# Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_k(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)}$$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. In Bruinier (p.6) he says one can use $E_k(\tau)$ for $k=4,6,8,10,14$. What are the other functions for higher $k$?

Edit: As pointed out by S. Carnahan below, there is already a known moonshine for $Th$. A partial(?) list of others from Griess' "Happy Family" can be found in Monstrous Moonshine, such as for the Higman-Sims $HS$ and so on. (Are there more?)

• 708938760=190373976+3*111321000+2*91171899+1707264+6*85995+30628+4123+1. – Jeff Harvey Feb 19 '14 at 22:44
• Aren't you tempted to get rid of the annoying factors of 240 by adding $120 \vartheta(0,\tau)$? – Jeff Harvey Feb 19 '14 at 23:03
• Probably, as 708938760+240 =190373976+3*91171899+44330496+2*3376737+2572752+957125+61256+248 which has lower multiplicities than needed without adding 240. – Jeff Harvey Feb 20 '14 at 13:32
• @F.C. The modular moonshine puzzle was solved by Griess and Lam last year: arxiv.org/abs/1308.2270 . As it happens, there is already a moonshine for the Thompson group, via the function $j(\tau/3)^{1/3} = q^{-1/9} + 248q^{2/9} + 4124q^{5/9} + \cdots$. This function is the character of the 3C-twisted module of $V^\natural$, and has a natural $Th$-representation structure. – S. Carnahan Feb 20 '14 at 21:15
• @TitoPiezasIII Borcherds defines $b(\tau)$ this way so that its Borcherds lift gives $E_4$. But the weight 12 discriminant function $\Delta(\tau)$ is the Borcherds lift of $12 \vartheta(\tau)$ so adding $120 \vartheta(\tau)$ gives a weight 1/2 modular form with Borcherds lift $E_4 \Delta^{10}$ and indeed one has the q expansion $E_4 \Delta^{10}=q^{10}-27000 q^{12}+4096000 q^{13} + \cdots$ exhibiting dimensions of Thompson irreps. However this lift only involves the coefficients $b(n^2)$ in the q expansion of $b$ while you see moonshine in $b(n)$ for n square free. – Jeff Harvey Feb 21 '14 at 13:40

This has since been explored and it appears as though there is an interesting type of moonshine for the Thompson group involving weakly holomorphic weight 1/2 modular forms over $\Gamma_0(N)$ with non-trivial multiplier systems.

To bring out the structure of this moonshine, we can consider the form \begin{align} \mathcal{F}_3(\tau) &\equiv 2b(\tau) + 240\theta(\tau) = \sum_{\substack{m\geq -3 \\ m\equiv 0,1~\mathrm{mod}~ 4}}c(m)q^m \\ &= 2q^{-3} + 248 + 2\cdot 27000q^4 - 2\cdot 85995q^5 + 2\cdot 1707264 q^8 \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ - 2\cdot 4096000q^9 + 2\cdot44330496q^{12} + \cdots \end{align}
where $$\theta(\tau) = \sum_{n\in \mathbb{Z}}q^{n^2}.$$ Inspection of the character table of Th suggests that we should interpret these first few coefficients as either single, real irreducible representations (with multiplicity 2) or representations of the form $V\oplus \overline{V}$. If we are right in making this association, it might further suggest the existence of a graded module $$W = \bigoplus_{\substack{m\geq -3 \\ m\equiv 0,1 ~\mathrm{mod}~ 4}}^{\infty}W_m$$ where we demand that the module be compatible with the Fourier coefficients of $\mathcal{F}_3$ in the sense that $\dim W_m = |c(m)|$. The alternating signs can be treated by endowing $W$ with a superspace structure. In other words, we can decompose each component into an odd and even part, $W_m = W_m^{(0)} \oplus W_m^{(1)}$. The multiple of $\theta$ in the definition of $\mathcal{F}_3$ was chosen so that each Fourier coefficient could be decomposed into dimensions of irreducibles of the same sign. Then, this super-module is special in that, for $m\geq 0$, $W_m$ has vanishing odd part when $m$ is even and vanishing even part when $m$ is odd. The coefficient of $q^{-3}$ in this sense has the 'wrong sign' in that $W_{-3}$ has vanishing odd part, a feature similar to one observed in Umbral moonshine.

One can take this idea farther and consider McKay-Thompson series as in Monstrous Moonshine, $$\mathcal{F}_{3,[g]}(\tau) = \sum \mathrm{str}_{W_m}(g)q^m$$ where the $\mathrm{str}$ is meant to denote the super trace. It was found that you can naturally identify these McKay-Thompson series with weight half modular forms (obtained via the formalism of Rademacher series) on $\Gamma_0(4\cdot o(g)\cdot m)$, i.e. of level 4 times a multiple of the order of $g$, with non-trivial multiplier system. It was verified that these candidates for the McKay-Thompson series produced a well-defined character for the components $W_m$ for $m\leq 52$. In other words, treating each MT series as a graded character, one can decompose the $W_m$ into irreducible representations of the Thompson group with positive integer multiplicities for $m\leq 52$ and it is conjecture that this holds more generally for all $m$.

As it turns out, this moonshine is also conjectured to enjoy a discriminant property similar to the one observed in Umbral moonshine, and also has a natural connection to Monstrous moonshine in that the Borcherds lift of $b(\tau)-4\theta(\tau)$ is the $T_{3C}$ McKay Thompson series from Monstrous moonshine, as was pointed out in earlier comments.

If you'd like to find out more about this moonshine, all this information was taken from the following paper, written by Jeff Harvey and me: http://arxiv.org/abs/1504.08179

Your observation, which was expanded into a concrete conjecture last year by Harvey and Rayhaun, is now a theorem. See M. Griffin, M. Mertens, "A proof of the Thompson Moonshine conjecture".

This is not to say that the story is complete. While we now know that a suitable Thompson module exists, there is still no explicit construction or conceptual (e.g., physical) explanation for its existence.