On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine
Miranda C. N. Cheng, John F. R. Duncan
TL;DR
This work establishes a holographic, Rademacher-sum framework for the Mathieu moonshine attached to the largest Mathieu group $M_{24}$ by proving that each McKay–Thompson series $H_g$ is the regularized Rademacher sum $H_g=-2\,R_{n|h}$ attached to the corresponding subgroup $\Gamma_0(n)$ with multiplier data $(n,h)=(n_g,h_g)$. The authors develop a detailed construction of Rademacher sums with twists, analyze their convergence via Selberg–Kloosterman zeta functions, and derive explicit Fourier coefficients, showing that the sums yield mock modular forms with shadow propped by $\chi(g)\,\eta(\tau)^3$. They then prove that the resulting $R_{n|h}$ reproduce the proposed $H_g$ for all conjugacy classes of $M_{24}$, using modular-function arguments and Riemann–Roch/Atkin–Lehner techniques to control poles at cusps. The results provide a natural analogue of genus zero for Mathieu moonshine and offer a physico-mhemical interpretation through AdS/CFT, suggesting the Rademacher machinery as a powerful tool to understand moonshine beyond the Monster.
Abstract
Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Mathieu case, and at first glance the answer appears to be negative since not all the discrete groups arising there have genus zero. On the other hand, in this article we prove that each (mock) modular form appearing in the Mathieu correspondence coincides with the Rademacher sum constructed from its polar part. This property, inspired by the AdS/CFT correspondence in physics, was shown previously to be equivalent to the genus zero property of monstrous moonshine. Hence we conclude that this "Rademacher summability" property serves as the natural analogue of the genus zero property in the Mathieu case. Our result constitutes further evidence that the Rademacher method provides a powerful framework for understanding the modularity of moonshine, and leads to interesting physical questions regarding the gravitational duals of the relevant conformal field theories.