Moonshine for Rudvalis's sporadic group II

From the Monster to Thompson to O'Nan

The commencement of monstrous moonshine is a connection between the largest sporadic simple group---the monster---and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.

Traces of Singular Moduli and Moonshine for the Thompson Group

We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay--Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Tho…

Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras $E_{8,7,6}$ and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of $PSL(2,R)$ for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.

K3 Elliptic Genus and an Umbral Moonshine Module

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of $K3$ string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the $K3$ elliptic genus. Inspired by the above two relations between moonshine and $K3$ string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the c…

O'Nan moonshine and arithmetic

Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay--Thompson series are weight $3/2$ modular forms. The coefficients of these series may be expressed in terms of class numbers, traces of singular moduli, and central critical values of quadratic twists of weight 2 modular $L$-functions. As a consequence, for primes $p$ dividing the order of the O'Nan group we obtain congruences between O'Nan group character values and class numbers, $p…

Moonshine for All Finite Groups

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group $G$, we give constructions of infinitely many graded infinite-dimensional $\mathbb{C}[G]$-modules where the McKay-Thompson series for a conjugacy class $[g]$ is a weakly holomorphic modular function properly on $Γ_0(\text{ord}(g))$. A…

Classical and Umbral Moonshine: Connections and $p$-adic Properties

The classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group $M$, the largest of the simple sporadic groups, and certain modular functions, called Hauptmodln. In particular, the $n$-th Fourier coefficient of Klein's $j$-function is the dimension of the grade $n$ part of a special infinite dimensional representation $V$ of the monster group. More generally the coefficients of Hauptmoduln are graded traces $T_g$ of $g \in M$ acting on $V$. Similar phenomena have been shown to hold for the Matthieu group $M_{24}$, but inste…

Thompson群的月光现象的两种新化身

Two New Avatars of Moonshine for the Thompson Group

本文证明了Thompson群月光现象的两种新化身的存在性，通过Borcherds乘积和奇异模迹揭示新旧模的关系，并扩展了广义魔群月光与半影月光的对应。

Moonshine, Superconformal Symmetry, and Quantum Error Correction

Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a $c=24$ two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a $c=12$ CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group $M_{24}$ has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving $(4,4)$ superconforma…