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Umbral Moonshine communications in number theory and physics Volume 8, Number 2, 101–242, 2014 Umbral moonshine Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A. Harvey We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterized naturally by the divisors of 12. The Mathieu group correspondence recently discov- ered by Eguchi–Ooguri–Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterizes the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight 2. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar– Murthy–Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moon- shine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an Umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjec- ture of Conway–Norton. Curiously, we find a number of Ramanu- jan’s mock theta functions appearing as McKay–Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homo- geneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay cor- respondence and McKay’s monstrous Dynkin diagram observation manifesting simultaneously and compatibly. 1 Introduction 104 2 Automorphic forms 110 2.1 Mock modular forms 111 2.2 Jacobi forms 114 101 102 Miranda C. N. Cheng et al. 2.3 Meromorphic Jacobi forms 116 2.4 Superconformal algebra 118 2.5 Extremal Jacobi forms 120 2.6 Siegel modular forms 133 3 Finite groups 137 3.1 Specification 137 3.2 Signed permutations 139 3.3 Realization part I 141 3.4 Realization part II 145 3.5 Dynkin diagrams part I 148 3.6 Dynkin diagrams part II 151 4 McKay–Thompson series 156 4.1 Forms of higher level 157 4.2 Lambency two 161 4.3 Lambency three 163 4.4 Lambency four 165 4.5 Lambency five 167 4.6 Lambencies seven and thirteen 169 4.7 Mock theta functions 169 4.8 Automorphy 171 5 Conjectures 174 Umbral Moonshine 103 5.1 Modules 174 5.2 Moonshine 176 5.3 Modularity 178 5.4 Discriminants 178 5.5 Geometry and physics 181 Acknowledgments 184 Appendix A Modular forms 185 A.1 Dedekind eta function 185 A.2 Jacobi theta functions 186 A.3 Higher level modular forms 186 Appendix B Characters 187 B.1 Irreducible characters 189 B.2 Euler characters 193 Appendix C Coefficients 197 C.1 Lambency two 198 C.2 Lambency three 199 C.3 Lambency four 201 C.4 Lambency five 204 C.5 Lambency seven 208 C.6 Lambency thirteen 214 Appendix D Decompositions 220 104 Miranda C. N. Cheng et al. D.1 Lambency two 221 D.2 Lambency three 222 D.3 Lambency four 223 D.4 Lambency five 224 D.5 Lambency seven 226 D.6 Lambency thirteen 228 References 230 1. Introduction The term monstrous moonshine was coined by Conway in order to describe the unexpected and mysterious connections between the representation the- ory of the largest sporadic group — the Fischer–Griess monster, M —and modular functions that stemmed from McKay’s observation that 196883 + 1 = 196884, where the summands on the left are degrees of irreducible rep- resentations of M and the number on the right is the coefficient of q in the Fourier expansion of the elliptic modular invariant [1] (1.1) J(τ)= a(m)qm = q−1 + 196884q + 21493760q2 + 864299970q3 + ··· . m≥−1 Thompson expanded upon McKay’s observation in [2] and conjectured the existence of an infinite-dimensional monster module (1.2) V = Vm m≥−1 with dim Vm = a(m) for all m. He also proposed [3] to consider the series, now known as McKay–Thompson series,givenby m (1.3) Tg(τ)= trVm (g) q m≥−1 for g ∈ M, and detailed explorations [1] by Conway–Norton led to the aston- ishing moonshine conjecture. Umbral Moonshine 105 For each g ∈ M the function Tg is a principal modulus for some genus zero group Γg. (A discrete group Γ < PSL2(R)issaidtohavegenus zero if the Riemann surface Γ\H is isomorphic to the Riemann sphere minus finitely many points, and a holomorphic function f on H is called a principal modulus for a genus zero group Γ if it generates the field of Γ-invariant functions on H.) Thompson’s conjecture was verified by Atkin, Fong and Smith (cf. [4, 5]). A more constructive verification was obtained by Frenkel–Lepowsky– Meurman [6, 7] with the explicit construction of a monster module V = V with graded dimension given by the Fourier expansion (1.1) of the elliptic modular invariant. They used vertex operators — structures originating in the dual resonance theory of particle physics and finding contemporaneous application [8, 9] to affine Lie algebras — to recover the non-associative Griess algebra structure (developed in the first proof [10] of the existence of the monster) from a subspace of V . Borcherds found a way to attach vertex operators to every element of V and determined the precise sense in which these operators could be given a commutative associative composition law, and thus arrived at the notion of vertex algebra [11], an axiomatization of the operator product expansion of chiral conformal field theory (CFT).The closely related notion of vertex operator algebra (VOA) was subsequently introduced by Frenkel–Lepowsky–Meurman [12] and they established that the monster is precisely the group of automorphisms of a VOA structure on V ; the Frenkel–Lepowsky–Meurman construction of V would ultimately prove to furnish the first example of an orbifold conformal field theory. Borcherds introduced the notion of generalized Kac–Moody algebra in [13] and by using the VOA structure on V was able to construct a par- ticular example — the monster Lie algebra — and use the corresponding equivariant denominator identities to arrive at a proof [14] of the Conway– Norton moonshine conjectures. Thus by 1992 monstrous moonshine had already become a phenomenon encompassing elements of finite group the- ory, modular forms, vertex algebras and generalized Kac–Moody algebras, as well as aspects of conformal field theory and string theory. Recently Eguchi–Ooguri–Tachikawa have presented evidence [15] for a new kind of moonshine involving the elliptic genus of K3 surfaces (the elliptic genus is a topological invariant and therefore independent of the choice of K3 surface) and the largest Mathieu group M24 (cf. Section 3.1). This connection between K3 surfaces and M24 becomes apparent only after decomposing the elliptic genus into characters of the N = 4 superconformal algebra (cf. [16– 18]). This decomposition process (cf. Section 2.4) reveals the presence of a 106 Miranda C. N. Cheng et al. mock modular form of weight 1/2 (cf. Section 2.1) satisfying ∞ (2) (2) n−1/8 (1.4) H (τ)= c1 (n − 1/8)q n=0 =2q−1/8(−1+45q + 231q2 + 770q3 + 2277q4 + ···) and one recognizes here the dimensions of several irreducible representations of M24 (cf. Table B.1). One is soon led to follow the path forged by Thompson in the case of the monster: to suspect the existence of a graded infinite-dimensional M24- module ∞ (2) (2) (1.5) K = Kn−1/8 n=0 (2) (2) − ≥ (2) with dim Kn−1/8 = c (n 1/8) for n 1, and to study the analogues Hg of the monstrous McKay–Thompson series obtained by replacing c(2)(n − (2) 1/8) = dim K −1 8 with tr (2) (g) in (1.4). This idea has been implemented n / Kn−1/8 successfully in [19–22] and provides strong evidence for the existence of such (2) (2) an M24-module K . A proof of the existence of K has now been estab- lished in [23] although no explicit construction is yet known. In particular, there is as yet no known analogue of the VOA structure which conjecturally characterizes [12] the monster module V . The strong evidence in support of the M24 analogue of Thompson’s conjecture invites us to consider the M24 analogue of the Conway–Norton moonshine conjectures — this will justify the use of the term moonshine in the M24 setting — except that it is not immediately obvious what the (2) analogue should be. Whilst the McKay–Thompson series Hg is a mock modular form of weight 1/2onsomeΓg < SL2(Z) for every g in M24 [22], it is not the case that Γg is a genus zero group for every g, and even if it were, there is no obvious sense in which one mock modular form of weight 1/2on some group can “generate” all the others, and thus no obvious analogue of the principal modulus property. A solution to this problem — the formulation of the moonshine conjec- ture for M24 — was found in [24] (see also Section 5.2) via an extension of the program that was initiated in [25]; the antecedents of which include Rademacher’s pioneering work [26] on the elliptic modular invariant J(τ), quantum gravity in three dimensions [27–29], the anti-de sitter (AdS)/CFT correspondence in physics [30–32], and the application of Rademacher sums Umbral Moonshine 107 to these and other settings in string theory [33–39] (and in particular [40]).
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