Mock modular Mathieu moonshine modules

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{Co}_0$$\end{document} that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}=4$$\end{document} superconformal algebra. Similarly, any subgroup of Co0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{Co}_0$$\end{document} that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}=2$$\end{document} superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}=4$$\end{document} (resp. N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}=2$$\end{document}) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}$$\end{document}-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.


Introduction
The investigation of moonshine connecting modular objects, sporadic groups, and 2d conformal field theories has been revitalized in recent years by the discovery of several new classes of examples. While monstrous moonshine [1][2][3][4][5][6] remains the best understood and prototypical case, a new class of umbral moonshines tying mock modular forms to automorphism groups of Niemeier lattices has recently been uncovered [7,8]. The best studied example, and the first to be discovered, involves the group M 24 and was discovered through the study of the elliptic genus of K3 [9]. The twining functions have been constructed in [10][11][12] and were proved to be the graded characters of an infinite-dimensional M 24 -module in [13]. Steps towards a better and deeper understanding of this Mathieu moonshine can be found in [14][15][16][17][18][19][20][21][22][23][24][25], and particularly in [26], where the importance of K3 surface geometry for all cases of umbral moonshine is elucidated. Possible connections to space-time physics in string theory have been discussed in [27][28][29][30][31].
In none of these cases, however, has a connection to an underlying conformal field theory (whose Hilbert space furnishes the underlying module) been established. The goal of this paper is to provide first examples of mock modular moonshine for sporadic simple groups G, where the underlying G-module can be explicitly constructed in the state space of a simple and soluble conformal field theory.
Our starting point is the Conway module sketched in [2], studied in detail in [32], and revisited recently in [33]. The original construction was in terms of a supersymmetric theory of bosons on the E 8 root lattice, but this has the drawback of obscuring the true symmetries of the model. In [32], a different formulation of the same theory, as a Z 2 orbifold of the theory of 24 free chiral fermions, is introduced. A priori, the theory has a Spin (24) symmetry. However, one can also view this theory as an N = 1 superconformal field theory. The choice of N = 1 structure breaks the Spin(24) symmetry to a subgroup. In [32] it was shown that the subgroup preserving the natural choice of N = 1 structure is precisely the Conway group Co 0 , a double cover of the sporadic group Co 1 . In [33] it is shown that this action can be used to attach a normalized principal modulus (i.e. normalized Hauptmodul) for a genus zero group to every element of Co 0 .
In this paper, we show that generalizations of the basic strategy of [32,33] can be used to construct a wide variety of new examples of mock modular moonshine. Instead of choosing an N = 1 superconformal structure, we choose larger extended chiral algebras A. The subgroup G A of Spin(24) that commutes with a given choice can be determined by simple geometric considerations; in the cases of interest to us, it will be a subgroup that preserves point-wise a 2-plane or a 3-plane in the 24 dimensional representation of Co 0 , or in plain English, a subgroup that acts trivially on two or three of the free fermions in some basis. In the rest of the paper, we will often use 24 to denote the unique non-trivial 24-dimensional representation of Co 0 , and an n-plane to refer to a n-dimensional subspace in 24.
It is natural to ask about the role in moonshine, or geometry, of n-planes in 24 for other values of n. One of the inspirations for our analysis here is the recent result of Gaberdiel-Hohenegger-Volpato [15] which indicates the importance of 4-planes in 24 for non-linear K3 sigma models. The relationship between their results and the Co 0 -module considered here is studied in [34], where connections to umbral moonshine for various higher n are also established.
We refer the reader to §9 for a discussion of the interesting case that n = 1.
In this work we will focus on the cases where A is an N = 4 or N = 2 superconformal algebra, though other possibilities exist. In the first case, we demonstrate that any subgroup of Co 0 that preserves a 3-plane in the 24 dimensional representation can give rise to an N = 4 structure. The groups that arise are discussed in e.g. Chapter 10 of the book by Conway and Sloane [35]. They include in particular the Mathieu group M 22 . In the second case, where the group need only fix a 2-plane and preserve an N = 2 superconformal algebra, there are again many possibilities (again, see [35]), including the larger Mathieu group M 23 . Note that the larger the superconformal algebra we wish to preserve, the smaller the global symmetry group is. Corresponding to specific choices of the N = 0, 1, 2, 4 algebras, we have the global symmetry groups Spin (24) We should stress again that there are other Co 0 subgroups that preserve N = 4 resp. N = 2 superconformal algebras arising from 3-planes resp. 2-planes in 24. Some examples are: the group U 4 (3) for the former case, and the McLaughlin (McL) and Higman-Sims (HS) sporadic groups, and also U 6 (2) for the latter case. However, only for the Mathieu groups do the twined mock modular forms arising from the twined partition function of the module display uniformly a special property, which we regard as an essential feature of the moonshine phenomena. Namely, all the mock modular forms obtained via twining by elements of the Mathieu groups are regular at all cusps inequivalent to the "low-temperature" cusp τ → i∞.
The importance of this property is its predictive power: armed with it we are able to write down trace functions for the actions of Mathieu group elements with no more information than a certain fixed multiplier system, and the levels of the functions we expect to find. A priori these are just guesses, but the constructions we present here verify their validity. For this reason, we focus on the two Mathieu groups as the two cases of mock modular moonshine arising from the present chiral conformal field theory.
This may be compared to the predictive power of the genus zero property of monstrous moonshine: if Γ < SL 2 (R) determines a genus zero quotient of the upper-half plane, and if the stabilizer of i∞ in Γ is generated by ± ( 1 1 0 1 ), then there is a unique Γ-invariant holomorphic function satisfying T Γ (τ ) = q −1 + O(q) as τ → i∞, for q = e 2πiτ . The miracle of monstrous moonshine, and the content of the moonshine conjectures of Conway-Norton [1], is that for suitable choices of Γ, the function T Γ is the trace of an element of the monster on some graded infinite-dimensional module (namely, the moonshine module of [3]). The optimal growth property formulated in [8] plays the analogous predictive role in umbral moonshine, and is very similar to the special property we formulate for the Mathieu moonshine considered here.
We mention here that although the moonshine conjectures have been proven in the monstrous case by Borcherds [4], and verified in [18] for the M 24 case of umbral moonshine, conceptual explanations of the genus zero property of monstrous moonshine, and of the analogous properties of umbral moonshine, and the Mathieu moonshine studied here, remain to be determined. An approach to establishing the genus zero property of monstrous moonshine via quantum gravity is discussed in [46].
The organization of the paper is as follows. We begin with a review of the module discussed in [32]. In §3, we describe how one can endow this module with an N = 4 or N = 2 structure.
In §4, we discuss what this does to the manifest symmetry group of the model -reducing the symmetry from Co 0 to a variety of other possible groups which preserve a 3-plane (respectively 2-plane) in 24 of Co 0 . In §4 and §5, we discuss the action of these Co 0 -subgroups on the modules and compute the corresponding twining functions. We identify M 23 , M 22 , McL, HS, U 6 (2) and U 4 (3) as some of the most interesting Co 0 -subgroups preserving some extended superconformal algebra. In §6 and §7, we discuss in some detail the decomposition of the graded partition function of our chiral CFT into characters of irreducible representations of the N = 4 and N = 2 superconformal algebra. In §8, we discuss the special property we require from a moonshine twining function, and show how this property singles out the Mathieu groups in our setup. We close with a discussion in §9. The appendices contain many tables: character tables for the various groups we discuss, tables of coefficients of the vector valued mock modular forms that arise in our class functions, and tables describing the decompositions of our modules into irreducible representations of the various groups.

The Free Field Theory
The chiral 2d conformal field theory that will play a starring role in this paper has two different constructions. The first is described in [3] and starts with 8 free bosons X i compactified on the 8-dimensional torus given by the E 8 root lattice, together with their Fermi superpartners ψ i .
One recognizes representations of the Co 1 sporadic group appearing in the q-series: apart In fact, this model has a (non-manifest) Co 0 ∼ = 2.Co 1 symmetry. In this paper, we will sometimes use n or Z n to denote Z/nZ depending on the context.
A better representation, for our purposes, was discussed in [32]. The E 8 orbifold theory is equivalent to a theory of 24 free chiral fermions λ 1 , λ 2 , . . . , λ 24 , also orbifolded by the Z 2 symmetry λ α → −λ α . This gives an alternative description of the Conway module above. The partition function from this "free fermion" point of view is more naturally written as This is equivalent to the answer (2.2) by non-trivial identities on theta functions. Note that The free fermion theory has a manifest Spin(24) symmetry, but not a manifest N = 1 supersymmetry. However, one can construct an N = 1 superalgebra as follows. There is a unique NS ground state, but there are 2 12 = 4096 Ramond sector ground states, constructed of products of the Ramond sector fermion zero modes. It will be convenient to label the 4096 states by a vector s ∈F 12 2 , whereF 2 = {−1/2, 1/2}. There are therefore 4096 spin fields of dimension 3 2 which implement the flow from the NS to the R sector. Denoting these as W a , one can try to find a linear combination which can serve as an N = 1 supercharge by taking linear As discussed in [32], and as we will review in the next section, there exists a set of c a satisfying the conditions for the operator product algebra of W and the stress tensor T to close properly onto an N = 1 superconformal algebra. This solution breaks the Spin(24) symmetry, and is stabilized precisely by the subgroup Co 0 .
In the rest of this paper, we extend this idea as follows. Instead of choosing a chiral N = 1 super-Virasoro algebra and viewing the theory as an N = 1 SCFT, we choose instead various extended chiral algebras. We will argue that N = 4 and N = 2 super-Virasoro presentations of the theory are in one to one correspondence with choices of subgroups of Co 0 which fix a 3-plane (respectively, 2-plane) in the 24 dimensional representation. This leads us naturally to super-modules with various interesting global symmetry groups, and whose twining functions are easily computed in terms of the partition function (or elliptic genus) of the free fermion conformal field theory. These functions in turn are expressed nicely in terms of mock modular forms, and thus we establish moonshine-like relations for subgroups of Co 0 via this family of modules.

The Superconformal Algebras
We first discuss the largest chiral algebra we will consider, which gives rise to the smallest global symmetry groups. We will construct an N = 4 super-conformal algebra (SCA) in the free fermion orbifold theory. Our strategy is to first construct the SU (2) R-currents, and act with them on an N = 1 supercurrent to generate the full N = 4 SCA. In this process, we break the Co 0 symmetry group down to a proper subgroup as we will discuss in §4.
We start with 24 real free fermions λ 1 , λ 2 , . . . , λ 24 . Picking out the first three fermions, we obtain the currents J i : They form an affine SU (2) current algebra with level 2 as may be seen from their OPE: The next step is to pick an N = 1 supercurrent and act with J i on it. As we reviewed in §2, an N = 1 supercurrent exists in this model and may be written as a linear combination of spin fields. Let us go through this construction in more detail, as we will extend it to find the N = 4 super-algebra. To write the N = 1 supercurrent explicitly, we first group the 24 real fermions into 12 complex ones and bosonize them: In terms of the bosonic fields H = (H 1 , . . . , H 12 ), an N = 1 supercurrent W may be written as where each component of s = (s 1 , s 2 , · · · , s 12 ) takes the values ±1/2, and the coefficients w s are C-numbers. We have introduced cocycle operators c s (p) to ensure that the operators with integer spins commute with all other operators, and the operators with half integral spins anticommute among themselves. The cocycle operators depend on the zero-mode operators p which are characterized by the commutation relation p, e ik·H = ke ik·H , (3.5) where k = (k 1 , . . . , k 12 ) is an arbitrary 12-tuple of C-numbers. The associativity and closure of the OPE of the "dressed" vertex operators requires that and that ǫ(k, k ′ ) satisfies the 2-cocycle condition Moreover, in order for V k to have the desired (anti)-commutation relation, a further condition is imposed: An explicit description of the cocycle for a general vertex operator e ik·H may be chosen as [36] c k (p) = e iπk·M·p . (3.10) In our case, M is a 12 × 12 matrix that has the block form and I 4 is a 4 × 4 matrix with all elements being 1.
Generically, the OPE of W with itself is where we have defined for α < β < γ < δ. Other components of Γ αβγδ are defined by the requirement that it is totally antisymmetrized.
For W to be an N = 1 supercurrent, the last term must vanish: 15) and the first two terms must have the correct normalization We may also write the SU (2) R-currents in (3.1) in the bosonized language: where we have included the cocycles e ±iπp1 . We may now act with the SU (2) currents J i on our N = 1 supercurrent W and extract the singular terms of their OPEs: where W i are slightly modified combinations of spin fields: where Rs ≡ (−s 1 , −s 2 , s 3 , · · · , s 12 ).
We claim that all three W i defined above are valid N = 1 supercurrents. This is because we may obtain, for instance, W 3 from W by rotating the 1-2 plane by π, and the conditions (3.15), (3.16) for being an N = 1 supercurrent are invariant under SO(24) rotations. We may obtain W 2 and W 3 similarly. This shows that each of the W i is an N = 1 supercurrent.
Furthermore, we can check that the OPEs of the W i are given by Therefore, the U (1)-graded NS-sector partition function becomes (3.37) In the above discussion, we have chosen the first three fermions out of a total of 24 to generate a set of SU (2) currents. Together with an N = 1 supercurrent they generate a full N = 4 SCA. It is clear that we are free to choose any three fermions for this purpose. In fact, we could choose an arbitrary three-dimensional subspace of the 24-dimensional vector space spanned by the fermions, and obtain an N = 4 SCA. For a given N = 1 supercurrent, not all choices of three-spaces are equivalent, as we will see in a moment.
Before moving on, we should also mention that we could instead have chosen to single out only two real fermions, and construct a U (1) current algebra instead of an SU(2) current algebra. Completely analogous manipulations then show that each such choice provides an N = 2 superconformal algebra; as a result we can associate N = 2 SCAs with G symmetry to subgroups G of Co 0 which stabilise 2-planes in 24.

Global Symmetries
Enhancing the N = 1 structure of the theory to N = 4 breaks the Co 0 symmetry. We now show that for a specific choice of a three-dimensional subspace of the 24-dimensional vector space, resulting in a specific copy of the N = 4 SCA, the stabilising subgroup of Co 0 is the sporadic group M 22 . Similarly, for a specific choice of a two-dimensional subspace of the 24-dimensional vector space, resulting in a specific copy of the N = 2 SCA, the stabilising subgroup of Co 0 is the sporadic group M 23 . This amounts to a proof that the model described in §2 results in an infinite-dimensional M 22 (resp. M 23 )-module underlying the mock modular forms described in §6 (resp. §7) arising from its interpretation as an N = 4 (resp. N = 2) module.
Recall that the theory regarded as an N = 0 theory has a Spin(24) symmetry resulting from the SO(24) rotations on the 24-dimensional space, and choosing an N = 1 supercurrent breaks the Spin(24) group down to its subgroup Co 0 . The group Co 0 is the automorphism group of the Leech lattice Λ Leech , and various interesting subgroups of Co 0 can be regarded as the subgroups of Co 0 which stabilise certain combinations of lattice vectors in Λ Leech . To study the automorphism group of the module when fixing more structure-more supersymmetries in this case-it will therefore be useful to describe the enhanced supersymmetries in terms of Leech lattice vectors. In chapter 10 of [35], it is shown that if we choose an appropriate tetrahedron in the Leech lattice whose edges have lengths squared 16 × (2, 2, 2, 2, 3, 3) in the normalisation described below, the subgroup of Co 0 that leaves all vertices of the tetrahedron invariant is M 22 .
To be more precise, let e γ , γ ∈ {1, 2, . . . , 24} be an orthonormal basis of R 24 and let Λ Leech be the copy of the Leech lattice generated by the vectors {2 γ∈C e γ | C is a special octad of the extended Golay code} and the vector −4e 1 + 24 γ=1 e γ . (One can show that all 24 vectors of the form −4e α + 24 γ=1 e γ are in Λ Leech .) Define the tetrahedron T {α,β} to be that whose four vertices are O = 0, X α = 4e α + 24 γ=1 e γ , X β = 4e β + 24 γ=1 e γ and P αβ = 4e α + 4e β , for any α, β ∈ {1, 2, . . . , 24} with α = β. For every such T {α,β} , the subgroup fixing every vertex is a copy of M 22 , a sporadic simple group of order 2 7 · 3 2 · 5 · 7 · 11 = 443, 520 and the subgroup of A natural question is: what is the symmetry group G that fixes a given choice of N = 2 superconformal structure? Given the above description of the M 22 action, we can choose the R 2 ⊂ R 3 generated by e α and 24 γ=1 e γ and use the two free fermions lying in the R 2 to construct the N = 2 sub-algebra of the N = 4 SCA. Specifically, the U (1) R-symmetry is simply the rotation of the R 2 . From the above discussion, it is not hard to see that there is a copy of M 23 fixing e α and hence stabilising the N = 2 structure. Recall that M 23 is a sporadic simple group of order 2 7 · 3 2 · 5 · 7 · 11 · 23 = 10, 200, 960. In terms of the Leech lattice, it corresponds to the fact that the stabiliser of the triangle in Λ Leech whose edges have lengths squared 16 × (6, 3, 2), with vertices chosen to be O, X α and 2 24 γ=1 e γ , is a copy of M 23 inside the copy of Co 0 stabilising Λ Leech .
This furnishes a proof that the theory considered in §2 leads to M 22 -and M 23 -modules underlying the mock modular forms defined in §6 and §7.
We should mention that by stabilising slightly different choices of geometric structure (instead of the tetrahedron and triangle which result in preservation of M 22 or M 23 , respectively), one can find instead other global symmetry groups G. We can therefore construct G-modules with N = 4 (N = 2) superconformal symmetry, leading to mock modular forms, for any G which arises by stabilizing an object contained in a 3-plane (2-plane). We will discuss as further examples, beyond M 22 and M 23 , the following cases. We stress however that the special properties of the twining functions enjoyed by the M 22 and M 23 moonshines do not extend to these other groups, as we will show in §8.

Another N = 4 super-module
The group U 4 (3), which has order 3,265,920, can arise as the stabilizer of a suitably chosen 3plane in the 24 of Co 0 [35]. The where we have introduced a chemical potential for the J 0 charges and written y = e(z). As we have seen in §2, the two different ways of writing this function, (5.2) and (5.3), are intuitively connected more closely with the free fermion and E 8 root lattice descriptions of the chiral CFT, respectively. Of course, the NS sector partition function (cf. (3.37), (2.2) and (2.6)) is related to the above graded Ramond-sector partition function by a spectral flow transformation There is a natural way in which one can twine the above function under certain subgroups of Co 0 . From the previous discussions, we see that the representation 24 plays a central role in the way various subgroups of Co 0 act on the model. Let's denote by {ℓ g,k ,l g,k }, k = 1, . . . 12, the 12 complex conjugate pairs of eigenvalues of g when acting on 24. This information is conveniently encoded in the so-called Frame shape of g, given by . . , and m n ∈ Z, m n = 0 , satisfying n L n m n = 24, through the fact that the 12 pairs {ℓ g,k ,l g,k } are precisely the 24 roots solving the equation As discussed in §3 and §4, in order to preserve at least N = 2 superconformal symmetry and hence be able to twine the graded R-sector partition function (5.1), the subgroup G must leave at least a 2-dimensional subspace in 24 pointwise invariant. In the graded partition function this corresponds to leaving the factor θ i (τ, 2z) in (5.2) invariant. As a result, for every conjugacy class [g] of such a group G we can choose ℓ g,1 =l g,1 = 1. It is easy to see that when acting on the twisted sector contributing to the terms involving θ i with i = 3, 4 in (5.2), the group element g simply replaces θ 11 i (τ, 0) with 12 k=2 θ i (τ, ρ g,k ) (5.5) where e(ρ g,k ) = ℓ g,k . When trying to do the same for the contribution from the untwisted sector contributing to the term involving θ 2 in (5.2), however, we see that the above simple consideration suffers from an ambiguity. This can be seen from the fact that θ 2 (τ, ρ) = −θ 2 (τ, ρ+ 1) and hence the answer cannot be determined simply by looking at the g-eigenvalues on 24. This of course is a reflection of the fact that the ground states in the untwisted sector transforms in the 2 12 -dimensional representation of Spin(24) (when the theory is treated as a Virasoro module; that is, when we do not keep any superconformal structure fixed), and the 2-fold ambiguity in taking the logarithm of the eigenvalues ℓ g,k is precisely the 2-fold covering structure of Spin (24) with respect to SO (24). As a result, to specify the twining of the untwisted sector contribution we also need to know the action of G on its 2 12 -dimensional representation, henceforth denoted 4096, which corresponds to 4096 = 1 + 276 + 1771 + 24 + 2024 in terms of the decomposition into irreducible representations of Co 0 .
Finally we will discuss the twining of the vanishing term 0 = θ1(τ,2z)θ 11 1 (τ,0) 2η 12 (τ ) in (5.2). It vanishes because the 2 12 ground states in the twisted sector come in pairs with opposite eigenvalues for (−1) F . Moreover, exchanging the pair corresponds to complex conjugation ψ a ↔ψ a , a = 1, . . . , 12 of the complex fermions. Recall that one of the complex fermions, called ψ 1 in (3.19), was used to construct the U (1)-current and we are interested in the graded partition function where we introduce a chemical potential z for the corresponding U (1)-charge. Because exchanging ψ 1 ↔ψ 1 also induces a flip of U (1)-charges, captured by z ↔ −z, the contribution of the first complex fermion does not vanish, corresponding to the fact that the identity only forces θ 1 (τ, z) to vanish at z ∈ Z. Consequently, the g-twining of 0 = θ1(τ,2z)θ 11 1 (τ,0) 2η 12 (τ ) in (5.2) is only non-zero if and only if ρ g,k ∈ Z for all k = 2, . . . , 12; in other words, when the cyclic group generated by g fixes nothing but a 2-plane. By inspection we find that, among the groups we consider, such group elements must be in the conjugacy classes 23AB ⊂ M 23 , The pairs of these conjugacy classes corresponding to the letter A and B (D and E) are mutually inverse, and so their respective traces, on any representation, are related by complex conjugation. In terms of our construction, choosing one over the other is the same as choosing what one labels ψ 1 andψ 1 , and the same as choosing an orientation on the 2-plane fixed by the group element in 24. As a result, from (5.6) we see that the θ 1 term in the partition functions twined by these conjugate "A" (D) and "B" (E) classes come with an opposite sign. Let us work with the principal branch of the logarithm, and choose ρ g,k ∈ [0, 1/2] in (5.5). Then, by direct computation-we must compute directly, for the choice of labels for mutually inverse conjugacy classes is not natural-there exists a copy of Co 0 in Spin (24) for which the signs in (5.8) are and ǫ g,1 = −1 for the inverse classes, 23B ⊂ M 23 , 20B ⊂ HS, &c.
Putting these different contributions together, we conclude that for every [g] ⊂ G where G is a Co 0 subgroup preserving (at least) a 2-plane in 24, the corresponding twined graded R-sector partition function reads where ǫ g,2 = Tr 4096 g The cycle shapes Π g and the values of Tr 4096 g for [g] ∈ G are collected for various G ⊂ Co 0 in Appendix B. In §6 and §7 we will see how the above twining leads to the mock modular forms playing the role of the McKay-Thompson series in our mock modular moonshine.

The N = 4 Decompositions
From the discussion in §3, it is clear that the theory discussed in §2 is a module of the N = 4 superconformal algebra. In this section we will study the decomposition of the Hilbert space V into irreducible representations of the N = 4 SCA and see how the decomposition leads to mock modular forms relevant for the M 22 moonshine which we will discuss in §8.
Recall that the N = 4 superconformal algebra contains subalgebras isomorphic to the affine Lie algebraŝl 2 and the Virasoro algebra. In a unitary representation the former of these acts with level m − 1, for some integer m > 1, and the latter with central charge c = 6(m − 1).
The unitary irreducible highest weight representations v N =4 m;h,j are labeled by the two quantum numbers h and j which are the eigenvalues of L 0 and 1 2 J 3 0 , respectively, when acting on the highest weight state [37,38]. In the Ramond sector of the superconformal algebra there are two types of highest weight representations: the short (or BPS, supersymmetric) ones with and j ∈ {0, 1 2 , · · · , m−1 2 }, and the long (or non-BPS, non-supersymmetric) ones with h > m−1 in the short and long cases, respectively, [37]. In the above formulas, the function µ m;j (τ, z) is given by and Ψ 1,1 is a meromorphic Jacobi form of weight 1 and index 1 given by Finally, we have used the theta functions defined for all 2m ∈ Z >0 and r − m ∈ Z, and satisfying The vector-valued theta function θ m = (θ m,r ), r − m ∈ Z/2mZ, is a vector-valued Jacobi form of weight 1/2 and index m satisfying where the S θ and T θ matrices are 2m × 2m matrices with entries We will take m ∈ Z for the rest of this section. When we consider N = 2 decompositions in the next section, we will use the theta function with half-integral indices.
From the above discussion, it is clear that the graded partition function of a module for the c = 6(m − 1) N = 4 SCA admits the following decomposition Furthermore, from the identity The rest of the components of , (6.15) studied in [39], for instance, has the following relation to the modular group SL 2 (Z): let the (non-holomorphic) completion of µ m;0 (τ, z) bê Here S m = (S m,r ) is the vector-valued cusp form under SL 2 (Z) with a non-trivial multiplier, whose components are given by the unary theta function For later use, note that the cusp form S m,r (τ ) is defined for all 2m ∈ Z and r − m ∈ Z/2mZ.
The way in which the functions Z (m) andμ m;0 transform under the Jacobi group shows that the non-holomorphic function r∈Z/2mZF (m) r (τ ) θ m,r (τ, z) transforms as a Jacobi form of In other words, ), r ∈ Z/2mZ is a vector-valued mock modular form with a vector-valued shadow c 0 S m , whose r-th component is given by S m,r (τ ), with the multiplier for SL 2 (Z) given by the inverse of the multiplier system of S (m) (cf. (6.8)).
Now we are ready to apply the above discussion to the graded partition function of the theory discussed in §2. Recall that in this case we have c = 12, or m = 3 in other words. The N = 4 decomposition of (5.1) gives where . . . stand for terms with expansion Ψ −1 1,1 q α y β with α−β 2 /12 > 3. More Fourier coefficients of the functions h r (τ ) are recorded in Appendix C, where h = h g for [g] = 1A. Note that all the graded multiplicities c ′ r (n − r 2 12 ) appear to be non-negative. Of course, this is guaranteed by the fact that V is a module for the N = 4 SCA as shown in §3. In particular, the Fourier coefficients of h r (τ ) appear to be all non-negative apart from that of the polar term −2q −1/12 in h 1 . From the above discussion, we see that h = (h r ), r ∈ Z/6Z is a weight 1/2 vectorvalued mock modular form with 6 components and 2 independent components (h 0 = h 3 = 0, , with the shadow given by 24 S (3) . This is to be contrasted with the elliptic genus of a generic non-chiral SCFT. For example, the sigma model of a K3 surface has c = 6, and the elliptic genus is given by In §3 we have shown that the theory under consideration, as a module for the N = 4 SCA, admits a faithful action via automorphisms by a group G, as long as G is a subgroup of Co 0 fixing at least a 3-plane. The graded partition function Z g (τ, z) twined by any such g ∈ G is given by (5.8), and from the fact that the action of g commutes with the N = 4 SCA, we expect 21) and the coefficients of to be given by characters of the G-module arising from the free field theory in §2.
We have explicitly computed the first 30 or so coefficients of each q-series h g,r (τ ) for all conjugacy classes [g] of G, for G = M 22 and G = U 4 (3). These can be found in the tables in Appendix C. Subsequently, we compute the first 30 or so G-representations V G r,n in terms of their decompositions into the irreducible G-representations. They can be found in the tables in Appendix D.
Next we would like to discuss the mock modular property of the functions h g = (h g,r ). The Hecke congruence subgroups are defined as We expect Z g to be a weak Jacobi form of weight zero, index 2 for the group Γ 0 (o g ) where o g is the order of the group element g ∈ G. This can be verified explicitly from the expression From now on we will concentrate on the case when ℓ is half-integral, and henceĉ is even. Their graded characters, defined as for the long multiplets, and for the short multiplets with Q =ĉ 2 , respectively. The character ch N =2 ℓ;c/24,Q (τ, z) for Q =ĉ 2 is given in (7.5). In the above formula, we have used the Appell-Lerch sum (6.15) and defined Note that the above characters satisfy under charge conjugation.
From the relation between the massless and massive characters as well as the charge conjugation symmetry of the theory, we expect the graded partition function of a module, invariant under charge conjugation, for the N = 2 SCA with even central charge c = 3(2ℓ + 1), to admit the following decomposition: In the last equation, we have defined Similar to the case of short N = 4 characters, through its relation to the Appell-Lerch sum,μ ℓ;0 admits a completion which transforms as a weight one, half-integral index Jacobi form under the Jacobi group. More precisely, define μ ℓ;0 by replacing µ m;0 withμ ℓ;0 and the integral m with the half-integral ℓ in (6.16). Then μ ℓ;0 transforms like a Jacobi form of weight 1 and index ℓ under the Jacobi group SL 2 (Z) ⋉ Z 2 . Following the same computation as in the previous section, we is a vector-valued mock modular form with a vector-valued shadow C 0 S ℓ = C 0 (S ℓ,j (τ )). Now we are ready to apply the above discussion to the graded partition function of the VOA discussed in §2. The N = 2 decomposition gives where . . . denote the terms with expansion Ψ −1 Again, we observe that all the multiplicities of the multiplets with characters ch N =2 3 2 ;h,Q appear to be non-negative, consistent with our construction of V as an N = 2 SCA module.
In general, from the previous sections we have seen that the graded partition function twined by any element g of a subgroup G of Co 0 should admit a decomposition into N = 2 characters.
We write Moreover, from the discussion in §5 we have seen that (7.14) and the coefficients of these functions (TrṼ G r,n g) q n−j 2 /6 (7.15) are given by characters of the G-modulẽ which descends from the free field theory in §2. In particular, for any n, the G-representatioñ V G −1/2,n is the complex conjugate ofṼ G 1/2,n .

Mathieu Moonshine
In the previous sections we have seen that the free field theory described in §2 leads to infinitedimensional G-modules underlying a set of vector-valued mock modular forms from its N = 4 (N = 2) structures for any subgroup G of Co 0 fixing at least a 3-plane (2-plane) in 24. In this section we will discuss a natural property of the vector-valued mock modular forms that To be more precise, denote by Similarly, in the 23 instances of umbral moonshine [7,8] involving mock modular forms, it has been conjectured that, for any Niemeier root system X, the graded characters H X g = (H X g,r ) of the (conjectural) umbral G X -modules K X , are all mock modular forms for certain subgroups Γ X g of SL 2 (Z), with the following behaviour at the cusps See [8] for more details. A conceptual explanation for such a striking special property of the functions involved in moonshine is still unsettled. However, following the precedent set in the monstrous case by [46], this property can be regarded as a consequence of the (conjectural for umbral cases other than X = A 24 1 ) construction of these functions via a uniform sum over modular images, sometimes referred to as the regularised Poincaré sum, or Rademacher sum. See for instance [49] for a review. In turn, such a sum over modular images has been given a physical interpretation in terms of the gravity dual of the 2d CFT [48]. See [46] for specific conjectures that relate the genus zero property of monstrous moonshine to gravity.
A natural question is therefore: among the subgroups of Co 0 fixing 2-or 3-planes for which we have constructed a module in this work, is there any group G whose corresponding twined mock modular forms satisfy a condition analogous to the preceding cases of moonshine, monstrous and umbral, described above? We will see that the Mathieu groups M 22 and M 23 , in the N = 4 and N = 2 cases respectively, indeed render mock modular forms satisfying (i) q 1/6 h g,r (τ ) = O(1) as τ → i∞ for all r, for all g ∈ M 22 , and To investigate the behaviour of h g,r andh g,j , let us look at the behaviour of the twined graded partition function Z g (τ, z) (cf. (5.8)) at other cusps. From the SL 2 (Z) transformation of the Jacobi theta functions (cf. Appendix A), we obtain Near the cusp τ → i∞, the different contributions have the following leading behaviour: where ⌊x⌋ denotes the largest integer that is not greater than x.
From the fact that the character of the short, uncharged (in the sense that j = 0 resp. Q = 0) multiplet of both the N = 2 and N = 4 SCAs approaches a constant at any cusp, we see that h g (resp.h g ) has a pole at the cusp τ → 1/N whenever f N,1 (Π g ) ≤ 0.
We see that, among all the groups we have considered, U 4 (3), U 6 (2) as well as the McL contain a conjugacy classes with Frame shape Π g = 3 9 /1 3 , and the HS group has a conjugacy class with Frame shape Π g = 5 5 /1 1 . One can explicitly check that f 1,1 (Π g ) = 0 for these classes and hence the corresponding twining function has a pole at the cusp τ → 1.
In fact, in all cases f N,1 (Π g )/2 is given by the inverse of the width of the cusp. This verifies that there is no pole in the twining function coming from the contribution of θ 1 and θ 2 to the twined graded partition function (5.8) at cusps different from i∞ and its images under Γ g . To make sure that there is no pole in the twining function coming from the contribution of θ 3 and θ 4 , we first note that f N,2 (Π g ) = 0 whenever g has at least one eigenvalue −1 and at least one eigenvalue whose N -th power is −1. In this case, the contribution from lim τ →i∞ Θ 3,N (τ, z) and The reader will note that many of the numbers which occur as dimensions of irreducible representations of M 23 also occur as dimensions of irreducible representations for M 24 . Indeed, looking at the tables in C, one is tempted to guess that there is an alternative construction, or hidden symmetry in our model, which yields an M 24 -module with the same graded dimensions.
In fact, the procedure we have explained for computing twinings can be carried out for any element of M 24 , regarded as a subgroup of Co 0 , for any such element fixes a 2-space in 24.
However, there is no 2-space that is fixed by every element of a given copy of M 24 , and explicit computations reveal that any M 24 -module structure on the module we have constructed for M 23 would have to involve virtual representations. This indicates that there is no natural extension to M 24 of the M 23 -module we have considered here. However, there is a certain modification of our method for which M 24 can be expected to play a leading role, and we refer the reader to the next, and final, section for a description of this.

Discussion
In this paper, we have demonstrated that, starting with the free field Co 0 module of [32], one can There are several future directions. We considered here the N = 2 and N = 4 extended chiral algebras, and the subgroups of Co 0 that they preserve. Other extended chiral algebras may also yield interesting results. For instance, supersymmetric sigma models with target a Spin (7) manifold give rise to an extended chiral algebra [42], whose representations were studied in [43].
It is an extension of the N = 1 superconformal algebra where instead of adding a U (1) current (which extends the theory to an N = 2 superconformal theory), one chooses an additional Ising factor. This has a natural implementation in our setup, and studying this algebra and the resulting moonshine is something we intend to do in the future. In particular, moonshine for the groups M 24 , Co 2 , and Co 3 (involving modular forms, rather than mock modular forms) can be expected to arise in this setting.
The motivation that led, eventually, to the present study was actually to find connections between geometrical target manifolds associated to c = 12 conformal field theories, and sporadic groups. The elliptic genera of Calabi-Yau fourfolds were computed in [44], for instance; their structure is reminiscent of some of the modules we have seen here, and we intend to further explore and describe some of these connections in a future publication. Likewise, hyperkähler fourfolds, as well as the Spin(7) manifolds mentioned above, provide a wide class of geometries where an analogue of the connections between M 24 and K3 may be sought.
Last but not least, there are suggestive connections between the trace functions in moonshine modules, and the path integral for quantum gravity. Both the CFT appearing in monstrous moonshine and the Co 0 module that played a starring role in this paper appear to play special roles also in AdS 3 quantum gravity, where they are candidates for CFT duals to pure (super)gravity [45]. The genus zero property of the twining functions in monstrous moonshine can be reformulated as a condition that these class functions should be expressed as Rademacher sums based on a fixed polar part [46,47]; this latter description then applies uniformly to monstrous moonshine and umbral moonshine. In this paper we conjecture and provide evidence that it also applies to our M 22 and M 23 mock modular moonshine. In particular, we have shown that the mock modular forms relevant for the M 22 and M 23 moonshine satisfy the specific pole condition which indicates a possible construction via Rademacher sums. It is tempting to associate this property with the existence of a "Farey tail"-like expansion for the partition function of a dual quantum gravity theory [48]. Making this connection more precise, and especially finding examples which extend to asymptotically large central charge (as opposed to the present modules, which are at c = 12), is an enticing direction for the future.

Acknowledgements
We
−D                       Tables   Table 29: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,1(τ ) into irreducible representations χn of M22.