"Not moonshine" was Conway's verdict when he first heard McKay's observation. He was wrong. — Simon Norton, recalling John Conway, 1979
In 1978 John McKay noticed something that should not have been possible. He was looking at the expansion of the modular $j$-function — a classical object in the theory of elliptic curves, with no apparent connection to group theory — and saw that its first nontrivial coefficient was 196,884. He had recently learned that the smallest nontrivial representation of the Monster group had dimension 196,883. The difference was 1 — the trivial representation.
McKay wrote to John Thompson, who found the same pattern in the next coefficient: $21,493,760 = 21,296,876 + 196,883 + 1$, where 21,296,876 is the dimension of the Monster's second nontrivial representation. Conway and Norton computed further, confirmed the pattern, called it Monstrous Moonshine, and published a conjecture in 1979: every element $g \in \mathbb{M}$ has a McKay-Thompson series $T_g(\tau)$ that is a Hauptmodul for some genus-zero congruence subgroup of $\text{SL}(2,\mathbb{Z})$.
The modular $j$-function is the unique (up to additive constant) holomorphic function on the upper half-plane $\mathbb{H} = \{\tau \in \mathbb{C} : \text{Im}(\tau) > 0\}$ that is invariant under the modular group $\Gamma = \text{SL}(2,\mathbb{Z})$, has a simple pole at the cusp $\tau \to i\infty$, and satisfies $j(i) = 1728$, $j(e^{2\pi i/3}) = 0$. Writing $q = e^{2\pi i\tau}$:
$$j(\tau) = \frac{1}{q} + 744 + 196884\,q + 21493760\,q^2 + 864299970\,q^3 + \cdots$$The coefficient 744 is conventional (often written as $j(\tau) - 744$). The $j$-function classifies complex elliptic curves: two elliptic curves $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$ are isomorphic as complex tori if and only if $j(\tau_1) = j(\tau_2)$, where $\tau_i = \omega_{2,i}/\omega_{1,i}$ is the period ratio of $\Lambda_i$.
red = Monster representation dimensions · black = j-function coefficients
The key object is the moonshine module $V^\natural$, a vertex operator algebra (VOA) constructed by Frenkel, Lepowsky, and Meurman in 1984–88. It is built by compactifying the bosonic string on the Leech lattice $\Lambda_{24}$ — the unique even self-dual lattice in $\mathbb{R}^{24}$ — and then orbifolding by the $\mathbb{Z}/2$ involution that sends $v \mapsto -v$. The result is a VOA whose partition function is exactly $j(\tau) - 744$.
The Monster group $\mathbb{M}$ acts on $V^\natural$ as a group of VOA automorphisms. For any $g \in \mathbb{M}$, the McKay-Thompson series is
$$T_g(\tau) = \text{tr}_{V^\natural}(g \cdot q^{L_0 - 1}) = \sum_{n \geq -1} \text{tr}_{V^\natural_n}(g) \cdot q^n$$The Borcherds proof (1992, Fields Medal 1998) showed that every $T_g(\tau)$ is indeed a Hauptmodul for a genus-zero group — confirming the Conway-Norton conjecture in full. The key tool was the Monster Lie algebra, an infinite-dimensional Lie algebra built from $V^\natural$ whose denominator formula encodes the moonshine.
The upper half-plane $\mathbb{H}$ carries a natural contact structure. Its unit tangent bundle $T^1\mathbb{H}$ (the space of unit-speed geodesics) is diffeomorphic to $\text{PSL}(2,\mathbb{R})$, and the contact form is the Liouville form of the geodesic flow — precisely the dm³ contact form $\alpha = dz - r^2\,d\theta$ in local coordinates adapted to the hyperbolic metric.
The modular group $\text{SL}(2,\mathbb{Z})$ acts on $T^1\mathbb{H}$ by the geodesic-flow-preserving symmetries of the lattice. It is the lattice symmetry of the fold operator $F$ in the dm³ chain: the irreversible commitment that the $j$-function encodes is the identification of the modular orbit — two points related by a Möbius transformation from $\text{SL}(2,\mathbb{Z})$ are the same elliptic curve, and the $j$-function is the Lagrangian that assigns a value to each orbit.
The moonshine module $V^\natural$ is the chiral algebra of a CFT whose target space contact structure is the dm³ contact form on $T^1\mathbb{H}$. The Monster $\mathbb{M}$ is the automorphism group of this contact structure — the symmetry group of the fold $F$ in its most general, complete expression. Every other chapter in this arc studies a system whose fold participates in the same algebraic structure at a smaller scale.
The Conway-Norton conjecture specifies that each McKay-Thompson series $T_g$ generates the function field of a genus-zero Riemann surface. Genus zero means the surface is topologically a sphere — there are no handles. The Hauptmodul condition means $T_g$ alone generates the full field of modular functions for its group, the way $j$ generates the field for the full modular group.
Genus zero is a highly restrictive condition. Most congruence subgroups of $\text{SL}(2,\mathbb{Z})$ give positive-genus quotient surfaces. The fact that every element of the Monster satisfies genus zero is the deepest constraint in the moonshine correspondence, and it remains somewhat mysterious even after Borcherds' proof: the proof verifies genus zero but does not make it inevitable from first principles. The dm³ reading offers a partial account: the genus-zero condition is the statement that the fold operator $F$ for these systems has no topological obstruction — the quotient contact manifold is simply connected.
The moonshine correspondence was the first in a series of "moonshines" — relationships between sporadic groups and modular/automorphic forms. Mathieu moonshine (2010, Eguchi-Ooguri-Tachikawa) relates the Mathieu group $M_{24}$ — a member of the Happy Family — to the elliptic genus of K3 surfaces. Umbral moonshine (2012, Cheng-Duncan-Harvey) extends this to all 23 Niemeier lattices. These are all subsystems of the Monster moonshine in the dm³ sense: each corresponds to a fold operator operating on a smaller contact structure, a sub-lattice of the Leech.
The Monster remains the master symmetry. Everything else is a projection.