula. However there is another solution in which the black hole is local-ized on the S . This solution carries SU (2) R charge but can nonetheless dominate overuncharged BTZ. (Multicenter localized black holes, including some with zero SU (2) R charge, are also discussed in [18] but these have lower entropy.) The localized solutionexists for − c < E L < c and at the BTZ threshold E L = E R = 0 it has entropy S enigma = πc √ . (4.11)The scaling of (4.11) with c indicates that this solution has more entropy than BTZ in23ome range just above the threshold. The transition point is [18] E critL ≈ . c . (4.12)Thus the microscopic entropy is greater than the Cardy formula for 0 < E L < E critL ,and falls within our CFT bounds (4.10). As expected from CFT, the localized blackhole never dominates the canonical ensemble.As a second example we turn to the two-center solution of IIB supergravity com-pactified on T constructed in [19]. This solution, which is described as a BMPV blackhole surrounded by a supertube, has near horizon geometry AdS × S so our resultsshould apply. The entropy of the new solution (spectral flowed to the NS sector) is S ( E L ) = 2 π (cid:18)(cid:114) c − (cid:114) c − E L (cid:19) (cid:114) E L + c , (4.13)and it exists for − c < E L < c . This dominates over the Cardy entropy in a smallwindow above E L = 0 up to the critical value E critL ≈ . c . (4.14)Once again these states obey (4.10) and never dominate the canonical ensemble.The gravity examples that we have considered here are supersymmetric, but ourCFT results suggest that entropy above the Cardy value at intermediate energies isa generic feature of large c CFTs. Since we did not find a universal answer for S CF T in this range, we cannot check the explicit formula for S enigma from CFT beyondconfirming that it obeys the bounds. Indeed, we expect that S enigma depends on thespecific microscopic theory, and in particular it may depend on the coupling constant. So far our discussion has been general, as it applies to any unitary, modular invariantCFT with large c and sparse low-lying spectrum. We now turn to a specific class ofexamples, symmetric orbifold CFTs, to illustrate how these theories fit into our generalpicture. Symmetric orbifold CFTs have been studied extensively in the context of theD1-D5 system. They were used in the original computation of [1], and underlie many24f the more recent successful precision tests of black hole microstate counting in stringtheory summarized for example in [30, 31]. We will show that all symmetric orbifoldtheories have the universal free energy (3.30), which of course implies that they satisfythe constraints on the spectrum (3.32) and (3.31). In fact symmetric orbifolds saturatethe bound (3.31). This shows that in a sense they are most dense theories that are stillcompatible with the universal free energy (3.30).Starting with any ‘seed’ theory C , the symmetric orbifold C N /S N consists of N copies of the original theory, orbifolded by the permutation group. If we take the seedtheory to be the sigma model with target space M , where M = K T , then thesymmetric orbifold CFT is holographically dual to IIB string theory on AdS × S × M . The seed theory has central charge c = 6 and the orbifold has c = N c . Theorbifold theory itself is the weak coupling limit and does not have a good geometricaldescription, but in principle we can turn on exactly marginal deformations in the CFTto reach a point in moduli space with a semiclassical gravity description.The spectrum of the D1-D5 CFT depends on the moduli, so the spectrum of thesymmetric orbifold need not match the spectrum of supergravity, while certain super-symmetric quantities (such as the elliptic genus) are protected and can be successfullymatched on the two sides of the duality. Relatively little is known about the non-supersymmetric features of the CFT at strong coupling, except what is fixed entirelyby symmetry or has been deduced from the gravity picture. On the other hand, theresults of sections 2 - 3 do not require supersymmetry, and apply to the D1-D5 CFTin the gravity limit (if our assumptions about the light spectrum are satisfied) as wellas at the orbifold point.In this section we will compute the density of states at the orbifold point, foran arbitrary seed theory. We show that it satisfies our assumptions about the lightspectrum (1.10, 1.15), and confirm that the heavy spectrum is consistent with ourresults. Symmetric orbifolds also saturate the upper bound (3.31) in the enigmaticrange 0 < E < c , demonstrating that this bound is optimal.Some of these results have previously been derived using the long string descriptionof the D1-D5 system, but the explicit orbifold CFT computation is instructive to makeprecise exactly when the long string picture is reliable. The result in section 5.2 forthe spectrum of light states appears to be new.25 .1 Partition function The partition function of a symmetric orbifold is determined by the seed theory. Letus choose a seed theory C and denote its partition function by Z = Tr q L − c ¯ q L − c = q − c / ¯ q − c / (cid:88) h, ¯ h ∈ I d ( h, ¯ h ) q h ¯ q ¯ h , (5.1)where the sum is over a discrete spectrum I of conformal dimensions, h, ¯ h ≥
0. TheEuclidean notation is related to the Lorentzian notation in the rest of the paper via q = e − β L , ¯ q = e − β R (5.2)i.e., q = e πiτ , ¯ q = e − πi ¯ τ , τ = iβ L π , ¯ τ = − iβ R π . The partition function Z N of thesymmetric orbifold C N /S N , Z N = q − c N/ ¯ q − c N/ (cid:88) h, ¯ h d N ( h, ¯ h ) q h ¯ q ¯ h , (5.3)is obtained as usual by projecting out states that are not invariant under permutations,and introducing twisted sectors. In practice it can be extracted from its generatingfunction, for which a relatively simple expression exists [32, 33]: Z ≡ (cid:88) N ≥ p N Z N = (cid:89) n> (cid:89) h, ¯ h ∈ I (1 − p n q ( h − c / /n ¯ q (¯ h − c / /n ) − d ( h, ¯ h ) δ ( n ) h − ¯ h . (5.4)Here roughly speaking n corresponds to the length of the twisted sectors, and δ ( n ) h − ¯ h = h − ¯ h = 0 mod n N symmetric orbifolds has universal thermodynamic behaviorfor τ in the upper half complex plane. In appendix C.1 we repeat this argument forreal angular potential to provelog Z N = c
24 max ( β L + β R , β (cid:48) L + β (cid:48) R ) + O (1) , (5.6)26or all β L,R >
0, where throughout this section c = c N . This is somewhat strongerthan (3.20) derived in section 3.3, because it also applies in the sliver S . Let us now discuss the spectrum of the theory. We established above that the freeenergy satisfies (3.30), from which it follows that the bound (3.31) is satisfied. Inappendix C.2, we prove that this bound is actually saturated, S ( E L , E R ) ∼ π (cid:114) ( E L + c
24 )( E R + c
24 ) for E L E R < c . (5.7)Together with (3.32) this fixes the spectrum of symmetric orbifold theories completely,and shows that it is completely universal, i.e., depends only on the central charge. Adetailed derivation of (5.7) can be found in the appendix. The general idea is that weare counting the excitations of N strings that can join into longer strings. Long stringshave Cardy entropy in the range (3.32). For a given ( E L , E R ), the entropy (5.7) comesfrom the sector with M short strings and one long string (made of N − M short ones),maximized over M ≤ N .The entropy at energy E = E L + E R is dominated by E L,R = E/
2, which gives S ( E ) ∼ πc πE (0 < E < c
12 ) . (5.8)Thus the symmetric orbifold saturates our upper bound in (2.23) in the enigmaticregime. Pure gravity, on the other hand, saturates the lower bound, while knownUV-complete theories of 3d gravity+matter appear to fall in between, as discussedin section 4.3. This implies that going to strong coupling in CFT lifts some of theenigmatic states (similar conclusions were reached in [18, 19]). Acknowledgments
We thank Dionysios Anninos, Daniel Friedan, Matthias Gab-erdiel, Alex Maloney, Don Marolf, Greg Moore, Hirosi Ooguri, Eric Perlmutter, An-drea Puhm, and Andy Strominger for useful discussions. TH is supported by theNational Science Foundation under Grant No. NSF PHY11-25915. CAK is supportedby the Rutgers New High Energy Theory Center and by U.S. DOE Grants No. DOE-SC0010008, DOE-ARRA-SC0003883 and DOE-DE-SC0007897. CAK thanks the Har-vard University High Energy Theory Group for hospitality. BS is supported in part by27 Dominic Orr Graduate Fellowship and by U.S. DOE Grant No. DE-SC0011632. BSwould like to thank the Kavli Institute For Theoretical Physics for hospitality.
A Density of states in the microcanonical ensemble
The exact density of states is a sum of delta functions, so to make equations like ρ ( E ) ≈ e S ( E ) precise requires averaging over an interval. For this we introduce n u,δ = N states ( c u − δ < E < c u + δ ) , (A.1)which counts the number of states in an interval around some energy. For the expo-nential dependence, the distinction between number n u,δ and number density ρ is notimportant. We will take u fixed and independent of c . The size of the interval δ on theother hand needs to increase with c . Choosing the correct scaling with c is actuallycrucial. It turns out that we need it to scale as δ ∼ c α with < α <
1. With thisscaling we can show thatlog n u,δ ≤ πc u ) + O ( c α ) : 0 < u < n u,δ = πc √ u O ( c α ) : u > , (A.3)that is, we show that (2.17) and (2.20) indeed hold microscopically. This already showswhy we needed to pick α <
1, since otherwise the density would obtain corrections oforder c or bigger. To prove (A.2) it will be useful to decompose the heavy spectrum H into H = (cid:110) (cid:15) < E < cu − δ (cid:111) , H = (cid:110) cu − δ ≤ E < cu
12 + δ (cid:111) , H = (cid:110) cu
12 + δ ≤ E (cid:111) . (A.4)Let us first construct the upper bound. For β < π we have β (cid:48) c
12 = log Z ( β ) + O (1) = log Z [ H ] + O (1) ≥ log Z [ H ] + O (1) ≥ log (cid:0) n u,δ e − β ( c u + δ ) (cid:1) + O (1) (A.5)28o that log n u,δ ≤ π c β + β ( c u + δ ) + O (1) . (A.6)We can optimize this bound by picking β = 2 π/ √ u if u >
1, or β = 2 π if u <
1. Using δ = O ( c α ) it follows thatlog n u,δ ≤ πc √ u O ( c α ) ( u > , (A.7)log n u,δ ≤ πc u ) + O ( c α ) ( u < . (A.8)To derive (A.3), we must show that (A.7) is saturated. The idea is again to pick aspecific β so that the main contribution to Z [ H ] comes from the states at u . Setting β = 2 π/ √ u , we first want to show thatlog Z [ H ] = log Z [ H ] + O (1) . (A.9)To this end we estimatelog Z [ H ] ≤ πc (cid:112) u + 12 δ/c − u + 12 δ/c √ u ) + O (log c ) = πc √ u − πδ u / c + o ( c α − ) , (A.10)where in the first equality we have used that the total sum differs from its maximalsummand only by a polynomial prefactor. Since the first subleading term comes witha negative sign and grows as c α − , it follows from Z [ H ] = πc √ u + O (1) that Z [ H ] /Z [ H ] → . (A.11)We can show a similar result for Z [ H ]: Here we split H into H = { (cid:15) < E < } and H = (cid:8) < E < c u − δ (cid:9) . The contribution from H we can estimate using (A.8) aslog Z [ H ] ≤ πc √ u (1 − (1 − u − / ) ) + O (log c ) , (A.12)and the contribution from from H using (A.7), which gives (A.10) but with − δ insteadof δ . Combining these three estimates, (A.9) follows, and then we can use Z [ H ] ≤ log n u,δ e − cπ √ u ( u − δ/c ) (A.13)29o obtain the lower bound that leads to (A.3). B Mixed temperature calculations
This appendix contains the details of the calculation discussed in section 3.3. Weassume β L > π > β R and β L ≥ β (cid:48) R , which in particular implies β L + β R ≥ π . Toestablish (3.18), we need to bound Z [ H ]. We decompose it into 4 terms T = Z [ c < E L , c < E R ] (B.1) T = Z [0 < E L < c , c < E R ] (B.2) T = Z [ c < E L , < E R < c
24 ] (B.3) T = Z [0 < E L < c , < E R < c
24 ] , (B.4)and then apply the various bounds (3.10), (3.12) and (3.13). For T we use (3.10), T (cid:46) (cid:90) ∞ c dE L (cid:90) ∞ c dE R exp (cid:20) π (cid:114) c E L + 2 π (cid:114) c E R − β L E L − β R E R (cid:21) ≈ exp (cid:104) c
24 (4 π − β L + β (cid:48) R ) (cid:105) (cid:28) exp (cid:104) c
24 ( β L + β R ) (cid:105) , (B.5)the leading contribution coming from E L = c , E R = π c β R > c . The term T is in therange where the bound (3.13) applies. Thus T (cid:46) (cid:90) ∞ c dE R (cid:90) c dE L exp (cid:20) πc
12 + 2 πE L + 2 π (cid:114) c E R − β L E L − β R E R (cid:21) ≈ e πc/ (cid:90) ∞ c dE R exp (cid:20) π (cid:114) c E R − β R E R (cid:21) ≈ exp (cid:104) πc
12 + c β (cid:48) R (cid:105) . (B.6)30he dominant term here comes from E L = 0 , E R = π c β R . For T we apply the flippedversion of (3.13), T (cid:46) (cid:90) ∞ c dE L (cid:90) c dE R exp (cid:20) πc
12 + 2 πE R + 2 π (cid:114) c E L − β L E L − β R E R (cid:21) ≈ e c (4 π − β R ) (cid:90) ∞ c dE L exp (cid:20) π (cid:114) c E L − β L E L (cid:21) ≈ exp (cid:104) c
24 (8 π − β L − β R ) (cid:105) (cid:28) exp (cid:104) c
24 ( β L + β R ) (cid:105) . (B.7)Finally for T we use (3.12) to get T (cid:46) (cid:90) c dE R (cid:90) c dE L exp (cid:104) πc π ( E L + E R ) − β L E L − β R E R (cid:105) ≈ exp (cid:104) πc − c β R (cid:105) (cid:28) T , (B.8)where the dominant contribution comes from E L = 0 and E R = c . In total we haveshown Z [ H ] (cid:46) exp (cid:104) πc
12 + c β (cid:48) R (cid:105) . (B.9) C Symmetric orbifold calculations
C.1 Free energy
In this appendix we use (5.4) to derive the large- N phases of the symmetric orbifold atreal angular potential claimed in (5.6). The argument parallels the Euclidean discussionin [32] so we will be brief. Suppose β L > β (cid:48) R , so the first term in (5.6) dominates. Definethe remainder R N = log (cid:0) Z N e − c ( β L + β R ) (cid:1) , (C.1)which gives the contribution to the free energy of all the states other than the vacuum.We will prove that this is a subleading contribution by showing that R ∞ is finite.Using (5.4), it is straightforward to derive (see [34] and in particular section 2.2.3 andappendix A.2 of [32]) R ∞ = (cid:88) n> (cid:88) k> (cid:88) h, ¯ h ∈ I (cid:48) k d ( h, ¯ h ) δ ( n ) h − ¯ h q kh/n + k c ( n − /n ) ¯ q k ¯ h/n + k c ( n − /n ) (C.2)31here the primed sum indicates that we skip the term with n = 1 , h = ¯ h = 0. Everyterm is positive so in checking convergence we can ignore the delta and exchange sumsat will. The n th term for n > (cid:88) k> k exp (cid:20) − c kn
24 ( β L + β R ) (cid:21) Z (cid:18) kn β L , kn β R (cid:19) . (C.3)To proceed we will bound the seed partition function Z that appears in this expressionby Z ( β L , β R ) ≤ p ( β L , β R ) e c ( β L + β R ) e c ( β (cid:48) L + β (cid:48) R ) , (C.4)where p ( β L , β R ) grows at most polynomially. To see this note that the standard Cardyformula tells us that for all h and ¯ hρ ( h + ¯ h ) ≤ N e π √ c ( h +¯ h ) / (C.5)for some constant N . (This follows from the fact that (C.5) holds asymptotically forlarge h + ¯ h , so we simply choose N large enough so that it holds everywhere.) It followsthat Z ( β L , β R ) = e c ( β L + β R ) (cid:90) dhd ¯ hρ ( h, ¯ h ) e − β L h e − β R ¯ h ≤ N e c ( β L + β R ) (cid:90) dhd ¯ he π √ ch/ − β L h e π √ c ¯ h/ − β R ¯ h ≤ p ( β L , β R ) e c ( β L + β R ) e c ( β (cid:48) L + β (cid:48) R ) (C.6)where we have used ρ ( h, ¯ h ) ≤ ρ ( h + ¯ h ). Plugging this into (C.3) we can bound theexponential factors in the terms for k > , n > e − nkc ( β L + β R − n ( β L + β R ) − k ( β (cid:48) L + β (cid:48) R ) ) ≤ e − nkc ( ( β L + β R ) − ( β (cid:48) L + β (cid:48) R ) ) . (C.7)Since by assumption β L + β R > β (cid:48) L + β (cid:48) R the double sum over k > , n > n = 1 , k > n = 1,so that the exponent of the first factor in (C.4) is given by the lowest state of thetheory instead. The sum for k = 1 , n > n large enough we canestimate Z (cid:18) β L n , β R n (cid:19) = Z ( nβ (cid:48) L , nβ (cid:48) R ) ≤ Ke nc ( β (cid:48) L + β (cid:48) R ) (C.8)32here we can use the last inequality if n is large enough so that nβ (cid:48) L , nβ (cid:48) R > π .Convergence then follows from β L + β R > β (cid:48) L + β (cid:48) R . It follows that when β L > β (cid:48) R ,the free energy is indeed given only by the vacuum contribution c ( β L + β R ), and bymodular invariance we obtain (5.6). C.2 Spectrum
We now derive the low-energy density of states (5.7). We have already argued thatthis is an upper bound, so the strategy is to find a contribution saturating this bound.For this we will use the fact that the generating function (5.4) can be reorganized as[33, 35] Z = exp (cid:32)(cid:88) L> p L L T L Z (cid:33) , (C.9)where T L is the (unnormalized) Hecke operator. The definition of T L can be foundin [32], but for our purposes we just need one basic fact: If Z is a modular-invariantpartition function with positive coefficients d ( h, ¯ h ) >
0, then T L Z is also modularinvariant, and can be expanded as T L Z = q − c L/ ¯ q − c L/ (cid:88) h, ¯ h d T L ( h, ¯ h ) q h ¯ q ¯ h (C.10)with non-negative weights h, ¯ h ≥ d T L > N , the degeneracy of states in the symmetric orbifold d N can be extracted from (C.9) by a minor extension of the argument in section 2.2.1 of[32]. Let ˜ p = p ( q ¯ q ) − c / . (C.11)Separating the contribution from the ground states in each sector, Z = exp (cid:88) L> ˜ p L L + (cid:88) L> ˜ p L L (cid:88) h, ¯ h> d T L ( h, ¯ h ) q h ¯ q ¯ h (C.12)= (cid:32)(cid:88) K ≥ ˜ p K (cid:33) (cid:88) L> ˜ p L L (cid:88) h, ¯ h> d T L ( h, ¯ h ) q h ¯ q ¯ h + · · · . (C.13)The corrections indicated by dots come with positive coefficients, so if we ignore the33orrections then the coefficient of ˜ p N gives a lower bound on the orbifold degeneracy: d N ( h, ¯ h ) ≥ N (cid:88) L =1 L d T L ( h, ¯ h ) . (C.14)In the effective string language, this equation has a simple interpretation. We arecounting the degeneracy at level ( h, ¯ h ) of N strings that are allowed to join into longerstrings. The L th term in (C.14) is the degeneracy in the sector with one long stringand N − L short strings.Suppose for a moment that the Cardy formula applies to T L Z , so d T L ( h, ¯ h ) ≈ exp (cid:34) π (cid:114) c L h − c L
24 ) + 2 π (cid:114) c L h − c L
24 ) (cid:35) . (C.15)The maximum in (C.14) occurs at L = 24 h ¯ hc ( h + ¯ h ) , (C.16)which as long as L ≤ N would give d N ( h, ¯ h ) (cid:38) exp (cid:104) π (cid:112) h ¯ h (cid:105) for h ¯ hh + ¯ h ≤ c . (C.17)To confirm that the argument given is reliable, we must show that the Cardy behavior(C.15) holds for (C.16). Note that d T L ( h, ¯ h ) ≤ Ld L ( h, ¯ h ) , (C.18) i.e., to leading order the L th Hecke transform does not have more states than the L thsymmetric orbifold. It is thus straightforward to show using (5.6) that it too has theuniversal free energy behaviorlog T L Z ∼ c L
24 max ( β L + β R , β (cid:48) L + β (cid:48) R ) (C.19) The Cardy formula applies to the density of states, not necessarily to the degeneracy at a particularlevel. To be precise, in these expressions we should average d N and d T L over a range ( h ± δ, ¯ h ± δ ) asin appendix A. We will not write this explicitly but it does not change the final answer. L → ∞ . Thus the Cardy formula (C.15) applies when E L E R > ( c L ) / i.e., h ¯ hh + ¯ h ≥ c L . (C.20)The choice (C.16) falls at the edge of this range, so the bound (C.17) is indeed valid.Translating to energies E L = h − c , E R = ¯ h − c , (C.17) implies that (3.31) is saturated,which implies (5.7). Finally if L > N , then d T N provides the optimal bound, d N ( h, ¯ h ) (cid:38) exp (cid:20) π (cid:114) c h − c
24 ) + 2 π (cid:114) c h − c
24 ) (cid:21) for h ¯ hh + ¯ h > c . (C.21)This is identical to the result we derived from the free energy (3.32). References [1] A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking En-tropy,” Phys. Lett. B , 99 (1996) [hep-th/9601029].[2] J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant The-ories,” Nucl. Phys. B , 186 (1986).[3] A. Strominger, “Black Hole Entropy from Near Horizon Microstates,” JHEP ,009 (1998) [hep-th/9712251].[4] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization ofAsymptotic Symmetries: An Example from Three-Dimensional Gravity,” Com-mun. Math. Phys. , 207 (1986).[5] M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Correspon-dence,” Phys. Rev. D , 124008 (2009) [arXiv:0809.4266 [hep-th]].[6] R. Dijkgraaf, J. M. Maldacena, G. W. Moore and E. P. Verlinde, “A Black HoleFarey Tail,” arXiv:hep-th/0005003.[7] J. Manschot and G. W. Moore, “A Modern Farey Tail,” Commun. Num. Theor.Phys. , 103 (2010) [arXiv:0712.0573 [hep-th]].[8] E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].[9] M. R. Gaberdiel, S. Gukov, C. A. Keller, G. W. Moore and H. Ooguri, “ExtremalN=(2,2) 2D Conformal Field Theories and Constraints of Modularity,” Commun.Num. Theor. Phys. , 743 (2008) [arXiv:0805.4216 [hep-th]].[10] A. Cappelli, C. Itzykson and J. B. Zuber, “Modular Invariant Partition Functionsin Two-Dimensions,” Nucl. Phys. B , 445 (1987).3511] S. Hellerman, “A Universal Inequality for CFT and Quantum Gravity,” JHEP , 130 (2011) [arXiv:0902.2790 [hep-th]].[12] T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955[hep-th].[13] T. Barrella, X. Dong, S. A. Hartnoll and V. L. Martin, “Holographic entanglementbeyond classical gravity,” JHEP , 109 (2013) [arXiv:1306.4682 [hep-th]].[14] B. Chen and J. -J. Zhang, “On short interval expansion of R´enyi entropy,” JHEP , 164 (2013) [arXiv:1309.5453 [hep-th]].[15] E. Perlmutter, “Comments on Renyi entropy in AdS /CFT ,” JHEP , 052(2014) [arXiv:1312.5740 [hep-th]].[16] A. L. Fitzpatrick, J. Kaplan and M. T. Walters, “Universality of Long-DistanceAdS Physics from the CFT Bootstrap,” arXiv:1403.6829 [hep-th].[17] T. Faulkner, “The Entanglement Renyi Entropies of Disjoint Intervals inAdS/CFT,” arXiv:1303.7221 [hep-th].[18] J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, “BlackHole Bound States in AdS(3) x S**2,” JHEP , 050 (2008) [arXiv:0802.2257[hep-th]].[19] I. Bena, B. D. Chowdhury, J. de Boer, S. El-Showk and M. Shigemori, “MoultingBlack Holes,” JHEP , 094 (2012) [arXiv:1108.0411 [hep-th]].[20] D. Friedan and C. A. Keller, “Constraints on 2d CFT Partition Functions,” JHEP , 180 (2013) [arXiv:1307.6562 [hep-th]].[21] J. D. Qualls and A. Shapere, “Bounds on Operator Dimensions in 2D ConformalField Theories,” arXiv:1312.0038 [hep-th].[22] M. Ba˜nados, C. Teitelboim and J. Zanelli, “The Black Hole in Three DimensionalSpace Time,” Phys. Rev. Lett. , 1849 (1992) [hep-th/9204099].[23] M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the 2+1Black Hole,” Phys. Rev. D , 1506 (1993) [gr-qc/9302012].[24] J. M. Maldacena and A. Strominger, “AdS3 Black Holes and a Stringy ExclusionPrinciple,” JHEP , 005 (1998) [hep-th/9804085].[25] S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in Anti-de SitterSpace,” Commun. Math. Phys. , 577 (1983).[26] E. Witten, “Anti-de Sitter Space, Thermal Phase Transition, and Confinement inGauge Theories,” Adv. Theor. Math. Phys. , 505 (1998) [hep-th/9803131].[27] A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three Di-mensions,” JHEP , 029 (2010) [arXiv:0712.0155 [hep-th]].[28] J. P. Gauntlett and J. B. Gutowski, “Concentric Black Rings,” Phys. Rev. D ,025013 (2005) [hep-th/0408010]. 3629] F. Denef and G. W. Moore, “Split States, Entropy Enigmas, Holes and Halos,”JHEP , 129 (2011) [hep-th/0702146 [HEP-TH]].[30] A. Sen, “Black Hole Entropy Function, Attractors and Precision Counting of Mi-crostates,” Gen. Rel. Grav. , 2249 (2008) [arXiv:0708.1270 [hep-th]].[31] A. Sen, “Logarithmic Corrections to Rotating Extremal Black Hole Entropy inFour and Five Dimensions,” Gen. Rel. Grav. , 1947 (2012) [arXiv:1109.3706[hep-th]].[32] C. A. Keller, “Phase Transitions in Symmetric Orbifold CFTs and Universality,”JHEP , 114 (2011) [arXiv:1101.4937 [hep-th]].[33] R. Dijkgraaf, G. W. Moore, E. P. Verlinde and H. L. Verlinde, “Elliptic Genera ofSymmetric Products and Second Quantized Strings,” Commun. Math. Phys. ,197 (1997) [hep-th/9608096].[34] J. de Boer, “Large N Elliptic Genus and AdS/CFT Correspondence,” JHEP ,017 (1999) [hep-th/9812240].[35] P. Bantay, “Symmetric Products, Permutation Orbifolds and Discrete Torsion,”Lett. Math. Phys.63