Umbral Moonshine and the Niemeier Lattices

The umbral moonshine module for the unique unimodular Niemeier root system

“…We show that canonically-twisted modules for V X , constructed explicitly in §2.5 (cf. (2.48)), furnish a bi-graded G X -module for which the graded trace functions are exactly compatible with the predictions of [4]. In other words, we construct the analogue of the moonshine module V ♮ , of Frenkel-Lepowsky-Meurman [19,20,9], for the X = E 3 8 case of umbral moonshine.…”

“…To prove that our construction is indeed the X = E 3 8 counterpart to V ♮ , we verify the X = E 3 8 analogue of the Conway-Norton moonshine conjecture, proven by Borcherds in [10] in the case of the monster, which predicts that the trace functions arising are uniquely determined by their automorphy and their asymptotic behavior near cusps. Thus we verify the X = E 3 To prepare for precise statements of results, recall that vector-valued functions H X g (τ ) = (H X g,r (τ )) on the upper half plane H are considered in [4], for g ∈ G X ≃ S 3 , where the components are indexed by r ∈ Z/60Z. Define o(g) to be the order of an element g ∈ G X .…”

“…Appell-Lerch sums appear in this analysis in the so called "massless" characters. This Mathieu moonshine connection was conjectured in [3,4] to be part of a much more general phenomenon, known as umbral moonshine, which attaches a vector-valued mock modular form H X , a finite group G X , and an infinite-dimensional graded G X -module K X to the root systems of each of the 23 Niemeier lattices. The analysis in [4] relied heavily on the construction of mock modular forms in terms of meromorphic Jacobi forms and built on the important work in [5] extending the analysis of [1] and characterizing special Jacobi forms in terms of growth conditions.…”

“…(1.2) from Ramanujan's last letter to Hardy [15,16]. The conjectures of [4] (see also [17]) imply the existence of a bi-graded super vector space K X = r K X r = r,d K X r,d that is a module for G X ≃ S 3 and satisfies sdim q K X 1 = −2q −1/120 + n>0 dim K X 1,n−1/120 q n−1/120 = 2q −1/120 (χ 0 (q) − 2), sdim q K X 7 = n>0 dim K X 7,n−49/120 q n−49/120 = 2q 71/120 χ 1 (q).…”

“…Define o(g) to be the order of an element g ∈ G X . The H X g are not uniquely determined in [4], except for the case that g = e is the identity, o(g) = 1. But it is predicted that H X g is a mock modular form of weight 1/2 for Γ 0 (o(g)), with shadow given by a certain vector-valued unary theta function S X g (cf.…”