Unravelling Mathieu Moonshine

The D1-D5-KK-p system naturally provides an infinite dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Phi_{10}(Z)$. We show that the Mathieu group, M_{24}, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M_{24} action. This establishes the correspondence (`moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M_{24} to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M_{24} that relates its conjugacy classes to eta-products and Jacobi forms.


Introduction
In ref. [1], a moonshine for the Mathieu group, M 24 , relating its conjugacy classes to genus-two Siegel modular forms was proposed. Let ρ = 1 a 1 2 a 2 · · · N a N be the cycle shape for an M 24 conjugacy class and (k+2) = 1 2 i a i . Then the moonshine correspondence proposed in [1] (see also [2]) is as follows: where Φ ρ k (Z) is a genus two Siegel modular modular at level N and Z = ( τ z z σ ) ∈ H 2 . We shall restrict the considerations of this paper to the situation when the conjugacy class reduces to a conjugacy class of M 23 : this implies that a 1 = 0. In these cases, it is known that the Siegel modular form counts 1 4 -BPS twisted dyons in the heterotic string compactified on a six-torus. 1 It is natural to ask whether there exists an M 24 -module V ♮ graded by three integers (n, ℓ, m) corresponding to the dyonic charges ( 1 2 q 2 e , q e · q m , 1 2 q 2 m ) i.e., STr g V (n,ℓ,m) q n r ℓ s m , (1.4) counts g-twisted 1 4 -BPS states in the toroidally compactified heterotic string qneρ is the conjugacy class of g ∈ M 24 that reduces to a conjugacy class of M 23 .
The D1 − D5 − KK − p system [3] provides an obvious candidate for V ♮ . It has a natural decomposition into three distinct parts: 1 Such dyons are invariant under the action of a finite abelian group of order N -this group maps to a symplectic automorphism of K3 in the dual type II picture. 2 The superdimension sdim of the graded vector space V = V + ⊕ V − is defined to be sdim(V ) = Tr V (−1) F := STr V (1) = (d + − d − ) where d + is the dimension of the bosonic (even) subspace, V + and d − is the dimension of the fermionic (odd) subspace, V − of V . with (n,ℓ) sdim W (n,ℓ,0) q n r ℓ s 0 = η(τ ) 6 θ 1 (τ, z) 2 , n sdim B (n,0,0) q n r 0 s 0 = 1 η(τ ) 24 , (n,ℓ,m) sdim S (n,ℓ,m) q n r ℓ s m = 1 E(K3, Z) , where E(K3, Z) is the second-quantized elliptic genus of K3 [4]. The problem thus reduces to showing the action of M 24 on these three modules. The paper is organized as follows. In section 2, we discuss the module V ♮ in terms of its components. When possible, we show how some parts are M 24modules connecting them to known moonshines for M 24 . We reduce the problem to proving that the module S, associated with the second-quantized elliptic genus of K3, is an M 24 module. In section 3, we focus on the details of the moonshine proposed in [1]. We show that the module S is an M 24 module by showing that the M 24 action implies replication formulae for twisted elliptic genera of symmetric products of K3. It is proposed that a twisted Hecke-like operator generates all these replication formulae. In section 4, we prove that the proposal provides a Borcherds product formula that incorporates the M 24 -action on the second-quantized elliptic genus of K3 and hence for the Siegel modular forms. This product formula agrees with existing formulae in the literature. We conclude with a few observations. Appendix A provides some information about the elliptic genus of Kähler and hyperKähler manifolds.

The module V ♮ and its parts
The microscopic counting of 1 4 -BPS states in four-dimensional CHL compactifications of the heterotic string with N = 4 supersymmetry was carried out by David and Sen [5]. Using a chain of dualities as well the 4D-5D correspondence [6], the counting was carried out in a configuration of D1 and D5 branes moving in K3 × S 1 × T N where T N is the Taub-NUT space. The geometry close to the center of T N is five-dimensional while the geometry far away from the center is four-dimensional with the appearance of an additional circle S 1 .
The result of David-Sen [5] for the type IIB dual of the toroidally compactified heterotic string expresses the Igusa cusp form, Φ 10 (Z), as a product of three terms As indicated above the states that contribute to the counting arise from three distinct sectors (labeled (i)-(iii)) in the type IIB description: (i) the overall motion of the D1-D5 branes in Taub-NUT space.
(ii) the excitations of the KK-monopole -following the chain of dualities, these excitations get mapped to the states of the heterotic string.
(iii) the motion of the D1-branes in the worldvolume of the D5-branes -this counting leads to the second-quantized elliptic genus of K3.
In terms of fermionic zero-modes, half of the supersymmetry (contributing sixteen zero-modes) is broken by the KK-monopole and its excitations. The remaining quarter of the supersymmetry is broken (contributing four zero-modes) by the center of mass motion. The motion of the D1-branes in the worldvolume of the D5-branes do not break any further supersymmetry. Let Q 1 and Q 5 denote the number of D1-branes and D5-branes respectively. Further, let n denote the momentum along S 1 (the KK monopole charge) and J the angular momentum in T N space. Then, the D1 − D5 − KK − p system carries the following T-duality invariant charges [5]: The module V ♮ is graded by these three charges.
We shall now consider each one of these terms separately to further elucidate the structure of the module V ♮ . The rest of this section is based on ref. [5] and we refer the reader to it for a detailed description.

The overall motion in Taub-NUT space
The overall motion of the D1-D5 system in Taub-NUT space is described by a 1+1-dimensional supersymmetric conformal field theory(CFT) with Taub-NUT as its target space. The charge associated with q 2 e gets related to the L 0 eigenvalue while the charge associated with q e · q m gets related to the charge due to a U(1) L symmetry in the field theory. The field theory consists of four free leftmoving fermions; four right-moving fermions charged under U(1) L and interacting with four scalars with Taub-NUT target space. The unbroken supersymmetry acts on the right-movers. Thus BPS states are obtained when the right-movers are in the ground state. The only permitted excitations are from the four leftmoving fermions as well as the left-movers of the four scalars. The index of the supersymmetric states in the field theory which we interpret as a supertrace over W is given by (n,ℓ) sdim W (n,ℓ,0) q n r ℓ s 0 = Z free (τ ) Z osc (τ, z) Z zero-mode (z) , where Z free (τ ) is the contribution of the free left-moving fermions (which are zindependent due to their U(1) L invariance); Z osc (τ, z) is the contribution from the left-moving oscillators of the four scalars and Z zero-mode (z) is the contribution from the bosonic zero-modes (which are τ independent). Thus, the space W is the direct product of three-spaces and the contributions are: (2.6) The action of M 24 on W Let g denote a symplectic automorphism of K3 of order N. Then, g acts trivially on the space W . This is easy to understand since W arises from the dynamics of motion in the Taub-NUT space and does not 'see' the K3. We identify symplectic automorphisms of K3 with elements of M 24 . Thus we see that M 24 has no action on W . Thus, Remarks: 1. Z zero-mode (z) has different Fourier expansions depending on whether |r| < 1 and |r| > 1: Z zero-mode (z) = r + 2r 2 + 3r 3 + · · · for |r| < 1 , r −1 + 2r −2 + 3r −3 + · · · for |r| > 1 . (2.8) Physically, this is related to the existence of a wall of marginal stability at r = 1, where there is a jump in dyon degeneracy due to additional contributions from two-centered black holes [7,8]. A mathematical interpretation can be given by using the connection of the square-root of the Igusa cusp form, ∆ 5 (Z), with a BKM Lie superalgebra [9]. The wall of marginal stability in the physical setting gets mapped to the wall of a Weyl chamber [10].
2. The module V ♮ is thus dependent on the Weyl chamber. As in the case of finite Lie algebras, the Weyl chamber is characterized by the choice of Weyl vector.

The KK monopole
The low-energy dynamics of the D1-D5 system localized at the center of the Taub-NUT space can carry momentum along S 1 but not along S 1 . 3 The momentum along S 1 contributes to the electric charge 1 2 q 2 e with other charges vanishing. Following the chain of dualities, BPS states of this system get mapped to BPS states of the fundamental heterotic string. The BPS condition requires the supersymmetric right-movers of the heterotic string to be in the ground state and the various electric charges take values in the Narain lattice for heterotic string compactified on a six-torus. In the light-cone gauge, the space of BPS states is that of the oscillator modes of 24 left-moving chiral bosons. We will denote this space by B.
The level matching condition relates the electric charge to the level, n, of the oscillator excitations [11]. One has where L ′ 0 is the contribution of the oscillator modes to L 0 .
The action of M 24 on B: the first moonshine Let g denote a symplectic automorphism of K3 of order N. Using the identification of the heterotic string as a NS5-brane wrapping K3, we can track the action of g to the fields in the 1+1 dimensional CFT of the heterotic string. In particular, one can show that g acts as a permutation of the 24 chiral bosons. Following the work of Mukai [12], it was shown in ref. [1] that the possible conjugacy classes are those of M 24 that reduce to conjugacy classess of M 23 . M 24 has precisely eight such conjugacy classes that we list in Table 1.
where ρ is the conjugacy class of g and g ρ (τ ) is the η-product given by the map Remarks: 1. The η-products that appear are multiplicative i.e., they are Hecke eigenforms. They also furnish a moonshine for the group M 24 [14,15]. This is the first moonshine for M 24 that we will encounter -this has been dubbed the additive moonshine. As we will later see, this is the first of an infinite set of moonshines for M 24 that imply that V ♮ is an M 24 -module.
2. B is obviously an S 24 -module and hence an M 24 -module. However, S 24 is not a symmetry of the full D1 − D5 − KK − p system. For instance, the requirement that g generate a symplectic automorphism needs us to choose special points in the moduli space of K3. This translates to a particular choice of the Narain lattice in the heterotic string -as the Narain lattice does not form part of B, we see the larger group i.e., S 24 .
3. The fact that g has to generate a symplectic automorphism of K3 also forces us to restrict to conjugacy classes of M 24 that reduce to M 23 conjugacy classes. Other conjugacy classes appear when one considers a pair of commuting symplectic automorphisms [2]. These lead to a generalized moonshine (in the sense of Norton [16, see appendix by Norton]) and will be discussed elsewhere.

The second-quantized elliptic genus
The third contribution arises from the relative motion of the D1-branes in the worldvolume of D5-branes (in the type IIB frame). Consider the situation when one has Q 1 D1-branes wrapping S 1 and Q 5 D5-branes wrapping K3 × S 1 . Recall that the magnetic charge of the system is 1 2 q 2 m = (Q 1 −Q 5 )Q 5 . In particular, note that when Q 5 = Q 1 = 1, the magnetic charge is vanishing. The other charges arise as momentum of the D1-brane along S 1 and S 1 . The BPS condition combined with level-matching in the associated CFT relates the dyonic charges to the L 0 and J eigenvalues in the RR sector.
In the U(Q 5 ) supersymmetric gauge theory on coincident D5-branes, the D1branes appear as instantons and the instanton number of the configuration is (Q 1 − Q 5 ) after taking into account the induced charge of a D5-brane wrapping K3 [17,18]. 4 The moduli space of this system is 4[(Q 1 −Q 5 )Q 5 +1] dimensional and is identified with the symmetric product of 4[(Q 1 − Q 5 )Q 5 + 1] copies of K3 [18]. We denote the symmetric product of m copies of K3 by S m (K3) ≡ K3 m /S m and its smooth resolution by K3 [m] .
The dynamics of this relative motion is captured by the N = (4, 4) super- It was conjectured in ref. [19] and proved in ref. [4] that the elliptic genus of this sigma model can be written in terms of the partition function of a secondquantized string theory on K3 × S 1 . (See ref. [20] for a review on the relation between second-quantized string theory and symmetric products.) Let us denote the ellliptic genus of a Kähler manifold, M, by χ(M; τ, z)(see appendix A for definitions) . Then, the result of ref. [4] leads to a Borcherds product formula for the generating function of the elliptic genera of symmetric products of K3: where c(nm, ℓ) are the Fourier-Jacobi coefficients of the elliptic genus of K3 which is a weight zero, index one Jacobi form that we denote by ψ 0,1 (τ, z) ψ 0,1 (τ, z) := χ(K3; τ, z) = n≥0,ℓ c(n, ℓ) q n r ℓ . (2.15) We refer to E(K3; Z) as the second-quantized elliptic genus of K3. Since we associate powers of s with the magnetic charge, we need to include an additional factor of s −1 in our definition of the second-quantized elliptic genus. Remark: In the limit z → 0, the elliptic genus, χ(M; τ, z) reduces to the Euler characteristic of M which we denote by χ[M]. One sees that 5 (2.16) The appearance of the η-product is easily understood by carrying out electricmagnetic duality and the magnetic η-product gets mapped to the electric ηproduct. The condition that the ground state have charge q 2 m = −1 then follows from level matching.

Decomposing the module S
We decompose the module S using the magnetic charge as (2.17) The m = −1 term is the ground state with Q 1 = Q 5 = 0 and has magnetic charge q 2 m = −1 with other charges vanishing. Thus, S −1 = C. The sub-module S 0 = H(K3) is special and we shall call it K and the elliptic genus of K3 is given by We will see that the modules S m for m ≥ 1 can be constructed from K [4]. Recall K is the Hilbert space of a D1-brane on K3 × S 1 with winding number one. Denote by K (n) the Hilbert space of a multiply wound D1-brane with winding number n -the oscillator modes of such a string have fractional moding 1/n. Thus one has The work of DMVV [4] uses the structure of pemutation orbifolds and shows that the module S m−1 (for m > 1) can be decomposed into a direct sum of twisted sectors of the orbifold CFT. Recall that twisted sectors of the S m -orbifold are labelled by conjugacy classes of S m . For instance, S 1 has only two sectorsthe untwisted sector (with conjugacy class 1 2 ) and a single twisted sector (with conjugacy class 2). Thus, one has where H 1 2 = K ⊗ K and H 2 = K (2) and the superscript 'inv' indicates that we need to project onto S 2 invariant subspaces. Thus the elliptic genus of S 2 (K3) can be written in terms of the elliptic genus of K3 and leads to the following doubling formula.
where P denotes the projection on to the S 2 -invariant sector.
More generally, one has . Thus, for a cycle shape [h] = 1 a 1 2 a 2 3 a 3 · · · (with j ja j = m), the centralizer group is Thus, it is easy to see that the above structure leads to replication formulae i.e., all elliptic genera are expressible in terms of the elliptic genus of K3 i.e., ψ 0,1 (τ, z). DMVV show that the various replication formulae implied by the structure of the various S m lead to the following remarkable formula: where V m is the Hecke-like operator that maps weak Jacobi forms of weight zero and index 1 to weak Jacobi forms of weight zero and index m It is easy to recover Eq. (2.21) by matching coefficients of s 2 on both sides of Eq. (2.24). On substituting the Fourier-Jacobi expansion of the elliptic genus of K3, we recover the Borcherds product formula given in Eq. (2.14).

The action of M 24 on K: the second moonshine
In CFT's on a torus on considers the following traces where g and h are symmetries of the CFT and H h is a module twisted by h (i.e., a module in the h-twisted sector in the orbifold by a group containing h). We will also denote the same object by [g, h], on occasion, in a more compact notation. There is growing evidence that K is an M 24 -module [22][23][24]. This is best seen by decomposing the elliptic genus of K3 in terms of characters of the N = 4 6 The Hecke-like operator is usually written as a sum over the coset Γ 1 \S m where S m is an GL(2, Z) with det(S m ) = m. In the following, we have written the Hecke-like operator for a particular parametrization of the coset. superconformal algebra [25,26]. This is the second moonshine for the Mathieu group M 24 . As before, let g be a symplectic automorphism of K3 of order N, then it has been shown that the g-twisted elliptic genus of K3 defined as follows can be expressed in terms of characters of M 24 corresponding to the conjugacy class, ρ, of g. Several aspects have been numerically verified to fairly high powers by several authors [27,28]. In this paper, we shall assume that K is indeed an M 24 -module though this has not been proven to the best of our knowledge. With this assumption, we shall proceed to show that S is then an M 24 module in the next section.

The moonshine correspondence
The generating function of 1 4 -BPS states that are invariant under a symplectic automorphism g (of order N) whose M 24 conjugacy class is ρ = 1 a 2 2 a 2 · · · N a N is a Siegel modular form of weight k and level N: Φ ρ k (Z) or more precisely Φ [g,1] k (Z) [1]. As with Φ 10 (Z), it can be written as product of three terms. 7

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(3.1) Term (i) remains unchanged while term (ii) becomes the multiplicative η-product as we discussed earlier. Term (iii) is the g-twisted second-quantized elliptic genus: is the g-twisted elliptic genus. The goal of this section is to provide evidence that S m is an M 24 -module graded by the L 0 and J L eigenvalues. Hence the above formula can interpreted as a trace over this module with g taken to be an element of M 24 .
We have already seen that the modules S m can be expressed in terms of K. The 'building blocks' of S m are S m (K) and K (m) . If g acts on K, then it acts naturally as ⊗ m g on the tensor product space K ⊗m . This induces an action on S m (K). 8 Given that K (m) differs from K only in the L 0 -grading and g commutes with L 0 , one may guess g acts exactly as it did on K. With this in mind, we shall consider S 1 and compute the g-twisted elliptic genus for S 2 (K3).
Notice the appearance of the g 2 -twisted elliptic genus of K3 in the above formulae. Thus we see that the g-twisted elliptic genus for S 2 (K3) satisfies a doubling formula as for the untwisted elliptic genus once we have included the action of g on the various building blocks. We have experimentally verified that the above formulae holds when g has order 2 and 3. It thus appears that replication formulae should appear for the g-twisted elliptic genus for all S m (K3). We propose that all such replication formulae can be written in terms of a twisted Hecke-like operator, which we denote by T m , that naturally generalizes the DMVV formula.
where the twisted Hecke-like operator, T m defined below, generates a Jacobi form of weight 0 and index m when acting on a weak Jacobi form of weight 0 and index 1.
The superscript [h, g] indicates the boundary conditions. This extends the considerations of Tuite [30], who defined such an operator for modular forms, to Jacobi forms. The important ingredient is that the boundary conditions also change under the action of GL(2, Z). One has We have experimentally verified that the replication formulae hold for twisted elliptic genera of S m (K3) for m = 2, 3 and for some choices of g and h. In fact, we first obtained these replication formulae experimentally before realizing that these are captured by the obvious generalization of Tuite's considerations to Jacobi forms. Such twisted Hecke-like operators have also appeared in the considerations of Ganter [31,32].

An infinite number of moonshines
We have seen that each of the spaces S m are M 24 -modules. Traces over them lead to Jacobi forms of weight zero and index (m + 1), one for each conjugacy class of M 24 appearing in Table 1. The first of these is the moonshine associated with the elliptic genus of K3. Of course, the Siegel modular form subsumes all these modular forms and hence the moonshine associated with Φ 10 (Z) subsumes all these moonshines for M 24 . It is important to note that on decomposing any of the Siegel modular forms into Jacobi forms and η-products, all of them are associated with the same conjugacy class.

Borcherds Product Formulae from moonshine
We shall now show that Eq. (3.5) leads to a Borcherds product product formula for second-quantized twisted elliptic genus and hence for the corresponding Siegel modular form. These product formulae will be shown to agree with known product formulae thus providing evidence for the conjectured form of the Hecke-like operator given in Eq. (3.6).
The Fourier-Jacobi expansion of the weak Jacobi form ψ c a (n, ℓ) q n r ℓ .
(4.1) When g has order N, a is defined modulo N since ψ c a ( n, ℓ) q a n/d r aℓ where β := exp(2πi/d) is a d-th root of unity. In the second line, we have used the following result to set n = dn q an r aℓ s am a , . (4.6) Taking the exponential on both sides, we get 1−ω α q n r ℓ s m cα(nm,ℓ) . (4.7) Note that the product over m runs from 1 to ∞. This implies a Borcherds product formula for the associated Siegel modular form after multiplying by s φ [g,1] k,1 (τ, z). We obtain 1 − ω α q n r ℓ s m cα(nm,ℓ) . (4.8) This formula is precisely the one obtained by David-Jatkar-Sen [33,see Eq. (3.17)]. All one needs is to observe that ψ [g a ,1] 0,1 (τ, z) = NF 0,a (τ, z) , in their notation. It can be shown that this formula is equivalent to alternate versions of the formulae due to Gritsenko-Nikulin [34] and Aoki-Ibukiyama [35] as well as an earlier formula also due to David-Jatkar-Sen [3]. This concludes the proof that the twisted Hecke operator and the formula (3.5) for the twisted second-quantized elliptic genus holds. Further, with the assumption that K is an M 24 -module, we see that all modules S m for m > 0 are also M 24 modules. Thus, we obtain a self-consistent picture that the module S is an M 24 -module.

M 23 vs M 24
The condition that any symplectic automorphism, g must be contained in a M 23 subgroup of M 24 might seem to suggest that V ♮ must be a M 23 -module rather than a M 24 module. Recall that in the realization of M 24 as a permutation group, M 23 is a subgroup of M 24 that preserves one element. The key point is that the action of a symplectic automorphism depends only on its conjugacy class -it is possible to find two different realizations of a symplectic automorphism, g, that preserve distinct elements of H * (K3, Z) -this is related to distinct symplectic structures.
There are two implications that arise from the claim that V ♮ is a M 24 -module. First, there are elements of M 23 with order > 8 -formally, one can insert such elements into the trace over V ♮ . Second, there exist elements of M 24 that are not in M 23 . While we do not have an answer in full generality, we will comment on specific instances of these two possibilities below.
1. For instance, let us consider the case when g has order 11. There is indeed a multiplicative eta product for the cycle shape 1 2 11 2 . The trace over the V ♮ leads to the product formula given in Eq. (4.8) for a possible Siegel modular form of weight 0. However, the naive formula for the additive lift does not work. Eguchi and Hikami have recently shown that there exists a modification due to the appearance of a new form for Γ 0 (11) and matches the terms appearing in the product formula [36].
2. Since Jacobi forms have been constructed for all conjugacy classes of M 24 [23,27,28], it appears that we must obtain a product formula for the corresponding elliptic genus and thence a product formula formula for a (potential) Siegel modular form for every conjugacy class of M 24 including those that do not reduce to conjugacy classes of M 24 . One such conjugacy class is 2 12 that has been recently shown to be a symmetry of the conformal field theory albeit not a symplectic one [37]. The Jacobi form as well as the eta product for the conjugacy class 2 12 are both modular forms of Γ 0 (4). Should one expect a Siegel modular form of weight 4 at level four? It has been argued in [2] that the same cycle shape is associated with a pair of symplectic automorphisms and that leads to a Siegel modular form of the paramodular group at level two. This appeared in the work of Clery and Gritsenko [38]. We do not have a definitive answer on this case. So we will conclude this discussion with a couple of questions that we hope to address in the future. Should the twisted Hecke-like operator defined in Eq. (3.6) be modified for these conjugacy classes? Indeed there is a modification that appears for the paramodular group in ref. [38] and one needs to study if that is relevant for these cases. Is there no factor that converts the secondquantized elliptic genus to a Siegel modular form for conjugacy classes of M 24 that do not reduce to M 23 conjugacy classes? It may be that the trace over the module V ♮ is not a modular form -recall that in all situations, the second-quantized elliptic genus needs to be multiplied by a factor (called the Hodge anomaly by Gritsenko [39]) to become a modular form. Note that an affirmative answer doesn't contradict the existence of V ♮ . However, if the factor exists and we do obtain a modular form, it must arise as one of the modular forms constructed by Clery and Gritsenko [38].

Conclusion
In this paper, we have shown that the D1 − D5 − KK − p system indeed provides M 24 -module. As a by-product of our investigation, we have implicitly obtained product formulae when the element g has orders six and eight. These were already obtained using the additive (Saito-Kurokawa-Maaß) lift in ref. [1]. It would be nice to see if there is Lie algebraic structure underlying these two modular forms of the kind when the order of g was less than six [40]. It is obvious that the replication formulae that we considered here also hold for the moonshine proposed by us in ref. [41] for the Mathieu group, M 12 . This more or less follows from the observation in that paper that the twisted elliptic genus of K3 decomposes into to a sum of two terms along with a similar decomposition for the Siegel modular form. This implies that there exists an M 12 -module, V ♮ such that V ♮ = V ♮ ⊗ V ♮ . An obvious extension of our considerations is to consider the modular forms that arise in CHL orbifolds -these were denoted by Φ k (Z) by Jatkar-Sen [29]. In the notation of this paper, these modular forms should be denoted by Φ where h is an element of M 24 . As argued by us elsewhere [2], these should be considered in the context of generalized moonshine in the sense of Norton [16, see appendix by Norton]. The analog of V ♮ in this context should be a twisted version that we shall call V ♮ h . Now consider insertions of another element g of M 24 that commutes with h into the various trace. This should give rise the modular forms that count twisted dyons in the CHL orbifold [2]. The details of the generalized moonshine will be discussed elsewhere [41]. These also provided realizations of conjugacy classes of M 24 that do not reduce to conjugacy classes of M 23 . For instance, the class ρ = 2 12 is associated with two commuting elements of M 24 , each of order two.