Discrete Information from CHL Black Holes

AdS_2/CFT_1 correspondence predicts that the logarithm of a Z_N twisted index over states carrying a fixed set of charges grows as 1/N times the entropy of the black hole carrying the same set of charges. In this paper we verify this explicitly by calculating the microscopic Z_N twisted index for a class of states in the CHL models. This demonstrates that black holes carry more information about the microstates than just the total degeneracy.


Introduction and Summary
CHL models [1,2] in four dimensions with N = 4 supersymmetry have proved to be a rich arena for studying the physics of black holes [3][4][5][6][7][8][9]. On the one hand they have as much supersymmetry and hence as much control as the toroidally compactified heterotic string theory.
On the other hand they have different effective actions beyond the supergravity approximation and hence make different predictions for the entropy of BPS black holes beyond the leading order result of [10,11]. Thus they provide us with more data points at which we can compare the macroscopic and microscopic predictions for the black hole entropy. This comparison has been remarkably successful at the level of four derivative corrections to the effective action, reproducing complicated non-trivial functional dependence of the entropy on the charges on both sides. 1 Indeed, most of the results on black holes in heterotic string theory on T 6 [6,[16][17][18][19][20][21][22][23][24] have now been generalized to the case of CHL models.
In this paper we shall make use of the CHL model to explore another aspect of black holes.
Based on AdS 2 /CF T 1 correspondence [25,26] it was argued in [27] that if a theory has a Z Z N symmetry that cannot be regarded as part of a U(1) gauge transformation, and if we pick a 1 We should add a note of caution that this comparison requires us to make assumption of certain nonrenormalization results which have not been proven. In particular it assumes that at the level of four derivative terms the Gauss-Bonnet terms (or their supersymmetric completion given in [12][13][14][15]) in the action are sufficient to calculate the correction to the black hole entropy. The analysis in this paper does not require us to make any such assumption. black hole carrying U(1) charges which are invariant under this Z Z N transformation, then the logarithm of the trace of the Z Z N generator over the microstates of the black hole grows as 1/N times the entropy of the black hole. 2 This can be made more concrete for BPS black holes in supersymmetric string theories by working with protected helicity trace index. In the context of N = 4 supersymmetric string theories in four dimensions the relevant twisted index is the 6th helicity trace index [27,30,31]: where the trace is taken over all states carrying a fixed set of charges q, h is the third component of the angular momentum of the state in its rest frame, and g is the generator of a Z Z N symmetry which leaves q invariant. This index receives contribution from 1/4 BPS states in this theory which describe large black holes with near horizon AdS 2 × S 2 geometry. In this case the analysis of [27] applies and tells us that where S BH ( q) is the entropy of an extremal black hole carrying charge q. We shall not review the arguments of [27] here; but the central idea is that in computing the contribution to (1.2) from the horizon of the black hole the leading saddle point corresponding to the AdS 2 × S 2 near horizon geometry does not contribute. However a Z Z N orbifold of AdS 2 × S 2 [26,32,33], whose asymptotic geometry coincides with that of the original near horizon geometry of the black hole, contributes and gives the answer exp[S BH /N] in the semiclassical limit. While (1.2) follows almost trivially from the AdS 2 /CF T 1 correspondence, it is quite striking from the point of view of the microscopic theory. For large black holes the right hand side of (1.2) is much smaller than the untwisted helicity trace index carrying the same charges, since the latter is given by exp[S BH ( q)]. What this tells us is that in a given charge sector the microstates of different g eigenvalues come in almost equal numbers so that the sum weighted by g is much smaller than the total number of states. This was explicitly verified in [27] by deriving the microscopic formula for this twisted index in toroidally compactified heterotic and type II string theories and then studying their asymptotic behaviour. 3 Given the unusual nature of this macroscopic prediction, it is important to test this in as many examples as possible. In this paper we shall verify this in the context of CHL models.
The construction of the CHL models that we shall analyze proceeds as follows. We begin with type IIB string theory on M × S 1 × S 1 where M is either K3 or T 4 and go to a special subspace of the moduli space of M where the theory has a geometric Z Z M × Z Z N symmetry that commutes with 16 supersymmetry generators of the theory. An extensive list of possible symmetries of this type can be found in [34,35]. Note also that a Z M N group with M and N relatively prime can be considered as a Z M × Z N group for the purpose of our analysis. Let us denote by g M and g N the generators of Z Z M and Z Z N respectively. We now take an orbifold of this theory by a symmetry that involves 1/M unit of translation along the circle S 1 accompanied by the transformation g M . This gives a theory with N = 4 supersymmetry in four dimensions and the Z Z N group generated by g N is a symmetry of this theory. We now consider a g N invariant charge vector q in this theory and define the index where the trace is taken over all states carrying the charge q. Eq.(1.2) now translates to for large charges. Our goal will be to verify this by explicit computation of d( q) in the microscopic theory.
Since the explicit counting of states involves technical details, we shall take this opportunity to summarize the results of our analysis. We use a convention in which the coordinate radius of the original circle S 1 before orbifolding is 2πM so that the orbifold action involves translation by 2π along S 1 accompanied by g M . In this convention the minimum amount of momentum along S 1 is 1/M. We focus on states carrying one unit of KK monopole charge associated with the circle S 1 , one unit of D5-brane charge wrapped on M × S 1 , Q 1 units of D1-brane charge wrapped on S 1 , left-moving momentum n/M along S 1 and J units of momentum along S 1 , and define Q 2 = 2n/M, In this case our result of d( q) is given by Here C is a three real dimensional subspace of the three complex dimensional space labelled by (ρ = ρ 1 + iρ 2 , σ = σ 1 + iσ 2 , v = v 1 + iv 2 ) given by M 1 , M 2 , M 3 being large but fixed positive numbers satisfying The function Φ(ρ, σ, v) is a modular form of a subgroup of Sp(2, Z Z), given by where the coefficients c (4n − j 2 )e 2πi(nτ +jz) The trace is taken over all the g r ′ M g r N twisted RR sector states in the (4,4) superconformal CFT 2 with target space M. L 0 andL 0 are the left and right-moving Virasoro generators and J L /2 and J R /2 are the generators of the U(1) L × U(1) R subgroup of the SU(2) L × SU(2) R R-symmetry group of this CFT 2 . An algorithm for explicitly computing the right hand side of (1.10) has been outlined in appendix A. The coefficients α, β, γ are given by where Q r ′ ,s ′ = MN c Eqs. (1.10), (1.11) define all the quantities which appear in the definition of Φ. The only ambiguity that remains in computing the right hand side of (1.6) is the choice of the integration contour encoded in the choice of (M 1 , M 2 , M 3 ). As is well known by now, this ambiguity is related to the phenomenon of wall crossing [36][37][38][39]. Different choices of M 1 , M 2 and M 3 give the value of d( q) for different values of the asymptotic moduli. However the ambiguity in the value of d( q) that it introduces is sufficiently small so as not to affect our analysis, and hence we shall ignore it in our subsequent discussion.
Given the result (1.6) for d( q) we can find its behaviour for large Q 2 , P 2 and Q.P by standard method [3,6,16,17]. The result is that d( q) behaves as (1.13) Since in this limit a black hole of charge q has entropy [10,11] S BH ( q) ≃ π Q 2 P 2 − (Q.P ) 2 , (1.14) we see that the microscopic result (1.13) is in perfect agreement with the macroscopic prediction (1.4).
Finally we would like to remark that even though we have presented our analysis for the index T r((−1) 2h (2h) 6 g N ), we can repeat the analysis with g N replaced by (g N ) b for any integer b. In this case the role of N is played by the order of (g N ) b , and in all the formulae we simply have to replace g N by (g N ) b . This in turn allows us to compute the index T r((−1) 2h (2h) 6 ) for states carrying a definite g N eigenvalue e 2πia/N using the combination (1.15) Thus our result can also be interpreted as the agreement between the macroscopic and the microscopic results for the helicity trace index over states carrying a definite g N charge.

The counting
The counting of states of the D1-D5-KK monopole system proceeds as in [6][7][8]40]. We take the circle S 1 to be large compared to the size of M and regard the world-volume theory as a 1+1 dimensional field theory living on S 1 . Denoting by d(Q 1 , n, J) the twisted index (1.3) of states carrying charge labeled by (Q 1 , n, J) in the convention of §1, we define Proving (1.6) is now equivalent to proving that Z = −1/ Φ. The twisted partition function Z is given by the product of three twisted partition functions, -that of the excitations living on the KK monopole, that of the dynamics describing the overall motion of the D1-D5 system in the background of the KK monopole and that of the motion of the D1-brane along the D5-brane. For each system we must keep the right-movers in their ground state and excite the left-movers in order to preserve supersymmetry. 4 Since the fermion zero modes associated with the broken supersymmetries are automatically removed while computing the helicity trace (which is B 6 in this case since the system breaks 12 of the 16 supersymmetries), we shall ignore their contribution during the rest of our analysis.
We begin by analyzing the partition function of the KK monopole. The massless bosonic modes on the world-volume of the KK monopole arise from the motion along the three transverse directions and the components of the p-form fields along the product of the harmonic (p − 2)-forms of M and the harmonic 2-form of the Taub-NUT space. The massless fermions are the goldstinos associated with the supersymmetries broken by the Kaluza-Klein monopole.
In general one can show that the left-moving bosons and fermions are in one to one correspondence with the even and odd degree harmonic forms on M [40]. Furthermore their (g M , g N ) quantum numbers are also given by the (g M , g N ) eigenvalues of the harmonic forms on M.
Since the harmonic (p, q) forms on M are in one to one correspondence with the RR sector ground states in the supersymmetric σ-model with target space M carrying L 0 =L 0 = 0, J L = (p − 1), J R = (q − 1), it follows from (1.10) that the number of left-moving bosons minus the number of left-moving fermions on the KK monopole world-volume, carrying g N quantum number e 2πir/N and g M quantum number e 2πik ′ /M , is given by [8,40] 1 MN In arriving at (2.2) we have used the fact that c (r,s;r ′ ,s ′ ) b (u) = 0 for u < −1. Now consider a mode carrying g M eigenvalue e 2πik ′ /M and left-moving momentum l/M along S 1 . The requirement of invariance under the simultaneous action of g M and 2π translation along S 1 gives us the requirement that l = k ′ mod M. Furthermore the contribution to the twisted index from these states will be weighted by the g N eigenvalue e 2πir/N . Thus the net contribution to the partition function from these modes is given by Here the term e −2πi αρ reflects the effect of the momentum carried by the ground state of the Kaluza-Klein monopole. The analysis of [8,40] gives Next we turn to the dynamics of the overall motion of the D1-D5 system in the KK monopole background. The dynamics in the transverse direction is independent of whether we are working with K3 or T 4 . Furthermore these modes do not carry any g N or g M quantum numbers; thus the contribution from these modes to the partition function is universal. The result is [8,40] −e −2πiv 1 − e −2πiv −2 The first line represents the contribution from the zero mode dynamics that binds the D1-D5 system to the KK monopole [6,41,42], and the second line represents the contribution from the oscillators. The last two terms in the second line of (2.5) represent the contribution from the four left-moving bosonic modes representing transverse oscillation of the D1-D5 system whereas the first factor in the same line represents contribution from the left-moving fermionic modes. 5 In arriving at (2.5) one needs to use the fact that in the presence of the KK monopole background, the momentum along S 1 appears as the angular momentum 2J L for the D1-D5 system where J L is the generator of the U(1) L ⊂ SU(2) L subgroup of the rotation group in transverse space [43]. The v dependence of (2.5) then follows from the fact that the bosonic modes, transforming as a vector of the transverse rotation group SU(2) L × SU(2) R , carry J L = ±1 while the fermionic modes are neutral under U(1) L as a consequence of footnote 5.
For M = T 4 we also have four additional bosonic modes arising from the Wilson lines on the D5-brane along T 4 and four additional fermionic modes. In order to find the contribution to the partition function from these modes we need to know the action of g M and g N on these modes. If z 1 and z 2 denote the complex coordinates on T 4 then in order to preserve supersymmetry both g M and g N must act as equal and opposite rotation of z 1 and z 2 , possibly accompanied by shifts. We shall assume for definiteness that g M and g N induce respectively 2π/M and 2π/N rotations on these coordinates: 1 − e −2πi/N e 2πilρ−2πiv . (2.7) The first two factors come from the bosonic modes and the last four factors arise from the fermionic modes whose contribution have not already been included in (2.5). The only new ingredient in this formula compared to that in [8,40] is the insertion of the factors of e ±2πi/N , -these arise from the insertion of g N into the trace.
The product of (2.5) and (2.7) can be written in a compact form using the coefficients c Note that the (1 − e −2πiv ) −2 factor has been absorbed into the l = 0 term in the last term.
Finally let us turn to the contribution to the partition function from the motion of the D1branes along the D5-branes. First we consider a single D1-brane wrapped w times along S 1 , and count the number of states n(w, j, l; r, k ′ ) of the system carrying left-moving momentum l/M along S 1 , g M eigenvalue e 2πik ′ /M , g N eigenvalue e 2πir/N and S 1 momentum j. Since the boundary condition on various fields are twisted by g M under 2π translation along S 1 , the CFT on a D1-brane wrapped w times along S 1 satisfies boundary condition twisted by (g M ) w . Furthermore since the effective length of the D1-brane is now 2πw, a momentum l/M along S 1 will appear as lw/M units of momentum in the CFT living on the D1-brane. It now follows from (1.10) that [8,40] n(w, j, l; r, k ′ ) = (2.10) The requirement that we only keep the modes which are invariant under the transformation g M accompanied by 2π translation along S 1 forces the constraint k ′ = l mod M. It is now straightforward to evaluate the contribution to the g N twisted partition function from multiple states of this type, carrying different (w, l, j) [44]: w>0,l≥0 1 − e 2πir/N e 2πi(σw/M +ρl+vj) −n(w,j,l;r,l) . The e −2πi γσ in (2.11) accounts for the fact that the actual number of D1-branes required to produce a total D1-brane charge Q 1 in the background of a D5-brane is given by Q 1 + 1 for M = K3 and Q 1 for M = T 4 . Multiplying (2.3), (2.9) and (2.11) we get the total partition function of the system: (4kl − j 2 ) , (2.14) with α, β, γ defined in (1.11). Note that the k = 0 term in this product gives the result for Z KK Z CM .

Asymptotic Growth
We now study the growth of the index for large Q 2 , P 2 and Q.P . This can be done by standard procedure described in [3,6,16,17]. We deform the three dimensional contour of integration over (ρ, σ, v) to small imaginary values of (ρ, σ, v). During this deformation we pick up contribution from the residues at various poles, given by the zeroes of Φ, which give the leading contribution to the index, -the contribution from the final contour can be shown to be subleading compared to the contribution from the residues at the poles [40]. Thus we need to first determine the location of the zeroes of Φ. This has been done in appendix C where it is shown that Φ has double zeroes on the subspaces: for values of (m 1 , n 1 , m 2 , n 2 , j) satisfying Now the analysis of [3,6,16,17] tells us that for large Q 2 , P 2 , Q.P the contribution from the residue at the pole (3.1) of 1/ Φ grows as exp π Q 2 P 2 − (Q.P ) 2 /|n 2 | for |n 2 | > 0 . Since for this charge the black hole entropy S BH is given by π Q 2 P 2 − (Q.P ) 2 [10,11], (3.4) is in precise agreement with the macroscopic prediction (1.2).

Conclusion
It is widely believed that since string theory provides us with a consistent quantum theory of gravity, black holes in string theory do not lead to a loss of information. If so, the black hole must represent an ensemble of microstates and the black hole entropy must have an interpretation as the logarithm of the degeneracy of microstates. Furthermore quantum string theory around a black hole background must contain all possible information about the microstates. It is therefore important to learn how we can extract information about the black hole microstates by studying quantum string theory around the black hole background.
The results of [27] and this paper provide a small step in this direction. In these papers we discuss how to extract information about one specific feature of the black hole microstates, namely distribution of the Z Z N charges among the microstates. Quantum string theory around the near horizon background leads to a specific algorithm for extracting this information. Our analysis shows that in the limit of large charges the results of the macroscopic analysis are in exact agreement with the microscopic results in a wide class of models where the latter is computable. While using the rules of AdS 2 /CF T 1 correspondence we can in principle compute the ensemble average of more general operators on the black hole side, in the absence of nonrenormalization results it is not clear how we might compare this with the microscopic results.
Acknowledgment: I wish to thank Nabamita Banerjee, Atish Dabholkar, Joao Gomes and Sameer Murthy for useful discussions. This work was supported in part by the JC Bose fellowship of the Department of Science and Technology, India and by the Blaise Pascal Chair, France.
Note added: I have been informed by Suresh Govandarajan that the modular forms of subgroups of Sp(2, Z Z) which appear here have also been constructed independently in [45,46].
In this appendix we shall describe the strategy for explicit computation of the right hand side of (1.10) . First of all F (0,0;0,0) is simply 1/MN times the elliptic genus of M and is given by [47] where ϑ i are the Jacobi theta functions. For non-vanishing r and/or r ′ the contribution to the right hand side of (A.1), coming from twisted sectors localized at the fixed points of g r ′ M g r N , can be computed by taking the size of K3 to be large so that the geometry near the fixed points is nearly flat. In this case the σ model near the fixed point can be regarded as the orbifold of a free field theory, and the action of g r N g r ′ M near the fixed point may be represented by a rotation by some angle 2πθ in one plane and rotation by an opposite angle −2πθ in the orthogonal plane. The action of g s ′ M g s N near such a fixed point can be of two types, -either it takes the fixed point to a different fixed point or it leaves the fixed point fixed. In the former case the contribution to the trace in (A.1) vanishes, whereas in the latter case the action of g s ′ M g s N near the fixed point can be represented by a rotation by 2πφ in one plane and a rotation by −2πφ in the orthogonal plane. This gives a contribution to (A.1) of the form where q ≡ e 2πiτ . The full contribution to F (r,s;r ′ ,s ′ ) (τ, z) is obtained by summing over the contribution from all the fixed points of g r N g r ′ M which are also fixed by g s ′ M g s N . Finally F (0,s;0,s ′ ) can be computed using the modular transformation rules [47] This gives a way to compute F (r,s;r ′ ,s ′ ) (τ, z) using purely geometric data, namely the fixed points of the different elements of Z Z M × Z Z N and the action of the elements of Z Z M × Z Z N near the fixed points. Note that F (r,s;r ′ ,s ′ ) (τ, z) constructed from (A.3), (A.4) is invariant under z → −z. This is a consequence of the SU(2) L R-symmetry of the underlying conformal field theory that allows us to change the sign of J L .

B Threshold Integral Representation of Φ
In this appendix we shall describe a threshold integral representation of Φ. For this we define We now consider the 'threshold integral' (0) have been chosen so that the integrand vanishes faster than 1/τ 2 in the τ → i∞ limit, rendering the integral finite.
Following the manipulations outlined in [4] following earlier work of [48][49][50]  where Φ has been defined in (1.9) and 6 Φ(ρ,σ,v) = e −2πi( αρ+ γσ+ βv) k,l≥0,j<0 f or k=l=0 Since the detailed analysis of a specific integral of this type has been carried out in [4], we shall only describe the basic steps leading to (B.7), focussing on the main differences between the general case and the special case analyzed in [4].
1. We first carry out Poisson resummation over m 1 and m 2 to express I r,s;r ′ ,s ′ ;b as  (B.14)

Using (A.4) one can show that
Under the map τ → (aτ +b)/(cτ +d) the fundamental region F of SL(2, Z Z) gets mapped to its image. With the help of (B.15) we can restrict the sum over the matrix A to the following ranges [48]: (a) Non-degenerate orbit: with the integration over τ ranging over two copies of the upper half plane.
We shall briefly discuss the computation of the contribution from the non-degenerate orbits. The analysis of the contribution from the other orbits follows [4].
4. The steps needed for evaluating the contribution from the non-degenerate orbits are as follows: (a) We first expand F (r,s;r ′ ,s ′ ) (τ, z) using (A.1) and change integration variable from τ 1 to τ ′ 1 : Since for non-degenerate orbit the τ 1 integration ranges from −∞ to ∞, τ ′ 1 integral will also range from −∞ to ∞. (B.20) The sum over n can now be performed using the Kronecker delta.
(c) Next we carry out the τ ′ 1 integral which is Gaussian and the τ 2 integral using the identity At this stage the p dependent part of the summand takes the form term, and since p 2 R is positive semi-definite, this requires p 2 R to vanish. Furthermore p 2 L is also positive semidefinite (although it is not directly apparent from its definition), and hence the only possible way to produce a divergence is to get a non-positive power of q from the expansion of h (r,s;r ′ ,s ′ ) b (τ ). 7 Such terms are quite restricted, and one finds that the possible divergences in I arise at [6,40] for values of (m 1 , n 1 , m 2 , n 2 , j) satisfying At this point p 2 R vanishes and p 2 L /2 = (1/4) − β so that the q p 2 L /2qp 2 R /2 factor becomes purely a function of τ and not ofτ . This has to be cancelled against a similar τ dependent factor in h b so that the τ 1 integral is finite and we get a logarithmically divergent τ 2 integral. The coefficient of the divergent term can be easily determined from (B There is a further restriction coming from the fact that for p 2 L = p 2 R = 0 all the m i 's, n i 's and j must vanish. The divergence in the τ 2 integral from such a term is removed by the subtraction term in (B.5). Thus a divergent τ 2 integral requires p 2 L to be strictly positive. This explains the strict inequality β < 1 4 in (C.2). 8 We need to account for the fact that the terms in the expression for I with (m i , n i , j) and (−m i , −n i , −j) give identical results.
Let us first focus on the r = r ′ = 0 terms, ı.e. configurations with n 2 ∈ N Z Z, n 1 ∈ Z Z. In this case it follows from (C.2) that β must vanish and so the right hand side of (C. given in (1.10) it follows that the exponent in (C.3) can be interpreted as the number of states weighted by (−1) J L +J R in the sector twisted by g r N g r ′ M , and carrying g M eigenvalue e −2πim 1 /M , g N eigenvalue e 2πim 2 , J L = ±1, L 0 = β and L 0 = 0. If g r ′ M g r N does not have a fixed point in M then this number is zero since by taking the size of M to be sufficiently large we can ensure that there will be no twisted sector state with L 0 = 0. If g r ′ M g r N has fixed points in M, then we can compute this number by taking the size of M to be large so that near the fixed points we can regard the space as almost flat, with g r N g r ′ Even after we apply left-moving oscillators to create states with L 0 = β, J L = ±1, the states will continue to have J R = 0. Thus the weight factor (−1) J L +J R is always −1 for J L = ±1 and hence the exponent of (C.3) is always negative. This shows that (C.3) never gives a zero of Φ for r and/or r ′ non-zero, and the only zeroes of Φ are of the form given in (C.6).