Symmetries of the refined D1/D5 BPS spectrum

We examine the large $N$ 1/4-BPS spectrum of the symmetric orbifold CFT Sym$^N(M)$ deformed to the supergravity point in moduli space for $M= K3$ and $T^4$. We consider refinement under both left- and right-moving $SU(2)_R$ symmetries of the superconformal algebra, and decompose the spectrum into characters of the algebra. We find that at large $N$ the character decomposition satisfies an unusual property, in which the degeneracy only depends on a certain linear combination of left- and right-moving quantum numbers, suggesting deeper symmetry structure. Furthermore, we consider the action of discrete symmetry groups on these degeneracies, where certain subgroups of the Conway group are known to play a role. We also comment on the potential for larger discrete symmetry groups to appear in the large $N$ limit.


Introduction
The study of BPS spectra of string compactifications with a large number of supercharges has furnished a laboratory for a number of exact results in a theory of quantum gravity, including a microscopic description of black hole entropy [1] and examples of the AdS/CFT correspondence [2]. If one considers the D1/D5 system compactified on M × S 1 , where M = K3 or T 4 , the worldsheet superconformal field theory (SCFT) of the resulting string in spacetime lies in the moduli space of the symmetric product conformal field theory Sym N (M ) where, in the case of Q 1 D1-branes wrapping S 1 and Q 5 D5-branes wrapping M × S 1 , N = Q 1 Q 5 . Furthermore, in the case of K3, the elliptic genus (EG) of the N = (4, 4) worldsheet SCFT, defined as, 1 Z EG (τ, z) = Tr RR (−1) F L +F R q L0− c 24 y J0 , 2 (1.1) which counts spacetime 1/4-BPS states, was shown to reproduce the Bekenstein-Hawking entropy of the corresponding five-dimensional black hole [1]. A similar accounting of spacetime 1/8-BPS states for M = T 4 coming from a modified worldsheet index was shown to govern the entropy of N = 8 black holes in [3].
In [4], a new quantity called the Hodge elliptic genus (HEG) was defined, and was proposed that unlike the EG, the HEG is not an index in the sense that it depends on the point in CFT moduli space at which it is computed. That is, by tracing over right-moving Ramond ground states graded by the U (1) charge that is part of the right-moving N = 2 superconformal algebra, this quantity can jump as one moves in CFT moduli space. Furthermore, the EG is known to have nice modular properties whenever the CFT has compact target space; i.e., for an N = (2,2) superconformal field theory with central charge c = 6m, the EG is known to be a weak Jacobi form of weight zero and index m. However, the HEG in general is not known to enjoy such nice modular properties.
Other recent work has employed the HEG to study the growth of refined BPS states at the supergravity point in the moduli space of Sym N (M ) for M = K3 and T 4 at large N [5], as well as the refined spectrum of the D1/D5 system on T 4 × S 1 [6]. We continue the analysis of refined BPS spectra at the supergravity point in the moduli space of Sym N (M ) at large N , analyzing the properties of the supergravity BPS spectrum upon decomposition into characters of the relevant worldsheet superconformal algebra of the dual CFT. The purpose of our paper is twofold: 1. First we examine the large N limit of 1/2-and 1/4-BPS degeneracies of the HEG at the 1 Below and throughout the text, we make use of the definitions q = e 2πiτ , y = e 2πiz , and u = e 2πiν . 2 Throughout the text we use J0 to denote the Cartan of the N = 4 SU (2). supergravity point; this applies to low-lying states of the dual CFT with conformal weight below the threshold corresponding to a black hole in the bulk. We show that in this limit the degeneracies "stabilize"; i.e. they only depend on certain combinations of left-and right-moving quantum numbers, which may be indicative of a new symmetry present. 3 We derive analytic expressions for the 1/2-and 1/4-BPS degeneracies in this limit and find an unexpected appearance of Rogers Ramanujan functions in the 1/4-BPS case. Note that because quantum numbers of the right-movers are involved, such stabilization is not visible from just the EG; 4 only upon refinement does one observe this phenomenon.
In particular, our results are as follows. Consider a 1/2-BPS state with left-and rightmoving NS-NS spins and conformal weights (i, j), respectively. At large N as long as j is sufficiently large, the degeneracy of the representation is unchanged when we increase both the left-moving SU (2) R spin and conformal weight by 1 2 , and increase both the right-moving SU (2) R spin and conformal weight by 3 2 . Furthermore, consider an NS-NS 1/4-BPS state with left-moving SU (2) R spin i and conformal weighth + i; and both rightmoving SU (2) R spin and conformal weight j (so that the right-moving part is BPS). At large N as long as j is sufficiently large relative to i andh, we find that the degeneracy of the representation is unchanged if we either: (a) Increase both the left-moving SU (2) R spin and conformal weight by 1 2 , and increase both the right-moving SU (2) R spin and conformal weight by 3 2 . (The same as observed for the 1/2-BPS degeneracies.) (b) Increase the left-moving conformal weight by 2, and increase both the right-moving SU (2) R spin and conformal weight by 5. (For the case of M = T 4 , this symmetry is even larger: we can increase the left-moving conformal weight by 1 and the rightmoving SU (2) R spin and conformal weight by 5 2 .) See equations (3.14) and (3.15) for the precise ranges of parameters in which these operations are symmetries.
2. Secondly, we consider the action of discrete symmetry groups on the refined supergravity BPS spectrum. In [5] it was observed that the coefficients in the character decomposition of the supergravity states counted by the HEG may have a connection to dimensions of irreducible representations of sporadic finite groups. 5 This is reminiscent of the famous observation [7] that the character decomposition of the EG of a K3 surface can be decomposed into representations of the sporadic group M 24 , a phenomenon now known as Mathieu moonshine. We show that there is indeed a connection between sporadic groups and the refined supergravity BPS spectrum, and it can be made precise for all four-plane preserving subgroups of the sporadic group Co 0 . We explicitly define and compute the HEG of the bulk supergravity twined under elements of such subgroups of Co 0 , and discuss the possibility of a larger discrete symmetry structure of the refined BPS spectrum at large N . We comment on a similar story for the case of T 4 .
The outline of the rest of the paper is as follows. In §2 we review the spectrum of BPS states of both half-maximal and maximal supergravity on AdS 3 × S 3 . In §3 we explain point 1, and describe curious properties both of the 1/2-BPS spectra and of the supergravity 1/4-BPS spectra. In §4, we discuss point 2 in detail. We present a general discussion of the action of discrete symmetry groups on states counted by the HEG and derive explicit actions of four-plane-preserving subgroups of Co 0 on the refined 1/4-BPS spectrum of half-maximal supergravity. Finally in §5 we conclude and raise several potentially interesting questions. Some special functions and character formulae used in the text are given in Appendix A, and a few long derivations of the results in §3 are presented in Appendix B.
2 The (refined) spectrum of supergravity on AdS 3 × S 3 Here we briefly review the spectrum of half-maximal (N = (2, 0)) and maximal (N = (2, 2)) supergravity on AdS 3 × S 3 (see [5,8,9] for more details). These supergravities arise upon compactification of type IIB string theory on AdS 3 × S 3 × M , where M = K3 and T 4 , respectively, and the radius of M is much smaller than that of AdS 3 and S 3 . These theories are holographically dual to a point in the moduli space of the CFT Sym N (M ) as N → ∞.
Following [8,9], in [5] this relation was used to compute the low-lying spectrum of the HEG of Sym N (M ) deformed to the supergravity point in the moduli space. In this section, we review the spectrum and decomposition of these supergravity states into characters of the relevant worldsheet superconformal algebra. We further study the symmetries of these degeneracies in §3 and §4.

Half-maximal supergravity
The half-maximal N = (2, 0) supergravity arises upon compactification of type IIB string theory on AdS 3 ×S 3 ×K3, where the size of the K3 is much smaller than that of the AdS 3 and S 3 . The dual 2d SCFT theory lies in the moduli space of Sym N (K3) and has N = (4, 4) superconformal symmetry with central charge c = 6N . This algebra has representations of two types: short (BPS) and long (non-BPS). We will specify the representations by the quantum numbers h and j, the eigenvalues of the highest weight states under the operators L 0 and J 0 (the SU (2) . These can be obtained from the NS characters via spectral flow. See Appendix A.1 for explicit formulas for these characters.
We now turn to a discussion of the bulk spacetime. The Kaluza-Klein (KK) spectrum N = (2, 0) supergravity on AdS 3 × S 3 can be organized into representations of the supergroup This group is generated by the global part of the 2d N = (4, 4) superconformal algebra. The KK spectrum consists of short representations of the supergroup which we denote by (j, j ) S . Such a representation corresponds to a chiral primary of the dual CFT in the NS-NS sector with (J 0 , J 0 ) eigenvalues (j, j ).
In [8] the KK spectrum of single-particle supergravity states was derived. Furthermore, in [9] it was understood that one can find a precise match at finite N between degeneracies of 1/4-BPS states in the bulk supergravity and those counted by the EG of the dual CFT. We review this relation now. First off, recall that the EG of a K3 non-linear sigma model (NLSM), which we refer to as Z K3 EG , has an expansion of the form where the coefficients c(m, ) ≡ c(4m − 2 ) only depend on the combination of 4m − 2 . In [10] a generating function for the elliptic genera of the N th symmetric product of a K3 NLSM, a(m, n, )q m p n y , (2.2) where the c(mn, ) are the coefficients in the expansion of the K3 EG (2.1). In fact, as the EG is independent of the CFT moduli space, the formula (2.2) holds for all points in the moduli space of the symmetric product theory.
From the supergravity side, one can reproduce the degeneracies a(m, n, ) of the EG of the dual CFT for states with left-moving conformal weight h ≤ N +1 4 , i.e. below the threshold corresponding to black hole states [9]. 6 This matching is achieved only after assigning a "degree" to each short multiplet and imposing an exclusion principle following [12] limiting the multiparticle states to have degree at most N . Labeling the short representations as (j, j ; d) where d is the degree, the spectrum of short multiplets including the degree is [9], (n − 1, n + 1; n) S (n + 1, n + 1; n) S (n, n; n + 1) S (n + 1, n − 1; n) S 20(n, n; n) S , where n runs over positive integers. We denote the single-particle Hilbert space of degree d as . For a given N , the exclusion principle is implemented by considering the Hilbert space, The supergravity EG is given by the trace, where we trace only over states that are chiral primaries on the right. We use a tilde to denote the fact that we are doing a supergravity computation. An explicit generating function for Z (2,0) EG,N (τ, z) can be found in [9]. One finds a matching of degeneracies in the relevant range of parameters after implementing a spectral flow of equation (2.2) to the NS-NS sector and comparing the coefficients of the power series expansion of the two generating functions. We refer to [9] for details.
In [5] a similar analysis was performed for the supergravity spectrum including a refinement by the right-moving U (1) current. In this case, one cannot compare degeneracies with the HEG of the dual CFT as it is not known independently. However, by analogous reasoning the trace 7 (where again we only trace over right-moving chiral primaries) captures degeneracies of the HEG of the dual CFT at finite N for conformal primaries with eigenvalues below the threshold corresponding to black hole states. An explicit generating function for such states was derived 7 Our convention is that in (2.6), J0 is taken in the R sector which differs by c/6 from the charge in the NS sector.

1
(1 − p n q m y u ) c (2,0) sugra (n,m, , ) , where the coefficients c (2,0) sugra are defined by Note that the generating function (2.7) corresponds to states in the NS-R sector of the dual CFT [5].
Given that the dual CFT has N = (4, 4) supersymmetry, it follows that Z where c j,j N and c j,h,j N denote the degeneracies of 1/2-and 1/4-BPS states, respectively. At large N the generating function (2.7) and character formulas are independent of N , and the decomposition "stabilizes." The first few such coefficients in this expansion are [5]:

Maximal supergravity
In this section we discuss the case of N = (2, 2) supergravity which arises after type IIB Explicit formulas for these characters are given in Appendix A.2.
The KK spectrum of N = (2, 2) supergravity on AdS 3 × S 3 was discussed in [8]. It can again be organized into representations of SU (1, 1|2) L × SU (1, 1|2) R ; however, there are now multiplets with fermionic highest weight states. Using the same labeling as in the previous section, the single-particle spectrum including the degree is then given by the data in Table 1, where n runs over positive integers.
The relation between the supergravity spectrum and that of the dual CFT is similar to the discussion in the previous section, so we will be brief. We again denote the single-particle Hilbert space by H (d),single (2,2) . The relevant supergravity Hilbert space at finite N including multi-particle states is given by where we have again implemented an exclusion principle limiting the spectrum at finite N to include (single-and multi-particle) states with degree at most N . The EG of the dual CFT vanishes due to fermionic zero modes; however, one can see matching of the supergravity BPS spectrum with the spectrum of the dual CFT at finite N by considering a modified index [3].
On the other hand, the refined supergravity EG does not vanish and can be organized into a generating function given by n≥0 Z (2,2) HEG,N (τ, z, ν)p n = n>0,m, ,
Given the superconformal symmetry of the dual CFT, it follows that Z

Stabilization of degeneracies at large N
In this section we point out some observations about the degeneracies of (supergravity) 1/2and 1/4-BPS states in the large N limit. For CFTs in the moduli space of Sym N (M ), denote by c i,j N ,č i,j N the degeneracies of 1/2-BPS states with (J 0 , J 0 ) eigenvalues (i, j) for M = K3 and T 4 , respectively. We find that after taking N → ∞, the degeneracies c i,j N ,č i,j N only depend on the combination 3i − j of the left-and right-moving U (1) charges when j is sufficiently large.
in the NS-NS sector for M = K3 and T 4 respectively. We find that after first taking the large N limit, the degeneracies c i,h;j N only depend on the combination 3i + 5h − j and the parity ofh, for sufficiently large j. Similarly the degeneraciesč i,h;j N depend only on the combination 3i + 5h − j. We refer to the above phenomena as "stabilization." Furthermore, we derive explicit formulas for these degeneracies in the range where they have stabilized. Note that this phenomenon only occurs upon considering the fully refined 1/4-BPS spectrum; the degeneracy of states counted by the EG never stabilizes but rather grows linearly with N due to right-moving ground state degeneracies [8]. In this section we simply state the results; for the derivations see Appendix B.

Half-BPS degeneracies
First we discuss the character decomposition of the refined spectrum of 1/2-BPS states at large N . As the spectrum of 1/2-BPS states is moduli-independent, the results of this section apply at any point in the moduli space of Sym N (M ). For M = K3 the generating function of 1/2-BPS states (in the RR sector) refined by both left-and right-moving U (1) charges can be obtained, e.g., from a formula in [14]: , We consider the decomposition of Z Hodge into N = 4 characters. Because the representations are RR ground states, the only characters that can contribute are those of the form (short, short).
Thus we can write where the coefficient c i,j N denotes the number of 1/2-BPS mutliplets with (J 0 , J 0 ) eigenvalues (i, j) for all CFTs in the moduli space of Sym N (K3).
In a certain regime, these degeneracies are independent of N and depend on a single combination of left-and right-moving spin. Let c i,j := lim N →∞ c i,j N . 8 Define k := 3 2 i − 1 2 j. In Appendix B.1.1 we show that the degeneracies c i,3i−2k only depend on k after first taking the limit N → ∞ and then the limit i → ∞. Moreover they are given by the generating function In fact, we find empirically that this generating function captures the degeneracies c i,3i−2k N at finite N as long as N ≥ 3i and i ≥ 2k (see Table 2).  Similarly, we study the large N limit of the degeneracies of 1/2-BPS states on Sym N (T 4 ) and its resolutions. The generating function for these states has the form [14] ∞ N =0 .
We decompose the Hodge polynomial of Sym N (T 4 ) into contracted large N = 4 characters in the following way: where nowč i,j N denotes the degeneracy of 1/2-BPS mutliplets with (J 0 , J 0 ) eigenvalues (i, j) for all CFTs in the moduli space of Sym N (T 4 ).
Definingč i,j := lim N →∞č i,j N , we again find that in a certain regime,č i,3i−2k only depends on the combination k as defined above. Furthermore, in Appendix B.2.1 we derive an explicit generating function for the degeneracies in this limit given by Again we find empirically that this generating function describes the degeneraciesč i,3i−2k N at finite N whenever N ≥ 3i + 1 and i ≥ 2k (see Table 3).

Quarter-BPS degeneracies
In this section we summarize the results of a similar analysis for 1/4-BPS states of the large N limit of Sym N (M ) deformed to the supergravity point in moduli space. As the refined spectrum of 1/4-BPS states is moduli-dependent, the results of this section only apply to the supergravity locus in moduli space. As in equations (2.9) and (2.14), we denote by c i,h,j N ,č i,h,j N the degeneracy of representations of the form for M = K3 and M = T 4 , respectively. Furthermore we find that, after taking the limit N → ∞, the degeneracies are independent of N , and at large j, they only depend on a linear combination of i,h, j (as well as the parity ofh for the case of M = K3). In particular, they depend only on the linear combination 3i + 5h − j, and, for K3, whetherh is even or odd (see (3.14) and (3.15) for the precise condition on the parameters necessary).
Note that this means that in, say, the NS-NS sector, if we (a) increase the left-moving spin and conformal weight by 1 2 and the right-moving spin and conformal weight by 3 2 ; or (b) increase the left-moving conformal weight by 2 and the right-moving spin and conformal weight by 5, the degeneracy remains the same, assuming j is large enough. 9 Note that (a) is the same symmetry that we found for the 1/2-BPS degeneracies in the previous section.
Then, explicitly, we find the following set of generating functions for the degeneracies in this limit. For the case of K3, we derive a formula for the difference of even and oddh degeneracies, given by, as well as a formula for the sum of even and oddh which is is given by where we make use of the following definitions, The functions G(τ ) and H(τ ) are called Rogers-Ramanujan functions; for more details see For the case of T 4 the generating function is independent of the parity ofh and is given by where we define, . (3.13) The above equations (3.9) and (3.10) are valid as long as and equation (3.12) holds as long as Detailed derivations can be found in Appendix B. 10 As an example, in Table 4 we write down some terms in the character decomposition of Sym N (K3) deformed to the supergravity point. We can see that as we increaseh, eventually the rows "stabilize" depending on the parity ofh. Similarly, in Table 5 we present an analogous table for T 4 ; note that in Table 5 there is no dependence on the parity ofh.    (3.15)). by 1 2 and the right-moving SU (2) spin by 3 2 , the same symmetry which occurs for the moduliindependent 1/2-BPS degeneracies. However, there does not appear to be a symmetry where we increase the left-moving conformal weight by 2 and the right-moving spin by 5. It would be interesting to explore this further, at the orbifold point or at other potentially interesting points in moduli space. Finally, a natural question is if at large N there is some enhanced symmetry beyond N = (4, 4) underlying this stabilization phenomenon. See §5 for further discussion on this point.

Discrete symmetries
In this section we consider the action of discrete symmetry groups on the refined BPS spectrum

Conformal field theory
We begin by considering the CFT with target space M . Such a NLSM has moduli space of the form [15][16][17]  these groups were classified in [18,19] and shown to be isomorphic to four-plane-preserving subgroups of the group Co 0 ("Conway zero"), the group of automorphisms of the Leech lattice.
For M = T 4 , these groups were classified in [20], and similarly shown to be isomorphic to fourplane-preserving subgroups of W + (E 8 ), the group of even Weyl transformations of the E 8 root lattice.
Suppose G is such a supersymmetry-preserving discrete symmetry group, occurring at a given point in the moduli space, X ∈ M(M ). Then for each conjugacy class g ∈ G we can define the following trace, . 12 Furthermore, all such conjugacy classes [19] and their associated twining genera [21] have been classified. For T 4 , a similar statement should hold with Γ 4,20 replaced by Γ 4,4 , though, to the best of our knowledge, the associated twining genera have not been classified in the same sense as for K3. See [20] for more details.
Similarly, given X ∈ M(M ) with symmetry group G, for each conjugacy class g ∈ G, one can define a twined version of the HEG as

Bulk supergravity
We would like to understand the action of discrete symmetry groups on the refined BPS spectrum at the supergravity point in the moduli space of Sym N (M ) when N → ∞. We focus only on the spectrum of single-and multi-particle states accessible via the analysis of AdS 3 ×S 3 supergravity as discussed in §2. Furthermore, we restrict ourselves to groups which preserve spacetime supersymmetry. At the supergravity point in moduli space, the theory has continuous symmetry group SO(4, n + 1), where n = 20, 4 for M = K3, T 4 , respectively, which is then broken to a discrete subgroup by charge quantization. 13 If we deform away from the supergravity locus to a generic point, it is plausible that the low-lying spectrum remains unchanged; thus it is possible that these discrete symmetries arise at some other point in the moduli space of Sym N (M ).
We will focus on the action of supersymmetry-preserving discrete subgroups of SO(4, n + 1) on the spectrum of refined BPS particle states. Our method will be to consider the action of discrete symmetry groups on the single-particle supergravity spectrum, and then use a formula analogous to (4.9) to lift this to an action on all multi-particle supergravity states. Consider a discrete symmetry group G of N = (2, 0) or N = (2, 2) supergravity, and let g ∈ G denote a conjugacy class. Given an explicit action of g on the supergravity Hilbert space H  For the same reasons as discussed in §2, this computation will only match the twined HEG of the dual CFT for states with conformal weight below the threshold corresponding to black holes in the bulk. We now explicitly describe this action for discrete groups which can arise as symmetries of N = (2, 0) supergravity, and briefly comment on the N = (2, 2) case.

Half-maximal supergravity
In this section we consider discrete symmetries of N = (2, 0) supergravity and their action on the refined spectrum. As mentioned earlier, charge quantization and assignment of a "degree" to states in the spectrum break the continuous SO(4, 21) symmetry group. However, it is still possible to define an action on the spectrum for all G ∈ Co 0 which preserve a four-plane. We now describe this action explicitly.
It will be convenient to introduce the notion of a Frame shape as follows. Given a conjugacy class g in Co 0 , the Frame shape, π g is defined as where M = o(g) is the order of g, runs over the positive divisors of M , and k ∈ Z are integers defined by the 24-dimensional irreducible representation of g as The Frame shape conveniently encodes the eigenvalues of g in its 24-dimensional representation, as these are the 12 complex-conjugate pairs { g,k , g, } which are the 24 roots of (4.15). Finally, we find it useful to define χ g as the trace of g in its 24-dimensional representation, Equipped with these definitions, we can now describe explicitly the action of all g ∈ Co 0 on the multi-particle spectrum of N = (2, 0) supergravity. At fixed degree d, the single-particle With this action, the coefficients c (2,0) sugra,g defined in (4.11) are explicitly given by, n,m, , c (2,0) sugra,g (n, m, , )p n q m y u = (1 − g,k q n ), (4.18) where the g,k are the 24 eigenvalues defined by the Frame shape of π g and we define g,k+12 := g,k . Then the twined version of equation (
As we have discussed, these twinings are defined for all conjugacy classes g ∈ Co 0 which preserve a four-plane. It follows that equations (4.19), (4.22), and (4.23) admit decompositions into characters of four-plane-preserving subgroups G ∈ Co 0 . However, one may wonder, given, e.g., the observation in [5], whether in the large N limit (or even for N > 1) there is any role for larger subgroups of Co 0 , i.e. subgroups G ∈ Co 0 which fix a subspace of dimension less than four. There are a couple of reasons one might ask this question. Firstly, the moduli space of NLSMs on Sym N (K3) is larger than than that of K3 NLSMs, and the symmetry groups haven't been classified. Therefore, there may be a role for larger discrete symmetry groups in the case of N > 1. Secondly, there are a number of subgroups of Co 0 which preserve, e.g., a two-or three-plane in the 24-dimensional representation where nevertheless, each individual conjugacy class preserves a four-plane and thus the action of each conjugacy class on the single-particle supergravity spectrum is well-defined. For example, the Mathieu groups M 22 and M 11 each preserve a three-plane, however, each individual element of these groups preserves a four-plane.
With this motivation in mind, in Appendix C, we use (4.22) and (4.23) to decompose the coefficients in (3.9) and (3.10) respectively into representations of the sporadic groups M 11 and M 22 . In fact, in the large N limit, any coefficient in the N = (4, 4) decomposition of the supergravity BPS spectrum given in equation (2.9) should admit a (virtual) decomposition into representations of these groups. Perhaps surprisingly, we find that at large N every coefficient in (2.9) can be decomposed into honest (non-virtual) M 22 representations. In Table 12 Such twinings were first conjectured in [25] for all four-plane-preserving conjugacy classes of Co 0 . Note that we can reproduce these conjectural twinings explicitly from the action of such symmetries on the supergravity Hilbert space described in this section. Furthermore, we find that for all N > 1, the decomposition of the refined 1/2-BPS degeneracies into M 22 representations is non-virtual, suggesting this group may play a role in the symmetries of the refined 1/2-BPS spectrum of symmetric product theories.

Maximal supergravity
Given the explicit description of four-plane-preserving conjugacy classes of W + (E 8 ) in [20], it should be possible to do an analysis for N = (2, 2) supergravity similar to that of the previous section. We expect that we would be able to define a twined version of the stabilized degeneracies of equation (3.12) at large N for all g ∈ W + (E 8 ) which preserve a four-plane. An interesting question is whether one can find evidence for a role of larger subgroups of W + (E 8 ) when N > 1, as we found in the case of half-maximal supergravity in the previous section. We leave a detailed analysis of the action of these groups on the refined N = (2, 2) spectrum to future work.

Discussion
In this paper we explored the BPS spectrum of the symmetric orbifold of K3 and T 4 deformed to the supergravity point in moduli space, including refinement under both left-and right-moving SU (2) R charges. We found interesting symmetry properties both of the 1/2-BPS and 1/4-BPS spectrum that potentially suggest deeper structure.
In §3 we find that the degeneracies of BPS states at the supergravity point satisfy an interesting stabilization phenomenon. In particular, if the SU (2) R spin on the right is large enough, we find that there are two minimal operations which leave the degeneracy unchanged: (a) increase the spin on the left by 1 2 and the spin on the right by Virasoro minimal model [26]; it would be interesting to have an explanation why they appear here.
Secondly, in §4 we discuss the action of supersymmetry-preserving discrete symmetry groups on the refined BPS spectrum of Sym N (M ) for M = K3 and T 4 . We explicitly derive an action for all four-plane-preserving subgroups of Co 0 on the multi-particle spectrum of N = (2, 0) supergravity at large N , and derive an analytic formula for the twined supergravity HEG in the regime where the degeneracies have stabilized. Furthermore, we comment on the potential action of larger (i.e. three-plane-preserving) symmetry groups on the large-N 1/4-BPS spectrum and provide (non-virtual) decompositions of the stabilized degeneracies into irreducible representations of the sporadic groups M 22 and M 11 as evidence.
Finally we end with a list of additional questions which we find interesting. 2, it appears that property (a) above holds at the orbifold point. Do we see similar structure at the point studied in [28], for instance?
• What happens if we include states that are dual to black holes 15 as opposed to just supergravity KK modes? Can we derive the full HEG including these states? Do these degeneracies have the same stabilization properties?
• Do similar stabilization phenomena hold for the refined BPS spectra other theories dual to AdS 3 supergravity, such as theories with, e.g., large N = 4 superconformal symmetry [30][31][32] or theories which arise from the MSW string [33]?
• Is there any relation between our results in §3 and the stabilization phenomena observed in [34], from studying orbifolds of NLSMs whose target space is an ADE surface singularity?
Is there a generalization of the connection discussed in [35] between the EG of ADE singularities, the K3 EG, and umbral moonshine [36] to the refined BPS spectrum?
• What is the classification of supersymmetry-preserving discrete symmetry groups of NLSMs in the moduli space of Sym N (M )? For M = K3 do subgroups of Co 0 which preserve less 15 Or black rings, see e.g. [29]. than a four-plane play a role?
• In [25] twinings of refined 1/2-BPS states counted by (3.1) under four-plane-preserving elements of Co 0 were proposed to be connected to symmetries of the chiral ring of an auxiliary conformal field theory with symmetry group Co 0 [37]. In [38] it was described how this conformal field theory naturally furnishes modules for a number of sporadic groups, including the Mathieu groups M 22 and M 11 . Is there any connection between the action of these groups on the states of this auxiliary conformal field theory and the M 22 and M 11 discussed in §4.2.1 which may play a role in the refined 1/4-BPS supergravity spectrum?
• The results of [25] regarding the twined 1/2-BPS spectrum have a natural geometric interpretation: they capture g-equivariant reduced refined Gopakumar-Vafa invariants of a K3 surface (c.f. Conjecture 3 of [25].) Furthermore, in [4] it is proposed that the refined 1/4-BPS spectrum captured by the HEG also has a geometric interpretation. A natural question is then whether our results for the stabilization of these (twined) 1/4-BPS degeneracies at large N can be understood in a geometric context.
• Do we see any hint of mock modularity when we further reduce to AdS 2 × S 2 (see e.g. We would also like to thank Shamit Kachru and Suvrat Raju for helpful comments on a draft. NB thanks the ICTS for their hospitality during which some of this work was completed, and is supported by a Stanford Graduate Fellowship and an NSF Graduate Fellowship. SMH is supported by a Harvard University Golub Fellowship in the Physical Sciences and DOE grant de-sc0007870. She acknowledges the kind hospitality of the Aspen Center for Physics, which is supported by NSF grant PHY-1066293, as this was being completed.

A Special Functions and Characters
In this appendix, we give definitions to various special functions used throughout the paper, including character formulas. In this section, and throughout the text, we define q = e 2πiτ , y = e 2πiz .
(A.1) 16 We thank the anonymous referee for pointing out this interesting question.
We define the following Jacobi theta functions as well as the Dedekind eta function and the modular discriminant Finally we define the Rogers-Ramanujan functions G(τ ), H(τ ) as .
and their ratio as Note that the function R(τ ) is the Hauptmodul of the congruence subgroup Γ(5) (see e.g. [40,41]), defined as

A.1 Small N = 4 Characters
In this section we present the characters for the small N = 4 superconformal algebra, following [42,43]. The representations of the algebra are labelled by the conformal weight and the spin of their highest weight state. The representations come in two classes: short (or BPS) and long (or non-BPS); the short multiplets satisfy that, in the NS sector, the conformal weight is equal to the spin; the long multiplets have the conformal weight greater than the spin. We define the character of a representation as where H is either the Ramond or NS Hilbert space and by J 0 we mean the Cartan of the SU (2) current algebra which is part of the N = 4 superconformal algebra. By convention we will label each representation in the following way. The NS sector characters are then given by We can obtain the Ramond-sector characters by spectral flowing by 1/2 unit 17 : (A.10) In the above, we continue to label the representations by the eigenvaluesh, j of the highest weight state in the NS sector. 17 For convenience, in (A.9) and (A.10), we define the characters with a relative shift so that both the NS vacuum and R vacua characters begin at q 0 . In other words, the NS characters are defined as TrNS (−1) F q L 0 y J 0 and the R characters are defined as TrR (−1) F q L 0 − m 4 y J 0 . Note that the NS character differs from the usual definition of characters by a factor of q −c/24 .

A.2 Contracted Large N = 4 Characters
In this paper we also consider the contracted large N = 4 superconformal algebra. In a 2d SCFT with target T 4 , the superconformal symmetry on the worldsheet is a Wigner contraction of the large N = 4 superconformal algebra [3,13]. The large N = 4 algebra, A k + ,k − is labelled by two parameters k + , k − , with central charge given by The characters for this algebra were computed in [44,45]; to obtain the Wigner contraction we Specifically, we have and L R (τ, z) = χ ,R 0,0;1 (τ, z) (A.16)

B Derivation of Stabilization
In this appendix, we derive the stabilization phenomena discussed in §3. In the following derivations, we make repeated use of the following fact. Suppose we are given an equation of the form where x is some set of additional variables, and we would like to extract the behavior of the function f n ( x) as n → ∞. Furthermore, suppose that g(p, x) has the following expansion in terms of p, Noting that the RHS of (B.1) has the form (1 + p + p 2 + . . .)g(p, x) = 1 + We will make repeated use of (B.2) below.
Finally we note that although we believe the stabilization phenomena hold whenever (3.14) or (3.15) is satisfied (for the case of half-maximal and maximal supergravity respectively), in this appendix we will only derive the stabilization when we take the limit i → ∞ and theñ h → ∞. It would be interesting to derive the stricter conditions (including when we change the order of limits of i and h).

B.1 Half-maximal supergravity
We begin with an analysis of the refined 1/2-and 1/4-BPS spectrum of the supergravity limit of Type IIB compactification on AdS 3 × S 3 × K3.

B.1.1 Half-BPS states
First let's derive the large N behavior of the refined degeneracies of half-BPS states. These are moduli-independent. The generating function for the half-BPS states given in equation (3.1) can be written N is the q 0 term of the short N = 4 Ramond character χ s,R i;N at central charge 6N , i.e.
We are interested in the behavior of the coefficients c i,j N , which count the number of half-BPS states of CFTs in the moduli space of Sym N (K3) with J 0 , J 0 eigenvalues i and j, respectively.
Let's calculate the large central-charge limit of (B.3); i.e. the limit .
We note that this is an expansion inp which has the form (B.1), so to take the large N limit, .
Finally, we are interested in the coefficient c i,3i−2k N in the large i limit (after we have taken N large). Redefiningỹ = y −1 u −3 and again using (B.1) and (B.2) for the variableỹ, we find that In the RHS of (B.

B.1.2 Quarter-BPS states
There is a similar phenomenon for 1/4-BPS states. The generating function of multiparticle 1/4-BPS states given by equation (2.7) can be decomposed into characters as 18 where c (2,0) sugra is defined in (2.8). In this formula c i,h,j N denotes the degeneracy quarter-BPS represenations of the dual CFT whose highest weight state has NS-NS eigenvalues of (L 0 , J 0 , J 0 ) given by (i/2 +h, i, j). The N = 4 characters (given in (A.9)) simplify at large central charge in the following way Using (B.10), we can write (B.9) as, . (B.11) Note that this is only valid at large N , so we need to take the N → ∞ limit of (B.9). We redefinep = pu, remove a factor of (1 −p) −1 and use (B.2) to get, where we have dropped the subscript of N on c i,h,j to indicate that we have taken the limit as N → ∞ and for convenience, we have defined a new set of coefficients by m, , − (uy + 2u 2 y + u 3 y + 2u 4 y + uy 3 + 2u 2 y 3 + u 3 y 3 + 2u 4 y 3 )q + (y 2 + 2uy 2 + 22u 2 y 2 + 2u 3 y 2 + u 4 y 2 )q 3/2 ≡ g (2,0) (q, y, u), (B.13) and f (2,0) (p, q, y, u) is as in (2.8).
Now let's take the large i limit of equation (B.12), corresponding to taking the left-moving spin large. We will show that the degeneracies c i,h,j only depend on the combination 3i + 5h − j (as well as the parity ofh) in this limit. First let's redefineq 2 = qy 2 u −6 , pull out a factor of (1 −q) −1 , and use (B.2) to rewrite (B.12) as where we have defined , d (2,0) ( , )y u = g (2,0) (y −2 u 6 , y, u) − 1. (B.15) The above formula can be unwieldy due to arbitrary high poles in y. For convenience let's define α ≡ uy −2 and, after unpacking the theta functions, rewrite (B.14) and (B.15) as Finally, we take the largeh limit of lim i→∞ (−1)h +k c i,h,3i+5h−k , corresponding to taking the left-moving conformal dimension of the state large. This is done by taking out a factor of 1 (1−α) (from the = 2, = 0 term in the product of (B.16)) and then using (B.2). Thus we get that where G(τ ), H(τ ), and ∆(τ ) are modular functions defined in Appendix A.
However, limh →∞ lim i→∞ c i,h,3i+5h−k depends on the parity ofh. Thus this generating function is computing (up to a factor of 2) the difference between the even and odd parityh degeneracies (see Table 4 for example).
We can also compute the sum without much difficulty. Let's first rewrite (B.16) as To extract the largeh limit, we again take out a factor of 1 1−α and use (B.2). The final answer is This computes the sum (up to a factor of 2) of the even and oddh degeneracies. Again, see Table 4 for examples.
Note we can also extract other information from (B.16) and (B.17). For example we don't have to take the largeh limit; we can extract e.g. theh = 1 limit easily from those equations; the final answer is (compare with Table 4):

B.2 Maximal supergravity
In this section we do a similar analysis for the spectrum of multiparticle supergravity states arising from IIB compactification on AdS 3 × S 3 × T 4 . Here we will make use of characters of the contacted large N = 4 superconformal algebra, discussed in Appendix A.2. As the derivations are extremely similar to those of the previous section, we will be brief.

B.2.1 Half-BPS States
In this section we derive the behavior of the degeneracies of 1/2-BPS states on Sym N (T 4 ) as N → ∞. The full generating function of half-BPS states is given in equation (3.5) has a character expansion given by where nowy To get the large N limit, we plug (A.13) into (B.25), redefinep = puy, extract a factor of (1 −p) −1 , and use (B.2) to get, . (B.27) Next we take the large i limit of by redefiningỹ = y −1 u −3 , extracting a factor of (1 −ỹ) −1 , and using (B.2) to obtain .
However, we can instead compute limh →∞ lim i→∞č i,h,3i+5h−k using the same techniques as before.