Traces of singular moduli and moonshine for the Thompson group

Abstract

We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight onehalf 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 Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.

A proof of the Thompson moonshine conjecture

“…• its Fourier expansion is of the form 2q −3 + χ 2 (g) + (χ 4 (g) + χ 5 (g))q 4 + O(q 5 ), where χ j is the jth irreducible character of Th as given in Tables 1, 2 [25]. In Sect.…”

“…Tables 1, 2 The notation (κ m,g ) m indicates the addition of the theta correction m κ m,g ϑ(mτ ) in the definition of F [g] (τ ) in (2.6), see Table 5 in [25] and also Remark 2.3 24 2 24…”

“…(2) It should be pointed out that the definition of modular forms resp. harmonic Maaß forms with multiplier is slightly different in [25], where the multiplier is included into the definition of the slash operator f | k γ , so that multipliers here are always the inverse of the multipliers there.…”

“…To this end, we first establish the exact modularity and integrality properties of the conjectured McKay-Thompson series F [g] (τ ), which are stated without proof in [25]. Table 5.…”

Ramanujan's influence on string theory, black holes and moonshine

“…Here the Jacobi theta function plays the role of the constant function with coefficients c(n) that are constant with n and the function with slowest exponential growth is f 3 (τ ) plus any multiple of θ(τ ). Following [10] we note that the combination with vanishing term linear in q,…”