Open Access

Proof of the umbral moonshine conjecture

Research in the Mathematical Sciences20152:26

https://doi.org/10.1186/s40687-015-0044-7

Received: 4 March 2015

Accepted: 16 October 2015

Published: 10 December 2015

Abstract

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay–Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here, we establish the existence of the umbral moonshine modules in the remaining 22 cases.

Mathematics Subject Classification

11F22 11F37

1 Introduction and statement of results

Monstrous moonshine relates distinguished modular functions to the representation theory of the Monster, \(\mathbb {M}\), the largest sporadic simple group. This theory was inspired by the famous observations of McKay and Thompson in the late 1970s [18, 51] that
$$\begin{aligned} 196884&=1+196883,\\ 21493760&=1+196883+21296876. \end{aligned}$$
The left-hand sides here are familiar as coefficients of Klein’s modular function (note \(q:=\mathrm{e}^{2\pi i \tau }\)),
$$\begin{aligned} J(\tau )=\sum _{n=-1}^{\infty }c(n)q^n:= j(\tau )-744 =q^{-1}+196884q+21493760q^2+\cdots . \end{aligned}$$
The sums on the right-hand sides involve the first three numbers arising as dimensions of irreducible representations of \(\mathbb {M}\),
$$\begin{aligned} 1, \ 196883, \ 21296876, \ 842609326,\ \dots , \ 258823477531055064045234375. \end{aligned}$$
Thompson conjectured that there is a graded infinite-dimensional \(\mathbb {M}\)-module
$$\begin{aligned} V^{\natural }=\bigoplus _{n=-1}^{\infty }V^{\natural }_n, \end{aligned}$$
satisfying \(\dim (V^{\natural }_n)=c(n)\). For \(g\in \mathbb {M}\), he also suggested [50] to consider the graded-trace functions
$$\begin{aligned} \begin{aligned} T_g(\tau ):=\sum _{n=-1}^{\infty }\mathrm{tr}(g|V^{\natural }_n)q^n, \end{aligned} \end{aligned}$$
now known as the McKay–Thompson series, that arise from the conjectured \(\mathbb {M}\)-module \(V^{\natural }\). Using the character table for \(\mathbb {M}\), it was observed [18, 50] that the first few coefficients of each \(T_g(\tau )\) coincide with those of a generator for the function field of a discrete group \(\Gamma _g<{\textit{SL}}_2({\mathbb R})\), leading Conway and Norton [18] to their famous Monstrous Moonshine Conjecture: This is the claim that for each \(g\in \mathbb {M}\) there is a specific genus zero group \(\Gamma _g\) such that \(T_g(\tau )\) is the unique normalized hauptmodul for \(\Gamma _g\), i.e., the unique \(\Gamma _g\)-invariant holomorphic function on \({\mathbb H}\) which satisfies \(T_g(\tau )=q^{-1}+O(q)\) as \(\mathfrak {I}(\tau )\rightarrow \infty \) and remains bounded near cusps not equivalent to the infinite one.

In a series of ground-breaking works, Borcherds introduced vertex algebras [2], generalized Kac–Moody Lie algebras [3, 4], and used these notions to prove [5] the Monstrous Moonshine Conjecture of Conway and Norton. He confirmed the conjecture for the module \(V^{\natural }\) constructed by Frenkel, Lepowsky and Meurman [3032] in the early 1980s. These results provide much more than the predictions of monstrous moonshine. The \(\mathbb {M}\)-module \(V^{\natural }\) is a vertex operator algebra, one whose automorphism group is precisely \(\mathbb {M}\). The construction of Frenkel, Lepowsky and Meurman can be regarded as one of the first examples of an orbifold conformal field theory (Cf. [23]). Here, the orbifold in question is the quotient \(\left( {\mathbb R}^{24}/\Lambda _{24}\right) /(\mathbb Z/2\mathbb Z)\), of the 24-dimensional torus \(\Lambda _{24}\otimes _{\mathbb Z}{\mathbb R}/\Lambda _{24}\simeq {\mathbb R}^{24}/\Lambda _{24}\) by the Kummer involution \(x\mapsto -x\), where \(\Lambda _{24}\) denotes the Leech lattice.

We refer to [24, 32, 35, 36] for more on monstrous moonshine.

In 2010, Eguchi, Ooguri and Tachikawa reignited moonshine with their observation [28] that dimensions of some representations of \(M_{24}\), the largest sporadic simple Mathieu group (cf., e.g. [20, 21]), are multiplicities of superconformal algebra characters in the K3 elliptic genus. This observation suggested a manifestation of moonshine for \(M_{24}\): Namely, there should be an infinite-dimensional graded \(M_{24}\)-module whose McKay–Thompson series are holomorphic parts of harmonic Maass forms, the so-called mock modular forms (see [45, 54, 55] for introductory accounts of the theory of mock modular forms).

Following the work of Cheng [10], Eguchi and Hikami [27], and Gaberdiel, Hohenegger, and Volpato [33, 34], Gannon established the existence of this infinite-dimensional graded \(M_{24}\)-module in [37].

It is natural to seek a general mathematical and physical setting for these results. Here we consider the mathematical setting, which develops from the close relationship between the monster group \(\mathbb {M}\) and the Leech lattice \(\Lambda _{24}\). Recall (cf., e.g. [20]) that the Leech lattice is even, unimodular and positive-definite of rank 24. It turns out that \(M_{24}\) is closely related to another such lattice. Such observations led Cheng, Duncan and Harvey to further instances of moonshine within the setting of even unimodular positive-definite lattices of rank 24. In this way, they arrived at the Umbral Moonshine Conjectures (cf. Sect. 5 of [15], Sect. 6 of [16], and Sect. 2 of [17]), predicting the existence of 22 further, graded infinite-dimensional modules, relating certain finite groups to distinguished mock modular forms.

To explain this prediction in more detail, we recall Niemeier’s result [43] that there are 24 (up to isomorphism) even unimodular positive-definite lattices of rank 24. The Leech lattice is the unique one with no root vectors (i.e. lattice vectors with norm-square 2), while the other 23 have root systems with full rank, 24. These Niemeier root systems are unions of simple simply-laced root systems with the same Coxeter numbers, and are given explicitly as
$$\begin{aligned}&\quad \quad \quad A_1^{24},\;A_2^{12},\;A_3^{8},\;A_4^6,\;A_6^4,\;A_{12}^2,\nonumber \\&A_5^4D_4,\;A_7^2D_5^2,\;A_8^3,\;A_9^2D_6,\;A_{11}D_7E_6,\;A_{15}D_9,\;A_{17}E_7,\;A_{24},\\&\quad \quad D_4^6,\;D_6^4,\;D_8^3,\;D_{10}E_7^2,\;D_{12}^2,\;D_{16}E_8,\;D_{24},E_6^4,\;E_8^3,\nonumber \end{aligned}$$
(1.1)
in terms of the standard ADE notation (cf., e.g. [20] or [39] for more on root systems).
For each Niemeier root system X,  let \(N^X\) denote the corresponding unimodular lattice, let \(W^X\) denote the (normal) subgroup of \({\text {Aut}}(N^X)\) generated by reflections in roots, and define the umbral group of X by setting
$$\begin{aligned} G^X:={\text {Aut}}(N^X) /W^X. \end{aligned}$$
(1.2)
(See Sect. A.1 for explicit descriptions of the groups \(G^X\).)

Let \(m^X\) denote the Coxeter number of any simple component of X. An association of distinguished \(2m^X\)-vector-valued mock modular forms \(H^X_g(\tau )=(H^X_{g,r}(\tau ))\) to elements \(g\in G^X\) is described and analyzed in [1517].

For \(X=A_1^{24}\) we have \(G^X\simeq M_{24}\) and \(m^X=2\), and the functions \(H^X_{g,1}(\tau )\) are precisely the mock modular forms assigned to elements \(g\in M_{24}\) in the works [10, 27, 33, 34] mentioned above. Generalizing the \(M_{24}\) moonshine initiated by Eguchi, Ooguri and Tachikawa, we have the following conjecture of Cheng, Duncan and Harvey (cf. Sect. 2 of [17] or Sect. 9.3 of [24]).

Conjecture

(Umbral Moonshine Modules) Let X be a Niemeier root system X and set \(m:=m^X\). There is a naturally defined bi-graded infinite-dimensional \(G^X\)-module
$$\begin{aligned} \check{K}^X=\bigoplus _{r\in I^X}\bigoplus _{\begin{array}{c} D\in \mathbb Z,\; D\le 0,\\ D=r^2\pmod {4m} \end{array}}\check{K}^X_{r,-D/4m} \end{aligned}$$
(1.3)
such that the vector-valued mock modular form \(H^X_g=(H^X_{g,r})\) is a McKay–Thompson series for \(\check{K}^X\) related1 to the graded trace of g on \(\check{K}^X\) by
$$\begin{aligned} \begin{aligned} H^X_{g,r}(\tau )=-2q^{-1/4m}\delta _{r,1} +\sum _{\begin{array}{c} D\in \mathbb Z,\; D\le 0,\\ D=r^2\pmod {4m} \end{array}}\mathrm{tr}(g|\check{K}^X_{r,-D/4m})q^{-D/4m} \end{aligned} \end{aligned}$$
(1.4)
for \(r\in I^X\).

In (1.3) and (1.4), the set \(I^X\subset \mathbb Z/2m\mathbb Z\) is defined in the following way. If X has an A-type component then \(I^X:=\{1,2,3,\ldots ,m-1\}\). If X has no A-type component but does have a D-type component, then \(m=2\mod 4\) and \(I^X:=\{1,3,5,\ldots ,m/2\}\). The remaining cases are \(X=E_6^4\) and \(X=E_8^3\). In the former of these, \(I^X:=\{1,4,5\}\), and in the latter case \(I^X:=\{1,7\}\).

Remark

The functions \(H^X_g(\tau )\) are defined explicitly in Sect. B.3. An alternative description in terms of Rademacher sums is given in Sect. B.4.

Here, we prove the following theorem.

Theorem 1.1

The umbral moonshine modules exist.

Two remarks
  1. 1.

    Theorem 1.1 for \(X=A_1^{24}\) is the main result of Gannon’s work [37].

     
  2. 2.

    The vector-valued mock modular forms \(H^X=(H^X_{g,r})\) have “minimal” principal parts. This minimality is analogous to the fact that the original McKay–Thompson series \(T_g(\tau )\) for the Monster are hauptmoduln, and plays an important role in our proof.

     

Example

Many of Ramanujan’s mock theta functions [46] are components of the vector-valued umbral McKay–Thompson series \(H^X_g=(H^X_{g,r})\). For example, consider the root system \(X=A_2^{12}\), whose umbral group is a double cover \(2.M_{12}\) of the sporadic simple Mathieu group \(M_{12}\). In terms of Ramanujan’s third-order mock theta functions
$$\begin{aligned} f(q)&=1+\sum _{n=1}^{\infty }\frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots (1+q^n)^2},\\ \phi (q)&=1+\sum _{n=1}^{\infty }\frac{q^{n^2}}{(1+q^2)(1+q^4)\cdots (1+q^{2n})},\\ \chi (q)&=1+\sum _{n=1}^{\infty }\frac{q^{n^2}}{(1-q+q^2)(1-q^2+q^4)\cdots (1-q^n+q^{2n})}\\ \omega (q)&=\sum _{n=0}^{\infty }\frac{q^{2n(n+1)}}{(1-q)^2(1-q^3)^2\cdots (1-q^{2n+1})^2},\\ \rho (q)&=\sum _{n=0}^{\infty }\frac{q^{2n(n+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots (1+q^{2n+1}+q^{4n+2})}, \end{aligned}$$
we have that
$$\begin{aligned} H^{X}_{2B,1}(\tau )=H^X_{2C,1}(\tau )=H^X_{4C,1}(\tau )&=-2q^{-\frac{1}{12}}\cdot f(q^2),\\ H^X_{6C,1}(\tau )=H^X_{6D,1}(\tau )&=-2q^{-\frac{1}{12}}\cdot \chi (q^2),\\ H^X_{8C,1}(\tau )=H^X_{8D,1}(\tau )&=-2q^{-\frac{1}{12}}\cdot \phi (-q^2),\\ H^X_{2B,2}(\tau )=-H_{2C,2}^X(\tau )&=-4q^{\frac{2}{3}}\cdot \omega (-q),\\ H^X_{6C,2}(\tau )=-H^X_{6D,2}(\tau )&=2q^{\frac{2}{3}}\cdot \rho (-q). \end{aligned}$$
See Sect. 5.4 of [16] for more coincidences between umbral McKay–Thompson series and mock theta functions identified by Ramanujan almost a 100 years ago.

Our proof of Theorem 1.1 involves the explicit determination of each \(G^X\)-module \(\check{K}^X\) by computing the multiplicity of each irreducible component for each homogeneous subspace. It guarantees the existence and uniqueness of a \(\check{K}^X\) which is compatible with the representation theory of \(G^X\) and the Fourier expansions of the vector-valued mock modular forms \(H^X_g(\tau )=(H^X_{g,r}(\tau ))\).

At first glance our methods do not appear to shed light on any deeper algebraic properties of the \(\check{K}^X\), such as might correspond to the vertex operator algebra structure on \(V^{\natural }\), or the monster Lie algebra introduced by Borcherds in [5]. However, we do determine, and utilize, specific recursion relations for the coefficients of the umbral McKay–Thompson series which are analogous to the replicability properties of monstrous moonshine formulated by Conway and Norton in Sect. 8 of [18] (cf. also [1]). More specifically, we use recent work [40] of Imamoğlu, Raum and Richter, as generalized [42] by Mertens, to obtain such recursions. These results are based on the process of holomorphic projection.

Theorem 1.2

For each \(g\in G^X\) and \(0<r<m\), the mock modular form \(H^X_{g,r}(\tau )\) is replicable in the mock modular sense.

A key step in Borcherds’ proof [5] of the monstrous moonshine conjecture is the reformulation of replicability in Lie theoretic terms. We may speculate that the mock modular replicability utilized in this work will ultimately admit an analogous algebraic interpretation. Such a result remains an important goal for future work.

In the statement of Theorem 1.2, replicable means that there are explicit recursion relations for the coefficients of the vector-valued mock modular form in question. For example, we recall the recurrence formula for Ramanujan’s third-order mock theta function \(f(q)=\sum _{n=0}^{\infty }c_f(n)q^n\) that was obtained recently by Imamoğlu, Raum and Richter [40]. If \(n\in \mathbb {Q}\), then let
$$\begin{aligned} \sigma _1(n):={\left\{ \begin{array}{ll} \sum \nolimits _{d\mid n} d \ \ \ \ \ &{}{\text{ if }}\ n\in \mathbb Z,\\ 0\ \ \ \ \ &{}{\text{ otherwise }}, \end{array}\right. } \end{aligned}$$
$$\begin{aligned} \mathrm{sgn}^+(n):={\left\{ \begin{array}{ll} \mathrm{sgn}(n)\ \ \ \ \ &{}{\text{ if }}\ n\ne 0,\\ 1 \ \ \ \ \ &{}{\text{ if }}\ n=0, \end{array}\right. } \end{aligned}$$
and then define
$$\begin{aligned} d(N,\widetilde{N}, t, \widetilde{t}):= \mathrm{sgn}^{+}(N)\cdot \mathrm{sgn}^{+} (\tilde{N})\cdot \left( |N+t|-|\widetilde{N}+\widetilde{t}|\right) . \end{aligned}$$
Then for positive integers n, we have that
$$\begin{aligned}&\sum _{\begin{array}{c} m\in \mathbb Z\\ 3m^2+m\le 2n \end{array}} \left( m+\frac{1}{6}\right) c_f\left( n-\frac{3}{2}m^2-\frac{1}{2}m\right) \\&\quad \quad =\frac{4}{3}\sigma (n)-\frac{16}{3}\sigma \left( \frac{n}{2}\right) -2 \sum _{\begin{array}{c} a,b\in \mathbb Z\\ 2n=ab \end{array}}d\left( N,\widetilde{N},\frac{1}{6},\frac{1}{6}\right) , \end{aligned}$$
where \(N:=\frac{1}{6}(-3a+b-1)\) and \(\widetilde{N}:=\frac{1}{6}(3a+b-1)\), and the sum is over integers ab for which \(N, \widetilde{N}\in \mathbb Z\). This is easily seen to be a recurrence relation for the coefficients \(c_f(n)\). The replicability formulas for all of the \(H_{g,r}^{X}(\tau )\) are similar (although some of these relations are slightly more complicated and involve the coefficients of weight 2 cusp forms).

It is important to emphasize that despite the progress which is represented by our main results, Theorems 1.1 and 1.2, the following important question remains open in general.

Question

Is there a “natural” construction of \(\check{K}^X\)? Is \(\check{K}^X\) equipped with a deeper algebra structure as in the case of the monster module \(V^{\natural }\) of Frenkel, Lepowsky and Meurman?

We remark that this question has been answered positively, recently, in one special case: A vertex operator algebra structure underlying the umbral moonshine module \(\check{K}^X\) for \(X=E_8^3\) has been described explicitly in [25]. See also [14, 26], where the problem of constructing algebraic structures that illuminate the umbral moonshine observations is addressed from a different point of view.

The proof of Theorem 1.1 is not difficult. It is essentially a collection of tedious calculations. We use the theory of mock modular forms and the character table for each \(G^X\) (cf. Sect. A.2) to solve for the multiplicities of the irreducible \(G^X\)-module constituents of each homogeneous subspace in the alleged \(G^X\)-module \(\check{K}^X\). To prove Theorem 1.1 it suffices to prove that these multiplicities are non-negative integers. To prove Theorem 1.2 we apply recent work [42] of Mertens on the holomorphic projection of weight \(\frac{1}{2}\) mock modular forms, which generalizes earlier work [40] of Imamoğlu, Raum and Richter.

In Sect. 2 we recall the facts about mock modular forms that we require, and we prove Theorem 1.2. We prove Theorem 1.1 in Sect. 3. The appendices furnish all the data that our method requires. In particular, the umbral groups \(G^X\) are described in detail in Sect. A, and explicit definitions for the mock modular forms \(H^X_g(\tau )\) are given in Sect. B.

2 Harmonic Maass forms and Mock modular forms

Here, we recall some very basic facts about harmonic Maass forms as developed by Bruinier and Funke [9] (see also [45]).

We begin by briefly recalling the definition of a harmonic Maass form of weight \(k\in \frac{1}{2}\mathbb Z\) and multiplier \(\nu \) (a generalization of the notion of a Nebentypus). If \(\tau =x+iy\) with x and y real, we define the weight k hyperbolic Laplacian by
$$\begin{aligned} \Delta _k := -y^2\left( \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}\right) + iky\left( \frac{\partial }{\partial x}+i \frac{\partial }{\partial y}\right) . \end{aligned}$$
(2.1)
Suppose \(\Gamma \) is a subgroup of finite index in \({\textit{SL}}_2(\mathbb Z)\) and \(k\in \frac{1}{2}\mathbb Z\). Then, a function \(F(\tau )\) which is real-analytic on the upper half of the complex plane is a harmonic Maass form of weight k on \(\Gamma \) with multiplier \(\nu \) if:
  1. (a)
    The function \(F(\tau )\) satisfies the weight k modular transformation,
    $$\begin{aligned} F(\tau )|_k\gamma =\nu (\gamma )F(\tau ) \end{aligned}$$
    for every matrix \(\gamma =\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\in \Gamma ,\) where \(F(\tau )|_k\gamma :=F(\gamma \tau )(c\tau +d)^{-k},\) and if \(k\in \mathbb Z+\frac{1}{2},\) the square root is taken to be the principal branch.
     
  2. (b)

    We have that \(\Delta _kF(\tau )=0,\)

     
  3. (c)

    There is a polynomial \(P_{F}(q^{-1})\) and a constant \(c>0\) such that \(F(\tau )-P_{F}(e^{-2\pi i \tau })=O(e^{-cy})\) as \(\tau \rightarrow i\infty \). Analogous conditions are required at each cusp of \(\Gamma \).

     
We denote the \({\mathbb C}\)-vector space of harmonic Maass forms of a given weight k, group \(\Gamma \) and multiplier \(\nu \) by \(H_k(\Gamma ,\nu ).\) If no multiplier is specified, we will take
$$\begin{aligned} \nu _0(\gamma ):=\left( \left( \frac{c}{d}\right) \sqrt{\left( \frac{-1}{d}\right) }^{-1}\right) ^{2k}, \end{aligned}$$
where \(\left( \frac{*}{d}\right) \) is the Kronecker symbol.

2.1 Main properties

The Fourier expansion of a harmonic Maass form F (see Proposition 3.2 of [9]) splits into two components. As before, we let \(q:=e^{2\pi i \tau }\).

Lemma 2.1

If \(F(\tau )\) is a harmonic Maass form of weight \(2-k\) for \(\Gamma \) where \(\frac{3}{2}\le k\in \frac{1}{2}\mathbb Z\), then
$$\begin{aligned} F(\tau )=F^+(\tau )+F^{-}(\tau ), \end{aligned}$$
where \(F^+\) is the holomorphic part of F, given by
$$\begin{aligned} F^+(\tau )=\sum _{n\gg -\infty } c_{F}^+(n) q^{n} \end{aligned}$$
where the sum admits only finitely many non-zero terms with \(n<0\), and \(F^{-}\) is the nonholomorphic part, given by
$$\begin{aligned} F^{-}(\tau ) = \sum _{n<0} c_{F}^-(n) \Gamma (k-1,4\pi y|n|) q^{n}. \end{aligned}$$
Here, \(\Gamma (s,z)\) is the upper incomplete gamma function.
The holomorphic part of a harmonic Maass form is called a mock modular form. We denote the space of harmonic Maass forms of weight \(2-k\) for \(\Gamma \) and multiplier \(\nu \) by \(H_{k}(\Gamma ,\nu ).\) Similarly, we denote the corresponding subspace of holomorphic modular forms by \(M_k(\Gamma ,\nu ),\) and the space of cusp forms by \( S_k(\Gamma ,{\nu })\). The differential operator \(\xi _{w}:=2iy^w\overline{\frac{\partial }{\partial \overline{\tau }}}\) (see [9]) defines a surjective map
$$\begin{aligned} \xi _{2-k}:H_{2-k}(\Gamma ,\nu )\rightarrow S_k(\Gamma ,\overline{\nu }) \end{aligned}$$
onto the space of weight k cusp forms for the same group but conjugate multiplier. The shadow of a Maass form \(f(\tau )\in H_{2-k}(\Gamma ,\nu )\) is the cusp form \(g(\tau )\in S_k(\Gamma ,\overline{\nu })\) (defined, for now, only up to scale) such that \(\xi _{2-k}f(\tau )=\frac{g}{||g||}\), where \(||\bullet ||\) denotes the usual Petersson norm.

2.2 Holomorphic projection of weight \(\frac{1}{2}\) mock modular forms

As noted above, the modular transformations of a weight \(\frac{1}{2}\) harmonic Maass form may be simplified by multiplying by its shadow to obtain a weight 2 nonholomorphic modular form. One can use the theory of holomorphic projections to obtain explicit identities relating these nonholomorphic modular forms to classical quasimodular forms. In this way, we may essentially reduce many questions about the coefficients of weight \(\frac{1}{2}\) mock modular forms to questions about weight 2 holomorphic modular forms. The following theorem is a special case of a more general theorem due to Mertens (cf. Theorem 6.3 of [42]). See also [40].

Theorem 2.1

(Mertens) Suppose \(g(\tau )\) and \(h(\tau )\) are both theta functions of weight \(\frac{3}{2}\) contained in \(S_{\frac{3}{2}}(\Gamma ,\nu _g)\) and \(S_{\frac{3}{2}}(\Gamma ,\nu _h),\) respectively, with Fourier expansions
$$\begin{aligned} g(\tau )&:=\sum _{i=1}^{s}\sum _{n\in \mathbb Z}n\chi _i(n)q^{n^2},\\ h(\tau )&:=\sum _{j=1}^{t}\sum _{n\in \mathbb Z}n\psi _j(n)q^{n^2}, \end{aligned}$$
where each \(\chi _i\) and \(\psi _i\) is a Dirichlet character. Moreover, suppose \(h(\tau )\) is the shadow of a weight \(\frac{1}{2}\) harmonic Maass form \(f(\tau ) \in H_{\frac{1}{2}}(\Gamma ,\overline{\nu _h}).\) Define the function
$$\begin{aligned} D^{f,g}(\tau ):=2 \ \sum _{r=1}^{\infty }\sum _{\chi _i,\psi _j}\sum _{\begin{array}{c} m,n\in \mathbb Z^+\\ m^2-n^2=r \end{array}}\chi _i(m)\overline{\psi _j(n)}(m-n)q^r. \end{aligned}$$
If \(f(\tau )g(\tau )\) has no singularity at any cusp, then \(f^{+}(\tau )g(\tau )+D^{f,g}(\tau )\) is a weight 2 quasimodular form. In other words, it lies in the space \({\mathbb C}E_2(\tau )\oplus M_2(\Gamma ,\nu _g\overline{\nu _h})\), where \(E_2(\tau )\) is the quasimodular Eisenstein series \(E_2(\tau ):=1-24\sum \nolimits _{n\ge 1}\frac{nq^n}{1-q^n}.\)
Two Remarks.
  1. 1.

    These identities give recurrence relations for the weight \(\frac{1}{2}\) mock modular form \(f^+\) in terms of the weight 2 quasimodular form which equals \(f^+(\tau )g(\tau )+D^{f,g}(\tau )\). The example after Theorem 1.2 for Ramanujan’s third-order mock theta function f is an explicit example of such a relation.

     
  2. 2.

    Theorem 2.1 extends to vector-valued mock modular forms in a natural way.

     

Proof of Theorem 1.2

Fix a Niemeier lattice and its root system X, and let \(M=m^X\) denote its Coxeter number. Each \(H_{g,r}^{X}(\tau )\) is the holomorphic part of a weight \(\frac{1}{2}\) harmonic Maass form \(\widehat{H}_{g,r}^{X}(\tau ).\) To simplify the exposition in the following section, we will emphasize the case that the root system X is of pure A-type. If the root system X is of pure A-type, the shadow function \(S_{g,r}^{X}(\tau )\) is given by \(\hat{\chi }_{g,r}^{X_A} S_{M,r}(\tau )\) (see Sect. B.2), where
$$\begin{aligned} S_{M,r}(\tau )=\sum _{\begin{array}{c} n\in \mathbb Z\\ n\equiv r \pmod {2M} \end{array}}n ~q^{\frac{n^2}{4M}}, \end{aligned}$$
and \(\hat{\chi }_{g,r}^{X_A}\) = \(\chi _{g}^{X_A}\) or \(\bar{\chi }_{g}^{X_A}\) depending on the parity of r is the twisted Euler character given in the appropriate table in Sect. A.3, a character of \(G^X.\) (If X is not of pure A-type, then the shadow function \(S_{g,r}^{X}(\tau )\) is a linear combination of similar functions as described in Sect. B.2).
Given X and g, the symbol \(n_g|h_g\) given in the corresponding table in Sect. A.3 defines the modularity for the vector-valued function \((\widehat{H}_{g,r}^{X}(\tau ))\). In particular, if the shadow \((S_{g,r}^{X}(\tau ))\) is nonzero, and if for \(\gamma \in \Gamma _0(n_g)\) we have that
$$\begin{aligned} (S_{g,r}^{X}(\tau ))|_{3/2}\gamma =\sigma _{g,\gamma }(S_{g,r}^{X}(\tau )), \end{aligned}$$
then
$$\begin{aligned} (\widehat{H}_{g,r}^{X}(\tau ))|_{1/2}\gamma =\overline{\sigma _{g,\gamma }}(\widehat{H}_{g,r}^{X}(\tau )). \end{aligned}$$
Here, for \(\gamma \in \Gamma _0(n_g)\), we have \(\sigma _{g,\gamma }=\nu _{g}(\gamma )\sigma _{e,\gamma }\) where \(\nu _{g}(\gamma )\) is a multiplier which is trivial on \(\Gamma _0(n_gh_g)\). This identity holds even in the case that the shadow \(S_{g,r}^X\) vanishes.

The vector-valued function \((H_{g,r}^{X}(\tau ))\) has poles only at the infinite cusp of \(\Gamma _0(n_g)\), and only at the component \(H_{g,r}^{X}(\tau )\) where \(r=1\) if X has pure A-type, or at components where \(r^2\equiv 1 \pmod {2M}\) otherwise. These poles may only have order \(\frac{1}{4M}.\) This implies that the function \((\widehat{H}_{g,r}^{X}(\tau )S_{g,r}^{X}(\tau ))\) has no pole at any cusp and is therefore a candidate for an application of Theorem 2.1.

The modular transformation of \(S_{M,r}(\tau )\) implies that
$$\begin{aligned} (\sigma _{e,S})^2=(\sigma _{e,T})^{4M}=\mathbf {I} \end{aligned}$$
where \(S=\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}\), \(T=\begin{pmatrix}1&{}\quad 1\\ 0&{}\quad 1\end{pmatrix}\), and \(\mathbf {I}\) is the identity matrix. Therefore, \(S_{M,r}^{X}(\tau )\), viewed as a scalar-valued modular function, is modular on \(\Gamma (4M),\) and so \((\widehat{H}_{g,r}^{X}(\tau )S_{g,r}^{X}(\tau ))\) is a weight 2 nonholomorphic scalar-valued modular form for the group \(\Gamma (4M)\cap \Gamma _0(n_g)\) with trivial multiplier.

Applying Theorem 2.1, we obtain a function \(F_{g,r}^{X}(\tau )\)—call it the holomorphic projection of \(\widehat{H_{g,r}^{X}}(\tau ) S_{e,r}^{X}(\tau )\)—which is a weight 2 quasimodular form on \(\Gamma (4M)\cap \Gamma _0(n_g).\) In the case that \(S_{g,r}^{X}(\tau )\) is zero, we substitute \(S_{e,r}^{X}(\tau )\) in its place to obtain a function \(\widetilde{F}_{g,r}^{X}(\tau )=H_{g,r}^{X}(\tau ) S_{e,r}^{X}(\tau )\) which is a weight 2 holomorphic scalar-valued modular form for the group \(\Gamma (4M)\cap \Gamma _0(n_g)\) with multiplier \(\nu _{g}\) (alternatively, modular for the group \(\Gamma (4M)\cap \Gamma _0(n_gh_g)\) with trivial multiplier).

The function \(F_{g,r}^{X}(\tau )\) may be determined explicitly as the sum of Eisenstein series and cusp forms on \(\Gamma (4M)\cap \Gamma _0(n_gh_g)\) using the standard arguments from the theory of holomorphic modular forms (i.e. the “first few” coefficients determine such a form). Therefore, we have the identity
$$\begin{aligned} F_{g,r}^X(\tau )=H_{g,r}^X(\tau )\cdot S_{g,r}^X(\tau )+D_{g,r}^X(\tau ), \end{aligned}$$
(2.2)
where the function \(D_{g,r}^X(\tau )\) is the correction term arising in Theorem 2.1. If X has pure A-type, then
$$\begin{aligned} D_{g,r}^X(\tau )=(\hat{\chi }_{g,r}^{X_A})^2\sum _{N=1}^{\infty }\sum _{\begin{array}{c} m,n\in \mathbb Z_+\\ m^2-n^2=N \end{array}}\phi _r(m)\phi _r(n)(m-n)q^\frac{N}{4M}, \end{aligned}$$
(2.3)
where
$$\begin{aligned} \phi _r(\ell )={\left\{ \begin{array}{ll}\pm 1 &{}\text { if } \ell \equiv \pm r \pmod {2M}\\ 0&{}\text { otherwise.}\end{array}\right. } \end{aligned}$$
Suppose \(H_{g,r}^{X}( \tau )=\sum _{n=0}^{\infty }A^X_{g,r}(n) q^{n-\frac{D}{4M}}\) where \(0<D<4M\) and \(D\equiv r^2 \pmod {4M},\) and \(F_{g,r}^{X}(\tau )=\sum \nolimits _{N=0}^{\infty }B_{g,r}^X(n)q^n.\) Then by Theorem 2.1, we find that
$$\begin{aligned} B^X_{g,r}(N)&=\hat{\chi }_{g,r}^{X_A}\sum _{\begin{array}{c} m\in \mathbb Z\\ m\equiv r \pmod {2M} \end{array}} m\cdot A^X_{g,r}\left( N+\frac{D-m^2}{4M}\right) \nonumber \\&\quad +\,(\hat{\chi }_{g,r}^{X_A})^2\sum _{\begin{array}{c} m,n\in \mathbb Z^+\\ m^2-n^2=N \end{array}}\phi _r(m)\phi _r(n)(m-n). \end{aligned}$$
(2.4)
The function \(F_{g,r}^{X}(\tau )\) may be found in the following manner. Using the explicit prescriptions for \({H_{g,r}^{X}}( \tau )\) given in Sect. B.3 and (2.2) above, we may calculate the first several coefficients of each component. The Eisenstein component is determined by the constant terms at cusps. Since \(D_{g,r}^X(\tau )\) (and the corresponding correction terms at other cusps) has no constant term, these are the same as the constant terms of \(\widehat{H_{g,r}^{X}}( \tau )S_{g,r}^{X}(\tau ),\) which are determined by the poles of \(\widehat{H_{g,r}^{X}}\). Call this Eisenstein component \(E_{g,r}^{X}(\tau ).\) The cuspidal component can be found by matching the initial coefficients of \(F_{g,r}^{X}(\tau )-E_{g,r}^{X}(\tau ).\)
Once the coefficients \(B^X_{g,r}(n)\) are known, Eq. (2.4) provides a recursion relation which may be used to calculate the coefficients of \(H^{X}_{g,r}(\tau ).\) If the shadows \(S_{g,r}^{X}(\tau )\) are zero, then we may apply a similar procedure to determine \(\widetilde{F}_{g,r}^{X}(\tau )\). For example, suppose \(\widetilde{F}_{g,r}^{X}(\tau )=\sum \nolimits _{N=0}^{\infty }\widetilde{B}_{g,r}^X(n)q^n,\) and X has pure A-type. Then, we find that the coefficients \(\widetilde{B}^X_{g,r}(N)\) satisfy
$$\begin{aligned} \widetilde{B}^X_{g,r}(N)=\hat{\chi }_{g,r}^{X_A}\sum \limits _{\begin{array}{c} m\in \mathbb Z\\ m\equiv r \pmod {2M} \end{array}} m\cdot A^X_{g,r}\left( N+\frac{D-m^2}{4M}\right) . \end{aligned}$$
(2.5)
Proceeding in this way we obtain the claimed results. \(\square \)

3 Proof of Theorem 1.1

Here, we prove Theorem 1.1. The idea is as follows. For each Niemeier root system X,  we begin with the vector-valued mock modular forms \((H^X_g(\tau ))\) for \(g\in G^X\). We use their q-expansions to solve for the q-series whose coefficients are the alleged multiplicities of the irreducible components of the alleged infinite-dimensional \(G^X\)-module
$$\begin{aligned} \check{K}^X=\bigoplus _{r\pmod {2m}}\bigoplus _{\begin{array}{c} D\in \mathbb Z,\;D\le 0,\\ D=r^2\pmod {4m} \end{array}}\check{K}^X_{r,-D/4m}. \end{aligned}$$
These q-series turn out to be mock modular forms. The proof requires that we establish that these mock modular forms have non-negative integer coefficients.

Proof of Theorem 1.1

As in the previous section, we fix a root system X and set \(M:=m^X\), and emphasize the case when X is of pure A-type.

The umbral moonshine conjecture asserts that
$$\begin{aligned} H_{g,r}^{X}(\tau ) =\sum _{n=0}^\infty \sum _{\chi }m_{\chi ,r}^{X}(n)\chi (g)q^{n-\frac{r^2}{4M}} \end{aligned}$$
(3.1)
where the second sum is over the irreducible characters of \(G^X\). Here, we have rewritten the traces of the graded components \(\check{K}^X_{r,n-\frac{r^2}{4M}}\) in (1.4) in terms of the values of the irreducible characters of \(G^X\), where the \(m_{\chi ,r}^{X}(n)\) are the corresponding multiplicities. Naturally, if such a \(\check{K}^X\) exists, these multiplicities must be non-negative integers for \(n>0\). Similarly, if the mock modular forms \(H_{g,r}^{X}(\tau )\) can be expressed as in (3.1) with \(m_{\chi ,r}^{X}(n)\) non-negative integers, then we may construct the umbral moonshine module \(\check{K}^X\) explicitly with \(\check{K}^X_{r,n-r^2/4m}\) defined as the direct sum of irreducible components with the given multiplicities \(m_{\chi ,r}^{X}(n).\)
Let
$$\begin{aligned} H_{\chi ,r}^{X}(\tau ):=\frac{1}{|G^X|}\sum _{g}\overline{\chi (g)}H_{g,r}^{X}(\tau ). \end{aligned}$$
(3.2)
It turns out that the coefficients of \(H_{\chi ,r}^{X}(\tau )\) are precisely the multiplicities \(m_{\chi ,r}^{X}(n)\) required so that (3.1) holds: if
$$\begin{aligned} H_{\chi ,r}^{X}(\tau )=\sum _{n=0}^\infty m_{\chi ,r}^{X}(n)q^{n-\frac{r^2}{4M}}, \end{aligned}$$
(3.3)
then
$$\begin{aligned} H_{g,r}^{X}(\tau ) =\sum _{n=0}^\infty \sum _{\chi }m_{\chi ,r}^{X}(n)\chi (g)q^{n-\frac{r^2}{4M}}. \end{aligned}$$
Thus, the umbral moonshine conjecture is true if and only if the Fourier coefficients of \(H_{\chi ,r}^{X}(\tau )\) are non-negative integers.
To see this fact, we recall the orthogonality of characters. For irreducible characters \(\chi _i\) and \(\chi _j,\)
$$\begin{aligned} \frac{1}{|G^X|}\sum _{g\in G^X} \overline{\chi _i(g)}\chi _j(g)= {\left\{ \begin{array}{ll} 1 &{} \mathrm{if} \chi _i=\chi _j,\\ 0 &{} \mathrm{otherwise}. \end{array}\right. } \end{aligned}$$
(3.4)
We also have the relation for g and \(h\in G^X\),
$$\begin{aligned} \sum _{\chi } \overline{\chi _i(g)}\chi _i(h)= {\left\{ \begin{array}{ll} |C_{G^X}(g)| &{} \mathrm{if}\; g \; \mathrm{and} \; h \; \mathrm{are\,\, conjugate},\\ 0 &{} \mathrm{otherwise}. \end{array}\right. } \end{aligned}$$
(3.5)
Here, \(|C_{G^X}(g)|\) is the order of the centralizer of g in \(G^{X}\). Since the order of the centralizer times the order of the conjugacy class of an element is the order of the group, (3.2) and (3.5) together imply the relation
$$\begin{aligned} H_{g,r}^{X}(\tau )=\sum _{\chi }\chi (g)H_{\chi ,r}^{X}(\tau ), \end{aligned}$$
which in turn implies (3.3).

We have reduced the theorem to proving that the coefficients of certain weight \(\frac{1}{2}\) mock modular forms are all non-negative integers. For holomorphic modular forms, we may answer questions of this type by making use of Sturm’s theorem [49] (see also Theorem 2.58 of [44]). This theorem provides a bound B associated to a space of modular forms such that if the first B coefficients of a modular form \(f(\tau )\) are integral, then all of the coefficients of \(f(\tau )\) are integral. This bound reduces many questions about the Fourier coefficients of modular forms to finite calculations.

Sturm’s theorem relies on the finite dimensionality of certain spaces of modular forms, and so it cannot be applied directly to spaces of mock modular forms. However, by making use of holomorphic projection we can adapt Sturm’s theorem to this setting.

Let \(\widehat{H^{X}_{\chi ,r}}(\tau )\) be defined as above. Recall that the transformation matrix for the vector-valued function \(\widehat{H^{X}_{g,r}}(\tau ))\) is \(\overline{\sigma _{g,\gamma }},\) the conjugate of the transformation matrix for \((S_{e,r}^{X}(\tau ))\) when \(\gamma \in \Gamma _0(n_gh_g),\) and \(\sigma _{g,\gamma }\) is the identity for \(\gamma \in \Gamma (4M).\) Therefore, if
$$\begin{aligned} N_\chi ^X:={{\mathrm{lcm}}}\{n_gh_g \mid g \in G,\chi (g)\ne 0\}, \end{aligned}$$
then the scalar-valued functions \(\widehat{H^{X}}_{\chi ,r}(\tau )\) are modular on \(\Gamma (4M)\cap \Gamma _0(N_\chi ^X).\)
Let
$$\begin{aligned} A_{\chi ,r}(\tau ):=H^{X}_{\chi ,r}(\tau )S_{e,1}^{X}(\tau ), \end{aligned}$$
and \(\tilde{A}_{\chi ,r}(\tau )\) be the holomorphic projection of \(A_{\chi ,r}(\tau ).\) Suppose that \(H^{X}_{\chi ,r}(\tau )\) has integral coefficients up to some bound B. Formulas for the shadow functions (cf. Sect. B.2) show that the leading coefficient of \(S_{e,1}^{X}(\tau )\) is 1 and has integral coefficients. This implies that the function
$$\begin{aligned} A_{\chi ,r}(\tau ):=H^{X}_{\chi ,r}(\tau )S_{e,1}^{X}(\tau ) \end{aligned}$$
also has integral coefficients up to the bound B. The shadow of \(H^{X}_{\chi ,r}(\tau )\) is given by
$$\begin{aligned} {S_{\chi ,r}^{X}(\tau )}:=\frac{1}{|G^X|}\sum _{g}\overline{\chi (g)}S_{g,r}^{X}(\tau ). \end{aligned}$$
If X is pure A-type, then \(S_{g,r}^{X}(\tau )=\chi _{g,r}^{X_A}S_{M,r}(\tau )=(\chi '(g)+\chi ''(g))S_{M,r}(\tau )\) for some irreducible characters \(\chi '\) and \(\chi ''\), according to “Twisted Euler characters” and “Shadows”. Therefore,
$$\begin{aligned} S_{\chi ,r}^{X}(\tau )= {\left\{ \begin{array}{ll} S_{M,r}(\tau ) &{} \mathrm{if} \chi =\chi ' \,\mathrm{or}\, \chi '',\\ 0 &{} \mathrm{otherwise}. \end{array}\right. } \end{aligned}$$
When X is not of pure A-type the shadow is some sum of such functions, but in every case has integer coefficients, and so, applying Theorem 2.1 to \(A_{\chi ,r}(\tau ),\) we find that \(\tilde{A}_{\chi ,r}(\tau )\) also has integer coefficients up to the bound B. In particular, since \(\tilde{A}_{\chi ,r}(\tau )\) is modular on \(\Gamma (4M)\cap \Gamma _0(N_\chi ^X)\), then if B is at least the Sturm bound for this group we have that every coefficient of \(\tilde{A}_{\chi ,r}(\tau )\) is integral. Since the leading coefficient of \(S_{e,1}^{X}(\tau )\) is 1,  we may reverse this argument and we have that every coefficient of \({H^{X}}_{\chi ,r}(\tau ).\) Therefore, in order to check that \(H^{X}_{\chi ,r}(\tau )\) has only integer coefficients, it suffices to check up to the Sturm bound for \(\Gamma (4M)\cap \Gamma _0(N_\chi ).\) These calculations were carried out using the sage mathematical software [47].

The calculations and argument given above show that the multiplicities \(m_{\chi ,r}^X(n)\) are all integers. To complete the proof, it suffices to check that they are also are non-negative. The proof of this claim follows easily by modifying step-by-step the argument in Gannon’s proof of non-negativity in the \(M_{24}\) case [37] (i.e. \(X=A_1^{24}\)). Here, we describe how this is done.

Expressions for the alleged McKay–Thompson series \(H^{X}_{g,r}(\tau )\) in terms of Rademacher sums and unary theta functions are given in Sect. B.4. Exact formulas are known for all the coefficients of Rademacher sums because they are defined by averaging the special function \({\text {r}}^{[\alpha ]}_{1/2}(\gamma ,\tau )\) [see (B.114)] over cosets of a specific modular group modulo \(\Gamma _{\infty }\), the subgroup of translations. Therefore, Rademacher sums are standard Maass–Poincaré series, and as a result we have formulas for each of their coefficients as convergent infinite sums of Kloosterman-type sums weighted by values of the \(I_{1/2}\) modified Bessel function. (For example, see [8] or [53] for the general theory, and [12] for the specific case that \(X=A_1^{24}\).) More importantly, this means also that the generating function for the multiplicities \(m_{\chi ,r}^X(n)\) is a weight \(\frac{1}{2}\) harmonic Maass form, which in turn means that exact formulas (modulo the unary theta functions) are also available in similar terms. For positive integers n, this then means that (cf. Theorem 1.1 of [8])
$$\begin{aligned} m_{\chi ,r}^X(n)=\sum _{\rho } \sum _{m<0} \frac{a_{\rho }^X(m)}{n^{\frac{1}{4}}}\sum _{c=1}^{\infty } \frac{K^X_{\rho }(m,n,c)}{c}\cdot \mathbb {I}^X\left( \frac{4\pi \sqrt{|nm|}}{c}\right) , \end{aligned}$$
(3.6)
where the sums are over the cusps \(\rho \) of the group \(\Gamma _0(N_g^X)\), and finitely many explicit negative rational numbers m. The constants \(a_{\rho }^X(m)\) are essentially the coefficients which describe the generating function in terms of Maass–Poincaré series. Here, \(\mathbb {I}\) is a suitable normalization and change of variable for the standard \(I_{1/2}\) modified Bessel function.
The Kloosterman-type sums \(K_{\rho }^X(m,n,c)\) are well known to be related to Salié-type sums (for example see Proposition 5 of [41]). These Salié-type sums are of the form
$$\begin{aligned} S^X_{\rho }(m,n,c)=\sum _{\begin{array}{c} x\pmod c\\ x^2\equiv -D(m,n)\pmod {c} \end{array}} \epsilon _{\rho }^X(m,n)\cdot e\left( \frac{\beta ^X x}{c}\right) , \end{aligned}$$
where \(\epsilon _{\rho }^X(m,n)\) is a root of unity, \(-D(m,n)\) is a discriminant of a positive definite binary quadratic form, and \(\beta ^X\) is a nonzero positive rational number.

These Salié sums may then be estimated using the equidistribution of CM points with discriminant \(-D(m,n)\). This process was first introduced by Hooley [38] and was first applied to the coefficients of weight \(\frac{1}{2}\) mock modular forms by Bringmann and Ono [7]. Gannon explains how to make effective the estimates for sums of this shape in Sect. 4 of [37], thereby reducing the proof of the \(M_{24}\) case of umbral moonshine to a finite calculation. In particular, in equations (4.6–4.10) of [37], Gannon shows how to bound coefficients of the form (3.6) in terms of the Selberg–Kloosterman zeta function, which is bounded in turn in his proof of Theorem 3 of [37]. We follow Gannon’s proof mutatis mutandis. We find for most root systems that the coefficients of each multiplicity generating function are positive beyond the 390th coefficient. In the worst case, for the root system \(X=E^3_8\), we find that the coefficients are positive beyond the 1780th coefficient. Moreover, the coefficients exhibit subexponential growth. A finite computer calculation in sage has verified the non-negativity of the finitely many remaining coefficients. \(\square \)

Remark

It turns out that the estimates required for proving nonnegativity are the worst for the \(X=E_{8}^3\) case which required 1780 coefficients.

Footnotes
1

In the statement of Conjecture 6.1 of [16], the function \(H^X_{g,r}\) in (1.4) is replaced with \(3H^X_{g,r}\) in the case that \(X=A_8^3\). This is now known to be an error, arising from a misspecification of some of the functions \(H^X_{g}\) for \(X=A_8^3\). Our treatment of the case \(X=A_8^3\) in this work reflects the corrected specification of the corresponding \(H^X_g\) which is described and discussed in detail in [17].

 

Declarations

Acknowledgements

The authors thank the NSF for its support. The first author also thanks the Simons Foundation (#316779), and the third author thanks the A. G. Candler Fund. The authors thank Miranda Cheng, Jeff Harvey, Michael Somos and the anonymous referee for helpful comments and corrections.

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