Corrections in the relative entropy of black hole microstates

Inspired by the recent work of Bao and Ooguri (BO), we study the distinguishability of the black hole microstates from the thermal state as captured by the average of their relative entropies: the Holevo information. Under the assumption that the vacuum conformal block dominates the entropy calculation, BO find that the average relative entropy vanishes on spatial regions less than half the size of the CFT. However, vacuum block dominance fails for some microstates of the $M=0$ BTZ black hole. We show that this renders the average relative entropy nonzero even on infinitesimal intervals at $O(c^0)$.


Introduction
Relative entropy is among the most powerful tools available for characterizing the distinguishability of two quantum states, and has recently found broad application in holography: its equality on bulk and boundary convincingly identifies the entanglement wedge as the dual to a given boundary region [1]; its positivity and monotonicity leads to proofs of the average null energy condition [2] and the generalized second law [3]; its expansion about the vacuum leads to a derivation of Einstein's equations from the dynamics of entanglement [4][5][6][7][8][9]. One might hope to use relative entropy to further probe questions related to the black hole information paradox, such as how easily the black hole microstates can be distinguished from the coarse-grained thermal state of the black hole.
Unfortunately, despite our ability to make general statements about its properties, relative entropy is not an easy quantity to compute in black hole microstates. Unlike its less-useful cousins -the entanglement and Renyi entropies -it lacks a simple replica expression as a fourpoint function, and instead requires calculation of n-point correlators of heavy operators [10,11].
Bao and Ooguri [12] have recently identified an analog quantity which can be more easily computed -the Holevo information χ. Given an ensemble ρ = i p i ρ i of density matrices ρ i , the Holevo information is the average entropy of the ρ i relative to ρ: (1.1) Since the average relative entropy χ is equal to the average difference in von Neumann entropy between ρ and the ρ i , computation of χ reduces to the computation of tractable (heavy-light) four-point functions. Bao and Ooguri estimated this quantity for the thermal density matrix of a black hole in anti-de Sitter space at temperature 1/β: Here the microstates ρ i are the energy eigenstates |E i E i | weighted by a Boltzmann factor p i ∝ e −βE i . They can be perfectly distinguished by an observer who has access to the entire dual CFT, but a less omniscient observer has a more difficult task. Within a ball-shaped subregion A of the CFT with diameter , average distinguishability is characterized by the Holevo information of the reduced density matrix ρ obtained by tracing over the complement of the region: ρ ≡ trĀ ρ. (1.4) In general dimension there is no known way to compute the S(ρ ) in field theory and so one must resort to holographic arguments, using the HRRT [13,14] formula in either the black hole background or the (generically unknown) background created by a heavy operator. However, in large-c CFTs the vacuum block contribution to the heavy-heavy-light-light four-point function can be computed using the bootstrap [15], which in turn makes it possible to compute S(ρ i, ) in CFT [16] if one makes the assumption that conformal blocks other than the vacuum do not contribute. Bao and Ooguri use this approximation to obtain [12] χ(ρ BH, ) ≈ sinh(π /β) sinh(π(2πR− )/β) πR < < crit S BH ≥ crit , (1.5) where c is the central charge of the CFT and crit is a critical length that follows from the homology constraint in the Ryu-Takayanagi formula. We have reintroduced the radius R of the spatial circle of the CFT and corrected several factors in this expression. The vacuum block does not dominate at subleading orders in 1/c [17] and so one would naturally expect (1.5) to receive corrections. However, we will show that 1/c corrections to (1.5) are also induced by competition with the vacuum block in S(ρ i, ) at leading order in certain states. This generates 1/c corrections to the average relative entropy once the rarity of these states is taken into account.
We will study these corrections in the β → ∞ limit corresponding to the M = 0 BTZ black hole [18,19], which has a microscopic description in terms of D-branes whose near-horizon geometry is AdS 3 × S 3 [20] (the S 3 is necessary in the microscopic formulation of the theory and will play an important role). We restrict to infinitesimal intervals → 0 in the CFT, where the structure of entanglement entropy can be made particularly explicit [21,22]. The correction to χ that we estimate comes from states |i whose R-charges J scale with c, corresponding to gravitational solutions with angular momentum on the S 3 . The expectation values J i grow with the central charge, violating the assumption of vacuum block dominance.
Our main result is an O(c 0 ) correction to the average relative entropy in the small interval limit → 0: where α is an O(c 0 ) coefficient. The ellipses denote further positive corrections: at O( 2 ) from BPS operators other than J with conformal weights h +h = 1, and at higher orders in from higher-dimension operators. We also resolve the minor discrepancy between the holographic and field-theoretic computations of [22]. The rest of this note is organized as follows. In § 2 we briefly review the work of Bao and Ooguri and their input assumptions. In § 3 we review the work of Giusto and Russo on entanglement entropy in black hole microstates, which leads to our main result. We conclude in § 4 with some broader context. In the appendix we give two derivations of the density of states at generic R-charges: the first is intuitive and can be visualized; the second, which extends the thermodynamic analysis of Balasubramanian et al [23], allows us to systematically compute finite-N corrections.

Review of the Bao-Ooguri Estimate
In the AdS 3 /CFT 2 correspondence the BTZ black hole is dual to the thermal ensemble (1.3). Following [12], we will be interested in characterizing the distinguishability 1 of the microstates ρ i from the ensemble (1.3) on an interval of length in the CFT, which we take to live on a circle of radius R. It is captured by the relative entropy where ρ i, and ρ BH, are the reduced density matrices defined in (1.4). The average of this quantity -the Holevo information -is equal to the average difference of von Neumann entropies: It was recently argued [12] that the Holevo information is zero on intervals smaller than half of the circle, < πR. We will argue in the next section that 1/c corrections make the Holevo information nonzero even on infinitesimal intervals but first, we review the arguments of [12], highlighting the assumption that will be violated in our scenario. In order to evalute (2.2) one needs the von Neumann entropies of the reduced density matrices. When the CFT is holographic, these entropies are determined by the HRRT formula [13,14], Here Γ is the extremal surface in the bulk dual of ρ, homologous to the boundary region A . When ρ = ρ BH the dual geometry is the BTZ black hole. The homology constraint then implies a critical length crit where the extremal surface splits into two disconnected pieces, one wrapping the black hole and another homologous to the complementary region: The bulk dual of a generic microstate is not known, but one can still study the S(ρ i, ) using field-theoretic methods. Bao and Ooguri [12] estimated the microstate entropies using the results of [16], which starts from the replica trick for entanglement entropy in 1+1d CFTs [24]: Here u and v are the endpoints of the interval, |Ψ i = |i ⊗n , the T n are twist operators with weights h T =h T = c 24 n 2 −1 n , and we have made explicit the renormalization factor Z. The four-point function in (2.5) can be expanded in conformal blocks 2 [16], using a scale transformation to move the operators to (0, 1, x, ∞): Here the sum runs over Virasoro primaries, a (i) p = C iip C p T T , and x = u/v is the conformal crossratio. We have chosen to expand in the t-channel to anticipate our eventual interest in the small-interval limit.
In large-c conformal field theories this expression can be evaluated at leading order in c under the additional assumption of vacuum block dominance [25], i.e. that the sum is well-approximated by the contribution of the identity and its descendants: This approximation is valid (in some range of x) only in states where a (i) p grows slowly with c for all light primaries O p [16,25]. In the next section we consider a class of black hole microstates that violate this condition and discuss their impact on χ.
If one simply assumes vacuum block dominance, the correlator in (2.5) can be computed in the limit n → 1 [16] using the known form of the heavy-heavy-light-light vacuum block [15] in the channel in which it dominates. The result is the vacuum block contribution to S(ρ i, ): where and is the UV cutoff. When > πR the vacuum block dominates in the channel obtained by replacing with 2πR − [16], leading to the min in (2.8). This is the expression that is used in [12] to obtain the estimate (1.5), with the weight h i of the microstate fixed by demanding β i = β. It agrees precisely with the holographic entropy (2.3) in the BTZ geometry if one relaxes the homology constraint [16]. Collecting (2.4) and (2.8), the vacuum block contribution to χ is (2.10) We will see that the failure of vacuum block dominance makes χ ∼ O(c 0 ) even when πR.

Entanglement entropy in the M = 0 BTZ microstates
We will study the Holevo information in a string theory setup where the microstates of the black hole can be identified explicitly: the near-horizon limit of the D1-D5 brane system, whose CFT description includes 1/4-BPS states that are dual to microstates of the M = 0 BTZ black hole (× S 3 × M 4 ). The study of these states and their identification with dual geometries [26] is a rich subject with a long and tempestuous history; 3 we will not make use of this identification here. Our results follow solely from the properties of 1/4-BPS operators in the CFT 4 and are independent of (but consistent with) the identification of certain BPS operators with bulk geometries. For more details of the CFT see e.g. [23,26,[37][38][39] and references therein. The field theory description of the microstates is obtained by considering type IIB string theory on S 1 × M 4 with N 1 D1s wrapping the S 1 and N 5 D5s wrapping all the compactified directions, with the closed string sector decoupled by the near-horizon limit. Taking M 4 much smaller than the S 1 , the theory becomes a 1+1d SCFT with N = (4, 4) SUSY, whose central charge c = 6N 1 N 5 ≡ 6N is determined by the number of D1 and D5 branes. At weak coupling the SCFT is conjectured to be a nonlinear sigma model on the symmetric product orbifold (M 4 ) N /S N [40]; at strong coupling it is dual to supergravity in AdS 3 × S 3 × M 4 [41].
The microstates of the M = 0 BTZ black hole correspond to the Ramond ground states of this D1-D5 CFT [42,43]. Supersymmetry protects their dimensions and SU (2) L × SU (2) R R-charges, which can therefore be extrapolated from the orbifold theory to the region of moduli space where the theory has a gravitational dual. The R-charges of the black hole microstates are bounded by − N 2 < J 3 L , J 3 R < N 2 [36] and correspond to angular momenta on the S 3 in the bulk [44]. Most states have J 3 , violating vacuum block dominance. As in any CFT corrections to the vacuum entropy are particularly easy to isolate in the smallinterval limit, where the physics is determined by the twist operator OPE [21]. Since T n T −n has vanishing twist, only untwisted operators 5 pn appear: with further terms subleading as w → w . Here r, s are replica indices and the expansion has been organized according to how many of the tensor factors in O are trivial (e.g. on the second line the operator is O p on the r th factor tensored with O q on the s th factor, tensored with the identity on the remaining factors). This expansion was studied by [22] in the Ramond ground states dual to the microstates of the M = 0 BTZ black hole; we review their treatment, amending two points.
In any state |i , the entanglement entropy of a small interval can be computed by plugging (3.1) into (2.5). We use coordinates w on the cylinder related to the plane coordinates z via z = e iw , with the parametrization The operators in (3.1) are inserted at φ − φ = on a constant-τ slice.
As → 0 the dominant contributions to (3.1) come from the lightest operators in the theory. The leading correction naively appears to be the first non-vacuum term in (3.1), but this term vanishes for all primaries since C T T pr = 0 if O p is primary. 6 In the M = 0 BTZ microstates, the leading correction to the entropy comes from a sum over the lightest untwisted primaries with two tensor factors nontrivial. When these operators have nontrivial expectation values they will contribute to the entanglement entropy. The OPE coefficients C q on the replica manifold via a conformal map [21], then using the inverse of the Zamolodchikov metric g ab = O a |O b to raise the index [22].
We would like to compute the microstate entropies S(ρ i, ) in the gravity regime, but the orbifold CFT is at zero coupling. Fortunately, there is strong evidence that the gravity computation of (3.1) can be extracted from the calculation in the orbifold theory. The idea is as follows [22]: most operators acquire O(c) anomalous dimensions at strong coupling and so will contribute negligibly to (3.1). However, 1/4-BPS operators are protected by supersymmetry: their dimensions and three-point functions are independent of the coupling [45]. To extrapolate results from the orbifold theory to gravity we therefore keep only the contributions from these BPS operators (we will leave the 1/4 implicit for brevity). It was shown in [22] that this prescription agrees exactly with the results obtained from a generalization of the HRRT formula, apart from the vacuum piece we will correct.
The lightest BPS operators in the D1-D5 CFT have h +h = 1 and thus appear at O( 2 ) in the twist OPE (3.1). Inserting (3.1) into the formula for the entropy (2.5) and evaluating the sum over r and s in this sector exactly as in [22], at O( 2 ) we find 7 where M is the contraction of the Zamolodchikov metric with the plane two-point function: Among the lightest BPS operators are the holomorphic and antiholomorphic SU (2) L × SU (2) R R-symmetry currents J α L and J α R . For simplicity we will focus on corrections to the Holevo information coming from the α = 3 components alone, neglecting the rest of the lightest BPS operators; we will abbreviate J 3 L/R ≡ J L/R . Our computation of these corrections will be enough to show that χ O(c 0 ) at small , since the other operators that contribute at O( 2 ) make manifestly positive contributions to χ.
To obtain S(ρ i, ) from (3.3) we must include the renormalization factor Z, which was omitted in [22]. In the theory on the plane, the renormalization factor Z plane = −4h T is introduced in order to make the entropy dimensionless [24]. However, there is no need to fix the units in (3.3) and so it seems we are free to choose Z cyl as we please. We will fix it here by demanding that the leading divergence in S(ρ i, ) approaches the vacuum entropy on the plane when R. This scheme is natural in light of the conformal symmetry: we should get the same results regardless of whether we do the computation directly in the theory on the cylinder, or by transforming the four-point function on the plane normalized by Z plane . Thus we take Taking Z cyl into account, focusing just on the contribution of J L/R and following the computation of M (J,J) = 12/c in [22], we find where . . . denotes the contribution of the other BPS and higher-dimension operators. We have found a different vacuum contribution than [22] due to our inclusion of the renormalization factor (3.5); with this minor correction, the result agrees exactly with the holographic computation in [22] of the entropy at O( 2 ) using the deformed HRRT surface in the gravity background dual to the microstate. We will not however make use of this interesting fact.
Reintroducing the contributions of the remaining BPS operators with h +h = 1, This is the exact microstate entropy at O( 2 ).

Average relative entropy of the M = 0 BTZ microstates
The microstates |J L , J R of the D1-D5 black hole are labelled by their left and right R-charges (J L , J R ). We compute the average relative entropy χ of the zero-temperature density matrix dual to the M = 0 BTZ black hole × S 3 [23]. We have left the other quantum numbers implicit as they do not affect the contribution we compute. The probability p (J L J R ) that a microstate has charges (J L , J R ) is the number of states d J L ,J R with those charges divided by the total number of states, i.e.
We will compute contributions to the average relative entropy of ρ BH restricted to an interval of length , (3.10) In the limit → 0 this inherits the spectral structure of (3.7): where . . . denotes subleading terms in . The first term vanishes as → 0 [12] because the vacuum block contribution is equal to the thermal entropy when < πR. The second is O( 2 ): We will ignore the (positive) contributions of the other BPS operators with h +h = 1 and so obtain only a lower bound. Specifying to M 4 = T 4 for concreteness, in the appendix we find the density of states 8 at leading order in N when J L/R ∼ O(N ) (recall that c = 6N ). 8 The density of states had previously been computed [23] in the regime where only one linear combinations of the R-charges, We can combine (3.15) with (3.6) and (3.9) to evaluate (3.12). Approximating the sum over R-charges by an integral, we obtain where the γ ± must be determined numerically but are smooth functions of J L/R . We also show that if we use this more accurate density of states instead of (3.15) in the appropriate region of the integral in (3.16), we still find that α ∼ O(c 0 ).

Discussion
At the outset we motivated our investigation of the Holevo information by a hope that it would aid in study of the information paradox. Let us outline some steps in that direction. Ultimately, one would like to calculate some bulk probe of the distinguishability between the black hole microstates and the naive black hole geometry dual to the thermal state. Relative entropy is such a probe, but it is hard to calculate. However, we have seen that the average relative entropy -the Holevo information -can be estimated quite easily under sufficiently controlled circumstances. Fortunately, it is a decent substitute for the relative entropy in addressing certain interesting questions. Suppose we are wondering whether or not most states of a particular black hole have a firewall near the horizon. In this case one can simply ask whether χ is large in a region that probes the horizon. Unfortunately, we do not know much about the microstates of black holes that may have firewalls, and our approach has restricted us to both supersymmetry and small .
One step towards a more realistic black hole is the M > 0 BTZ black hole dual to the D1-D5-p system. Its decoupling limit is the same CFT we have studied but the microstates have nonzero momentum N p along the common D1-D5 direction. There are additional light BPS operators with nontrivial expectation values in these states, and since T ∼ N p = 0 the stress tensor also contributes to χ at O( 2 ) [46]. Despite much recent progress (see e.g. [47][48][49][50][51][52][53][54]) knowledge of the D1-D5-p microstates is not comprehensive; one would still need to know how the expectation values are distributed in order to compute contributions to χ. At the end of the 9 There are also states with J L/R < O( √ N ) but in such states the vacuum block dominates over J-exchange, so they make negligible contributions to the part of χ that we are estimating. day one would like to break supersymmetry and study evaporating black holes, but knowledge of their microstates is even more sparse [55][56][57][58][59].
There are also prospects in higher dimensions. In AdS 5 there is a close analog of the D1-D5 system, the LLM geometries [60], which are dual to 1/2-BPS states of N = 4 super-Yang Mills. The analog of the black hole for these geometries is the superstar solution [61][62][63]. While the LLM geometries are explicitly known, computation of the entanglement entropies seems to be difficult, 10 though there has been some recent progress in a different duality frame [64].
What about the vacuum block approximation? It is crucial to much of the recent technology that has been developed to study both correlation functions and bulk reconstruction in AdS 3 [65][66][67][68][69][70][71]. We do not know whether or not this affects the ability of these methods to make predictions about what happens near black hole horizons -one might suspect that the gravitational physics is sufficiently universal -but it could be useful to extend their approach to WZW models. With such techniques one can probe much further into the bulk than we have been able to here.
It is also natural to ask if χ itself has some interpretation in terms of ordinary (non-replica) correlation functions. While the relative entropy can be expressed in terms of the expectation value of the modular hamiltonian as ∆ K − ∆S, the first term drops out of the average. From the fact that two density matrices have a nonzero relative entropy we can conclude that there is some operator such that tr(ρ i O) = tr(ρO), which must even be local since we consider small , but the average expectation values are necessarily equal: p i tr(ρ i O) = tr(ρO). Last we discuss corrections to our result. First, χ receives further positive contributions at O( 2 ) from other BPS operators with h +h = 1. It seems reasonable to conjecture that the contributions of these operators do not increase α beyond O(c 0 ), though they might. One would need to know the distribution of their expectation values.
The second source of correction is more interesting. Our O(c 0 ) result comes at the order typically associated with quantum effects in the bulk. For instance, at O(c 0 ) one must account for the entanglement of bulk fields, which generically contains state-dependent pieces as in (3.7) and so χ will be further corrected. Of course this is just the usual story of quantum corrections in the bulk -the same effect gives rise to mutual information at O(c 0 ) between distant boundary intervals -but it is intriguing that there is an interplay here with the sum over geometries.

A Density of states at generic R-charges A.1 Intuitive derivation of the density of states
Supersymmetry protects the ground states of the D1-D5 CFT dual to the microstates of the M = 0 BTZ black hole. This is relevant for our purposes because it implies that the number of states with R-charges J L/R remains constant as we go from the strongly-coupled regime, where the system is dual to gravity in AdS 3 × S 3 × M 4 , to zero coupling, where the system is described by an orbifold theory on (M 4 ) N /S N . We can therefore use the orbifold description to obtain a particularly simple derivation of the density of states. We will need just one fact [26] about the ground states |ψ in the orbifold theory: they are completely characterized by a set of integers, n is a twist operator permuting n out of the N copies of the seed theory (which is an SCFT on M 4 ) and i runs over all the fields in the seed. Note that n here has no connection to the replica index used in the entropy calculations in the main text.
The integers N n,i can be chosen freely up to the constraint which implies that the ground states |ψ are in one-to-one correspondence with colored partitions of N , with i labelling the different colors. This is depicted in figure 1. 11 Figure 1: The black hole microstates correspond to colored partitions of N , which can be illustrated as a collection of multiwound colored strings: N n,i is the number of strings with winding number n and color i.
An asymptotic estimate of the number of uncolored partitions was derived by Hardy and Ramanujan [72]: to leading order in N . We have one color for each boson and half a color for each fermion (due to bosonization), so the total number of colors is the central charge of the seed theory, The figures in this appendix can be read quite literally: the Φ (i) n map to the usual string creation operators α i −n under U-duality [26].
The total number of microstates is then the number of partitions with c s colors: at large N . Notably, the logarithm of this quantity is the Bekenstein-Hawking entropy of the M = 0 BTZ black hole [73] (once α corrections have been taken into account). One can extend this reasoning to obtain (3.15). 12 First, note that the fields i are labeled by their R-symmetry charges. Focus on the four R-charged bosons, which carry (J L , J R ) = (±1/2, ±1/2), (±1/2, ∓1/2).
(A. 6) In terms of the linear combinations J ± = J L ± J R , they carry Suppose we want to estimate the number of states with fixed J + ∼ O(N ). This is a small fraction of the total number of states (A.5), since a typical colored partition will not carry much R-charge; we must fix some of the coloring so that the corresponding state carries J + . 13 At large N , the dominant contribution comes from the configurations depicted in figure 2. These configurations have |J + | singly-wound strings whose color corresponds to either (J + , J − ) = (1, 0) or (−1, 0), depending on the sign of J + ; the remaining N 1 = N − |J + | elements can be in an arbitrary configuration. The number of these configurations is the number of colored partitions of the unfixed remainder, P cs (N 1 ) = e 2π √ cs at large N . Configurations with some of the J + carried in multi-wound strings are subleading, since fewer elements will be left unfixed. Likewise subleading are configurations with J + carried by fermions, since the Pauli exclusion principle makes it impossible to carry all of J + in singlywound fermionic strings. The collection of strings carrying J + can be thought of as a Bose-Einstein condensate [23]. To estimate the density of states at generic J L/R we must estimate the number of colored partitions with both J + and J − fixed. Following the same logic as above, most of these partitions will have the structure depicted in figure 3, with two condensates: one made of |J + | singly-wound strings carrying (J + , J − ) = (±1, 0), and another made of |J − | singly-wound strings carrying (J + , J − ) = (0, ±1), with the signs again determined by those of J ± . The entropy then comes from colored partitions of the remaining N 2 = N − |J + | − |J − | elements: The theory on M 4 = T 4 has c s = 12, which yields (3.15).

A.2 Thermodynamic derivation of the density of states
Eq. (3.15) can be confirmed by a direct calculation of the density of states, which also enables us to study corrections. In this section we extend Appendix B of [23] to general J L,R O( √ N ). Before proceeding to our extension we warm up with a review of their original calculation of d J L ,J R , valid in the regime where J L + J R ∼ O(N ) and J L − J R ∼ 0. Our presentation is largely self-contained but the reader may also find it useful to consult their more detailed exposition. 14 Appendix B of [23] computes the density of states for J + ∼ O(N ) by starting with the partition function of the D1-D5 CFT at the orbifold point with a chemical potential µ + , When N is large this can be evaluated via the saddle point approximation, leading to an expression for the entropy at fixed J + . Note that in this expression, unlike in (1.3), β is not the inverse of the physical black hole temperature but rather an auxiliary chemical potential conjugate to the left-moving excitation number N ≡ N L (i.e. it is the inverse "effective temperature" β L of the left-movers); in particular, the physical Hawking temperature T H = T −1 The seed of the orbifold has N B left-moving bosons and N F left-moving fermions. Out of these states, n B of the bosons have J + = 1 and another n B have J + = −1. Similarly, there are n F fermions each with J + = ±1/2. We will follow [23] in leaving the numbers of species general to faciliate comparison with their expressions. 15 With these charge assigments, the partition function (A.10) is [23] Z(β, µ + ) = tr e −β(N −µ + J + ) where q = e 2πiτ = e −β and z = e 2πiν = e βµ + . This can be rewritten in terms of special functions as .12) and In the thermodynamic limit β → 0, corresponding to N 1, the partition function simplifies: Here c s is the central charge of the seed of the orbifold, c s = N B + N F /2, which is O(1). The first factor is the least singular as β → 0 and can be neglected. The saddle point approximation leads to the thermodynamic relations From these expressions one obtains the entropy and the expectation value of J + , J + = 3µ + g(µ + ) 2 c s π 2 1 + 2c s N π 2 3µ 2 + g(µ + ) 2 − 1 .
(A. 18) In the last two expressions we used n B = 1. The magnitude of J + is controlled by the behavior of g(µ + ). In [23] µ + is determined as follows: first, since N − µ + J + is not a positive operator when |µ + | > 1, we must restrict to |µ + | ≤ 1 in order to obtain a well-defined partition function. 16 Next, one demands that J + ∼ O(N ). As µ + → 0, g vanishes linearly, but it diverges as µ + → ±1: up to subleading corrections in N . This completes our review of [23].
To study the density of states at general J L/R we must also turn on a chemical potential for J − : where q = e 2πiτ = e −β , z = e 2πiν = e βµ + andz = e 2πiν = e βµ − . The second line follows from the fact that the theory has n B states with charges (J + , J − ) = (±1, 0), (0, ±1), and similarly for the fermions. We find and In the thermodynamic limit β → 0 corresponding to large N , the partition function once again simplifies: The saddle point analysis leads to 26) and the entropy where we again used n B = 1. Similarly, one finds the R-charges We have used time-reversal invariance to fix µ ± = γ ± sign(J ± ) for some 0 < γ ± < 1, but the γ ± (J ± ) must be determined numerically by inverting the transcendental relation (A.28). We will now quantify the error in our computation of χ in (3.16) caused by our use of (A.30) over the whole range of J L/R . 17 We should have used (A.31) instead of (A.30) in the appropriate region of the integral in (3.16) when we computed χ. Consider the fractional error we made in computing χ, . (A.32) 17 One might have worried about logarithmic corrections to the density of states, but they are subleading compared to modifications of the form (A.31).
To be complete we have integrated over the entire range of J ± , though the contribution from J ± ∼ O(N ) vanishes since (A.30) approaches (A.31) in that regime. Our conclusion in the main text that α ∼ O(c 0 ) will hold true so long as the fractional error does not scale with N .
We will analyze the simpler quantity ε(m) ≡ This quantity misses contributions from |J + | > m √ N but such contributions will be negligible when m is sufficiently large. Here γ + ≡ µ + sign(J + ), obtained by inverting (A. 18), is plotted in the left panel of figure 4. In the large-N limit and for sufficiently large m, ε is equal to the fractional error (A.32) up to an overall O(1) multiplicative factor (see the remarks in footnote 8). In the right panel of figure 4 we plot ε(m) numerically for several different values of N . There are two salient features. The first is the rapid convergence in m to an asymptotic value ε : we do not need to take m very large in order to obtain a good approximation of ∆χ/χ, once the O(1) multiplicative factor is restored. This follows from the fact that γ + is monotonic and approaches 1 quite rapidly above J + ∼ O( √ N ). The second feature is the crucial one: convergence ε → 0.1249 at large N . This leads to our conclusion that α ∼ O(c 0 ).