The Largest Mathieu Group 1 and (Mock) Automorphic Forms Miranda C. N. Cheng (cid:91),(cid:92),(cid:93) and John F. R. Duncan † (cid:91)Department of Mathematics, Harvard University, 2 Cambridge, MA 02138, USA 1 (cid:92)Department of Physics, Harvard University, 0 2 Cambridge, MA 02138, USA n (cid:93)Universit´e Paris 7, UMR CNRS 7586, Paris, France a † Department of Mathematics, Case Western Reserve University, J Cleveland, OH 44106, USA 9 1 ] T Abstract R We review the relationship between the largest Mathieu group and various modular objects, . h includingrecentprogressontherelationtomockmodularforms. Wealsoreviewtheconnections t between these mathematical structures and string theory on K3 surfaces. a m [ Contents 1 v 1 Cusp Forms and M24 1 0 4 2 Mock Modular Forms and M 7 24 1 4 3 Weak Jacobi Forms and M 11 . 24 1 0 4 Siegel Modular Forms and M 15 2 24 1 : 5 String Theory and M 19 v 24 i X A Modular Forms 21 r a B Character Table 24 C Tables of Decompositions 25 1 Cusp Forms and M 24 We start by recalling some facts about the largest Mathieu group, M . The group M may be 24 24 characterised as the automorphism group of the unique doubly even self-dual binary code of length 1Submitted to the AMS Proceedings of Symposia in Pure Mathematics (String-Math 2011). 1 24 with no words of weight 4, also known as the (extended) binary Golay code. In other words, there is a unique (up to permutation) set, G say, of length 24 binary codewords (sequences of 0’s and 1’s) such that any other length 24 codeword has even overlap with all the codewords of G if and only if this word itself is in G, and the number of 1’s in each codeword of G is divisible by but not equal to 4. The group of permutations of the 24 coordinates that preserves the set G is the sporadic group M . See, for instance, [1]. 24 [g] cycle shape η (τ) k n N h g g g g g 1A 124 η24(τ) 12 1 1 1 2A 1828 η8(τ)η8(2τ) 8 2 2 1 2B 212 η12(2τ) 6 2 4 2 3A 1636 η6(τ)η6(3τ) 6 3 3 1 3B 38 η8(3τ) 4 3 9 3 4A 2444 η4(2τ)η4(4τ) 4 4 8 2 4B 142244 η4(τ)η2(2τ)η4(4τ) 5 4 4 1 4C 46 η6(4τ) 3 4 16 4 5A 1454 η4(τ)η4(5τ) 4 5 5 1 6A 12223262 η2(τ)η2(2τ)η2(3τ)η2(6τ) 4 6 6 1 6B 64 η4(6τ) 2 6 36 6 7AB 1373 η3(τ)η3(7τ) 3 7 7 1 8A 12214182 η2(τ)η(2τ)η(4τ)η2(8τ) 3 8 8 1 10A 22102 η2(2τ)η2(10τ) 2 10 20 2 11A 12112 η2(τ)η2(11τ) 2 11 11 1 12A 214161121 η(2τ)η(4τ)η(6τ)η(12τ) 2 12 24 2 12B 122 η2(12τ) 1 12 144 12 14AB 112171141 η(τ)η(2τ)η(7τ)η(14τ) 2 14 14 1 15AB 113151151 η(τ)η(3τ)η(5τ)η(15τ) 2 15 15 1 21AB 31211 η(3τ)η(21τ) 1 21 63 3 23AB 11231 η(τ)η(23τ) 1 23 23 1 Table 1: The cycle shapes, weights (k ), levels (N ) and orders (n ) of the 26 conjugacy classes of the sporadic g g g groupM . Thelengthoftheshortestcycleish =N /n . ThenamingoftheconjugacyclassesfollowstheATLAS 24 g g g convention [2]. We write 7AB, for example, to indicate that the entries of the incident row are valid for both the conjugacy classes 7A and 7B. As such, M naturally admits a permutation representation of degree 24, which we denote R 24 and call the defining representation of M , and via R we may assign a cycle shape to each of its 24 elements. For example, to the identity element we associate the cycle shape 124, to an element of M that is a product of 12 mutually commuting transpositions we associate the cycle shape 212, 24 and so on. More generally, any cycle shape arising from an element of M (or S , for that matter) 24 24 is of the form r (cid:88) i (cid:96)1i (cid:96)2···i (cid:96)r, (cid:96) i = 24 , 1 2 r s s s=1 for some (cid:96) ∈ Z+ and 1 ≤ i < ··· < i ≤ 23 with r ≥ 1. Clearly, the cycle shape of an element s 1 r g ∈ M depends only on its conjugacy class, denoted by [g], although different conjugacy classes 24 can share the same cycle shape. We write 2 χ(g) = tr g R for the character attached to this 24-dimensional representation of M . The decomposition of R 24 into irreducible representations of M is given by 24 R = ρ ⊕ρ 1 23 (cf. Table 3). The value of the character χ(g) equals the number of fixed points of g in the action on the set of 24 points. In particular, note that χ(g) = (cid:96) in case i = 1 and χ(g) = 0 otherwise. 1 1 It turns out that the cycle shapes of M have many special properties that will be important 24 for the understanding of the modular properties of the associated McKay–Thompson series which wewilldiscussshortly. First, theM cycleshapesareprivilegedinthattheyarealloftheso-called 24 balanced type [3], meaning that for each g ∈ M24 there exists a positive integer Ng such that if (cid:81)i(cid:96)ss is the cycle shape associated to g then (cid:89)i(cid:96)s = (cid:89)(cid:18)Ng(cid:19)(cid:96)s . s i s s s We will refer to the number N as the level of the g. g If g has cycle shape i(cid:96)11···i(cid:96)rr then the order of g is the least common multiple of the is’s. A secondspecialpropertyof[g] ⊂ M isthattheorderofg coincideswiththelengthi ofthelongest 24 r cycle in the cycle shape. Henceforth we will denote n = i . g r Finally,observethatforallg ∈ M thelevelN definedaboveequalstheproductoftheshortest 24 g and the longest cycle. Hence we have n h = N where h denotes the length of the shortest cycle g g g g in the cycle shape. Moreover, we also have the property h |n and h |12. This is reminiscent of g g g the fact, significant in monstrous moonshine [3], that the normaliser of Γ (N)/(cid:104)±Id(cid:105) in PSL (R) 0 2 (cf. (1.2)) is a group containing a conjugate of Γ (n/h)/(cid:104)±Id(cid:105) where n = N/h and h is the largest 0 divisor of 24 such that h2|N. We also set k = (cid:80)r (cid:96) /2 to be half of the total number of cycles g s=1 s and call it the weight of g. Of course, N , n , h and k depend only on the conjugacy class [g] g g g g containing g. All the values of these parameters can be found in Table 1. To each element g ∈ M we can attach an eta-product, to be denoted η , which is the function 24 g on the upper half-plane given by (cid:89) η (τ) = η(i τ)(cid:96)s (1.1) g s s where (cid:81) i(cid:96)s is the cycle shape attached to g, and η(τ) is the Dedekind eta function satisfying s s η(τ) = q1/24(cid:81) (1−qn) for q = e(τ), where here and everywhere else in the paper we use the n∈Z+ shorthand notation e(x) = e2πix . As was observed in [4, 5], the eta-product η associated to an element g ∈ M (or rather, to its g 24 conjugacy class [g]) is a cusp form of weight k for the group Γ (N ), with a Dirichlet character ς g 0 g g that is trivial if the weight k is even and is otherwise defined, in terms of the Jacobi symbol (n), g m by (cid:16) (cid:17)  Ng (−1)d−21, d odd, d ςg(γ) = (cid:16)Ng(cid:17), d even, d 3 in case d is the lower right entry of γ ∈ Γ (N ). Let’s recall that 0 g (cid:26) (cid:12) (cid:18)a b(cid:19) (cid:27) Γ (N) = γ(cid:12) γ = ∈ SL (Z), c ≡ 0 mod N , (1.2) 0 (cid:12) c d 2 and a cusp form is a modular form which vanishes in the limit as τ → i∞. Interestingly,thesecuspformspointtotheexistenceofaninfinite-dimensionalZ-gradedmodule for M ; a fact we will now explain. It is by now a routine task to construct a module for M 24 24 whose graded dimension is the reciprocal of the cusp form η24(τ). To see this, let us consider the defining 24-dimensional representation R. To obtain a q-series we also need an extra Z-grading. Hence we will take a copy of this representation, denoted by R , for each positive integer k and k consider the direct sum ∞ (cid:77) V = R ⊕R ⊕··· = R . 1 2 k k=1 Furthermore we would like to consider the graded Fock space H built from the above infinite- dimensional vector space V ∞ ∞ (cid:77) (cid:77) H = SnV = H , SnV = V⊗n/S , k n n=0 k=0 where the Z-grading is naturally inherited from that of V. The associated partition function is then given by ∞ (cid:88) 1 Z(τ) = qk−1dimH = . k η24(τ) k=0 Inthelanguageofvertexoperatoralgebras,thisissimplythepartitionfunctionoftheconformal field theory (CFT) with 24 free chiral bosons. Namely, we consider 24 pairs of creation and (i) (i) annihilation operators a ,a ,i = 1,···,24 for each positive integer k, which furnish the 24- −k k dimensional representation R and satisfy the commutation relation k (i) (j) [a ,a ] = δ δ . k (cid:96) ij k+(cid:96),0 The quantum states of this theory are built from the unique vacuum state |0(cid:105), characterised by the (i) condition a |0(cid:105) = 0 for all n ≥ 0,i = 1,···,24, and take the form n a(i1)···a(iN)|0(cid:105) , n > 0, 1 ≤ i ≤ 24 , −n1 −nN k k while the Hamiltonian of the theory is given by the operator 24 ∞ Hˆ = (cid:88)(cid:88)na(i) a(i)−1 . −n n i=1n=1 Hence the partition function of this theory is indeed given by the modular form 1 Z(τ) = tr qHˆ = . H η24(τ) 4 Notice that, physically the (−1)-shift in the Hamiltonian operator corresponds to the ground state energy in the quantum theory with H as the space of quantum states (the Hilbert space). From the above construction, in particular the fact that the M -action commutes with the 24 Z-grading, we conclude that not only H but also each of its components H are M -modules. k 24 Hence, for each element g ∈ M we can consider the following twisted partition function, or 24 McKay-Thompson series ∞ Z (τ) = tr gqHˆ = (cid:88)qk−1(cid:0)tr g(cid:1) . (1.3) g H H k k=0 To be more precise, in the above formula g really stands for the corresponding linear map acting on the representation in question. Note that taking the identity element g = 1, we recover the usual partition function tr qHˆ = Z(τ) encoding the graded dimension of the representation. Moreover, H it is clear from their definition that these McKay-Thompson series depend only on the conjugacy class [g] of the group element. Namely, we have Z (τ) = Z (τ) . g hgh−1 From the description of H, it is now easy to prove that they are given by the eta-products as Z (τ) = q−1exp(cid:32)(cid:88)∞ tr (qHˆg)k(cid:33) = q−1exp(cid:32)(cid:88)∞ (cid:88)∞ qnk trR(gk)(cid:33) = 1 . (1.4) g V k k η (τ) g k=1 n=1k=1 In the path integral language, the summing over k in the expression for logZ can be thought of k as a sum over how many times the time circle of the world-volume wraps around the time circle in the target space. In other words, we can equate the McKay-Thompson series of the infinite-dimensional M - 24 moduleH,givenbythetwistedpartitionfunctionofthephysicaltheoryZ (τ),toaweight−k ,level g g N modular form 1/η (τ). The first few Fourier coefficients of the twisted partition functions and g g the corresponding M -modules H in terms of its decomposition into irreducible representations 24 k can be found in Table 4. A few comments are in order here. First, historically the relation between the η (τ) and M g 24 was phrased in terms of the eta-products themselves as opposed to their reciprocals as we described above. In that case, underlying the Fourier coefficients is an infinite-dimensional, Z-graded super (or virtual) module of M , corresponding to considering instead the following Z/2-graded Hilbert 24 space ∞ (cid:77) H˜ = H˜ ⊕H˜ , H˜ = Λ2n+kV ¯0 ¯1 k¯ n=0 which is the exterior (super)algebra of the infinite-dimensional vector space V, where V is regarded ashavingoddparity[4,6]. AsuperG-moduleisjustaG-moduleW withaG-invariantZ/2-grading W = W ⊕W . Given such a module we may consider the operator F : W → W for which the ¯0 ¯1 subspaces W are eigenspaces with eigenvalue k for F. Then the super trace of an operator A on k¯ W is defined by setting str A = tr (−1)FA. W W In physical terms such a Z/2-grading often indicates the presence of fermions, and the operator F is called the fermion number operator. 5 Second, as the alert reader may have noticed, starting from η24(τ) we can build not only an infinite-dimensional module H for M but this same space (or vertex operator algebra) admits an 24 action by the full symmetric group S , or even O (C)2. In this sense, the relation between these 24 24 21 eta-products and M is not as distinguished as one might have hoped from the point of view of 24 moonshine. Nevertheless, as we will see later in §4, together with various weak Jacobi forms, mock modular forms and Siegel modular forms, they do tie together and form a closely-knitted net of modular objects attached to the sporadic group M . 24 Third, while we have seen that the eta-products η (τ) are cusp forms of level N , from the g g following standard CFT arguments we expect a further, more refined modular property. To explain this, let us first discuss a certain natural generalisation of the above discussion: Given a vertex operator algebra H and two commuting automorphisms g and h we expect there to exist a so-called h-twisted module H(h) for H and one can consider the twisted twisted sector partition function (cid:18) (cid:19) Z τ;g(cid:3) (1.5) h defined as the (rationally) graded trace of g on H(h), thereby generalizing the twisted partition function Z (τ) which is recovered by taking h to be the identity. The significance of the twisted g partition functions (1.5) in the context of monstrous moonshine was first observed by S. Norton in [7] and dubbed generalised moonshine. It is conjectured in loc. cit. that any such function is either constant or a generator for the field of functions on some discrete subgroup of PSL (R) in case H 2 is the moonshine module vertex operator algebra and g and h are a commuting pair of elements of the Monster simple group. An analogue of generalised moonshine for commuting pairs of elements of M is discussed in [8]. 24 Recall that the equivalence between the Hamiltonian and Lagrangian formulations of quantum field theory implies that the insertion of the elements g and h as in (1.5) has an interpretation as altering the boundary condition of the path integral over the maps from an elliptic curve world- sheet into the target space. Identifying the modular group SL (Z) as the mapping class group 2 acting on the elliptic curve world-sheet, we conclude that the following must hold (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) a b Z τ;g(cid:3)h = ε(γ,g,h)Z τ|γ;gahbgc(cid:3)hd , γ = c d ∈ SL2(Z), where ε(γ,g,h) is a phase. Specializing the above equation to the case that h is the identity we conclude that the McKay-Thompson series Z must be invariant up to a phase under the group g Γ (n ) where n is the order of the group. 0 g g Indeed, the eta-product η also defines a cusp form of weight k on the larger (or equal) group g g Γ (n ) ⊇ Γ (N ) if we allow for a slightly more subtle multiplier system. Note that this change in 0 g 0 g multiplier system only comes into play when g is in one of the fixed-point-free classes of M with 24 χ(g) = 0, for otherwise Γ (n ) = Γ (N ). Our convention is to say that a function ξ : Γ → C∗ is a 0 g 0 g multiplier system for Γ of weight w in case the identity ξ(γσ)jac(γσ,τ)w/2 = ξ(γ)ξ(σ)jac(γ,στ)w/2jac(σ,τ)w/2 (1.6) 2Althoughingeneralthecorrespondingtwistedpartitionfunctionwon’tbemultiplicativeifg∈S butg∈/ M . 24 24 Ofthe1575possiblepartitionsof24only30correspondtomultiplicativeeta-products[5]. Allofthe30multiplicative eta-products are recovered from a permutation action of a group of the form 224M and 21 of them arise from the 24 action of M ; these are the 21 eta-products listed in Table 1. 24 6 holds for all γ,σ ∈ Γ and τ ∈ H. Writing γ = (cid:0)a b(cid:1), we have in the above formula γτ = aτ+b, and c d cτ+d the Jacobian jac(γ,τ) = (cτ +d)−2 . In detail, for γ ∈ Γ, we define the slash operator | of weight w associated with the multiplier ξ ξ,w by setting (f| γ)(τ) = ξ(γ)f(γτ)jac(γ,τ)w/2 (1.7) ξ,w so that a holomorphic function f : H → C is a modular form of weight w and multiplier ξ for Γ if and only if (f| γ)(τ) = f(τ) for all γ ∈ Γ. In the present case we have ξ,w (cid:16) 1 (cid:12) (cid:17) 1 (cid:12) γ (τ) = , for all γ ∈ Γ (n ), (1.8) ηg(cid:12)ξg,kg ηg(τ) 0 g where ξ (γ) = ρ (γ)ς (γ) with g ng|hg g ρ (γ) = e(− 1 ) = e(−cd/nh) . (1.9) n|h (γ∞−γ0)nh Note that ρ is actually a character on Γ (n ) since we have that xy ≡ 1 (mod h ) implies ng|hg 0 g g x ≡ y (mod h ) by virtue of the fact that h = N /n is a divisor of 24 for every g ∈ M [3, §3]. g g g g 24 2 Mock Modular Forms and M 24 In the previous section we have seen the relation between M and η-quotients. Recently, there 24 has been an unexpected observation relating M and a weight 1/2 mock modular form. Namely, 24 consider the q-series (cid:32) ∞ (cid:33) H(τ) = 2q−81 (cid:0)−1+45q+231q2+...(cid:1) = q−18 −2+(cid:88)tnqn . n=1 Then the observation made in [9] is that the first few t ’s read n 2×45, 2×231, 2×770, 2×2277, 2×5796, ... and the integers 45, 231, 770, 2277 and 5796 are dimensions of irreducible representations of M . 24 ThisfunctionH(τ),definedin(2.4),enjoysaspecialrelationshipwiththegroupSL (Z);namely, 2 itisaweakly holomorphic mock modular form of weight 1/2onSL (Z)withshadow24η(τ)3 [10,11]. 2 This means that if we define the completion Hˆ(τ) of the holomorphic function H(τ) by setting (cid:90) ∞ Hˆ(τ) = H(τ)+24(4i)−1/2 (z+τ)−1/2η(−z¯)3dz, (2.1) −τ¯ then Hˆ(τ) transforms as a modular form of weight 1/2 on SL (Z) with multiplier system conjugate 2 to that of η(τ)3. In other words, we have (cid:0)Hˆ(τ)| γ(cid:1)(τ) = (cid:15)(γ)−3Hˆ(γτ)jac(γ,τ)1/4 = Hˆ(τ) (cid:15)−3,1/2 for γ ∈ SL (Z), where (cid:15) : SL (Z) → C∗ is the multiplier system for η(τ) satisfying 2 2 (cid:0) (cid:1) η| γ (τ) = η(τ) . (cid:15),1/2 7 See §A for an explicit description of (cid:15). Moregenerally,aholomorphicfunctionh(τ)onHiscalleda(weaklyholomorphic)mockmodular form of weight w for a discrete group Γ (e.g. a congruence subgroup of SL (R)) if it has at most 2 exponential growth as τ → α for any α ∈ Q, and if there exists a holomorphic modular form f(τ) of weight 2−w on Γ such that hˆ(τ), given by (cid:90) ∞ hˆ(τ) = h(τ)+(4i)w−1 (z+τ)−wf(−z¯)dz, (2.2) −τ¯ is a (non-holomorphic) modular form of weight w for Γ for some multiplier system ψ say. In this case the function f is called the shadow of the mock modular form h and ψ is called the multiplier system of h. Evidently ψ is the conjugate of the multiplier system of f. The completion hˆ(τ) satisfies interesting differential equations. For instance, completions of mock modular forms were identified as Maass forms in [12] and this led to a solution of the longstanding Andrews–Dragonette conjecture (cf. loc. cit.). As was observed in [11] we have the identity ∂hˆ(τ) 21−wπ(cid:61)(τ)w = −2πif(τ) ∂τ¯ when f is the shadow of h. Given the observation regarding the first few Fourier coefficients of H(τ) indicated above one would like to conjecture that the entire set of values t for n ∈ Z+ encode the graded dimension n of a naturally defined Z-graded M module K = (cid:76)∞ K with dimK =t . Of course, this 24 n=1 n n n conjecturebyitselfisanemptystatement, sinceallpositiveintegerscanbeexpressedasdimensions of representations of any group, since we may always consider trivial representations. However, the factthatthefirstfewt canbewrittensonicelyintermsofirreduciblerepresentationssuggeststhat n the K should generally be non-trivial, and given any particular guess for a M -module structure n 24 on the K we can test its merit by considering the twists of this mock modular form H obtained by n replacing the identity element in dimK = tr 1 with an element g of the group M . We would n Kn 24 then call the resulting q-series (cid:32) ∞ (cid:33) q−81 −2+(cid:88)trKngqn (2.3) n=1 the McKay–Thompson series attached to g. A non-trivial connection between mock modular forms andM arisesifallsuchMcKay–ThompsonseriesoftheM -moduleK displayinteresting(mock) 24 24 modular properties. In fact, since a function with good modular properties is generally determined by the first few of its Fourier coefficients, it is easier in practice to guess the McKay–Thompson series than it is to guess the representations K . Not long after the original observation was n announced in [9] candidates for the McKay–Thompson series had been proposed for all conjugacy classes [g] ⊂ M in [13, 14, 15, 16], and with functions T˜ (τ) defined as in Table 2 the following 24 g result was established. Proposition 2.1. Let H : H → C be given by (2) H(τ) = −2E2(τ)+48F2 (τ) = 2q−18 (cid:0)−1+45q+231q2+...(cid:1) (2.4) η(τ)3 8 [g] χ(g) T˜ (τ) g 1A 24 0 2A 8 16Λ 2 2B 0 −24Λ +8Λ = 2η(τ)8/η(2τ)4 2 4 3A 6 6Λ 3 3B 0 2η(τ)6/η(3τ)2 4A 0 4Λ −6Λ +2Λ = 2η(2τ)8/η(4τ)4 2 4 8 4B 4 4(−Λ +Λ ) 2 4 4C 0 2η(τ)4η(2τ)2/η(4τ)2 5A 4 2Λ 5 6A 2 2(−Λ −Λ +Λ ) 2 3 6 6B 0 2η(τ)2η(2τ)2η(3τ)2/η(6τ)2 7AB 3 Λ 7 8A 2 −Λ +Λ 4 8 10A 0 2η(τ)3η(2τ)η(5τ)/η(10τ) 11A 2 2(Λ −11η(τ)2η(11τ)2)/5 11 12A 0 2η(τ)3η(4τ)2η(6τ)3/η(2τ)η(3τ)η(12τ)2 12B 0 2η(τ)4η(4τ)η(6τ)/η(2τ)η(12τ) 14AB 1 (−Λ −Λ +Λ −14η(τ)η(2τ)η(7τ)η(14τ))/3 2 7 14 15AB 1 (−Λ −Λ +Λ −15η(τ)η(3τ)η(5τ)η(15τ))/4 3 5 15 21AB 0 (7η(τ)3η(7τ)3/η(3τ)η(21τ)−η(τ)6/η(3τ)2)/3 23AB 1 (Λ −23f +23f )/11 23 23,1 23,2 Table 2: In this table we collect the data that via equation (2.5) and (3.8) define the weight 1/2 (mock) modular forms H (τ) and the weak Jacobi forms Z (τ,z). For N ∈ Z , we denote by Λ the weight 2 modular form on g g + N Γ (N) given by Λ = Nq d (logη(Nτ)) (cf. (A.11)). For N = 23, there are two newforms and we choose the basis 0 N dq η(τ) f =η2 and f given in (A.15). 23,1 23AB 23,2 where F(2)(τ) = (cid:88) (−1)rsqrs/2 = q+q2−q3+q4+... . 2 r>s>0 r−s=1mod2 Then for all g ∈ M , the function 24 χ(g) T˜ (τ) g H (τ) = H(τ)− , (2.5) g 24 η(τ)3 is a (mock) modular form for Γ (N ) of weight 1/2 with shadow χ(g)η(τ)3. Moreover, we have 0 g Hˆ (τ) = ψ(γ)jac(γ,τ)1/4Hˆ (γτ), g g for γ ∈ Γ (n ) where 0 g (cid:90) ∞ Hˆ (τ) = H (τ)+χ(g)(4i)−1/2 (z+τ)−1/2η(−z¯)3dz. g g −τ¯ and the multiplier system is given by ψ(γ) = (cid:15)(γ)−3ρ (γ). ng|hg 9 (See [11] for more information regarding the particular expression for H(τ) given in (2.4).) Noticethattheextramultiplierρ thatappearswhenh (cid:54)= 1isthesameasthatoftheinverse n|h g eta-product 1/η (τ) (cf. (1.9)) which are also related to M , a fact that is in accordance with the g 24 1/2- and 1/4-BPS spectrum of the N = 4, d = 4 theory obtained by K3×T2 compactification of the type II string theory, as will be discussed in more detail in §5. Our discussion above leads to the following conjecture. Conjecture 2.2. The weight 1/2 (mock) modular forms H defined in (2.5) satisfy g ∞ Hg(τ) = q−81(cid:0)−2+(cid:88)qn(trKng)(cid:1) (2.6) n=1 for a certain Z-graded, infinite-dimensional M module K = (cid:76)∞ K . 24 n=1 n Moreover, the representations K are even in the sense that they can all be written in the form n K = k ⊕k∗ for some M -modules k where k∗ denotes the module dual to k . n n n 24 n n n ThefirstfewFouriercoefficientsoftheq-seriesH (τ)andthecorrespondingM -representations g 24 are given in Table 7. A proof of for the first part of the above conjecture, namely the existence of an M -module 24 K = (cid:76)∞ K such that (2.6) holds, has been attained very recently [17]. However, it is important n=1 n to stress that, unlike the eta-product moonshine described in §1, the nature and origin of the M -module K remains mysterious. 24 The compelling modular properties of the functions H constitute strong evidence for the exis- g tence of the M -module K. Yet stronger evidence would be furnished by a uniform construction 24 of the (mock) modular forms H . Such a construction was established recently in [18]. The con- g struction presented there, in terms of Rademacher sums, is of further interest in that it points to an extension of the connection between M and K3 surfaces, expressed in CFT terms via the H (cf. 24 g §3), to a connection in the context of quantum gravity. Also, by comparison with the Rademacher sum expressions for the functions of monstrous moonshine [19], the Rademacher sum construction of the H suggests a reformulation of the crucial genus zero property of monstrous moonshine that g incorporates, and emphasises the distinguished nature of, the functions H attached to M . g 24 Theorem 2.3. [18] Let g ∈ M . Let reg(γ,τ) be the regularisation factor given by reg(γ,τ) = 1 24 in case γ is upper triangular (i.e. γ ·∞ = ∞) and reg(γ,τ) = e(γτ−γ∞) e(γ∞−γτ, 1) (2.7) 8 8 2 otherwise, where e(x,s) is the following generalisation of the exponential function: (cid:88) (2πix)m+s e(x,s) = . (2.8) Γ(m+s+1) m≥0 Given Γ < SL (Z) containing the group Γ of upper triangular matrices in SL (Z), and given a 2 ∞ 2 character ρ on Γ define the Rademacher sum (cid:88) R (τ) = lim ψ(γ)e(−γτ)reg(γ,τ)jac(γ,τ)1/4 Γ,ρ 8 K→∞ γ∈(Γ∞\Γ)<K whereψ(γ) = ρ (γ)(cid:15)(γ)−3 and(Γ \Γ) denotes the set of cosets of Γ inΓhaving a represen- ng|hg ∞ <K ∞ tativewith lowerrow(c,d)satisfyingthe bounds0 ≤ c < K and|d| < K2. ThenH (τ) = −2R (τ) g Γ,ρ when Γ = Γ (n ) and ρ = ρ . 0 g ng|hg 10