Some irrational Generalised Moonshine from orbifolds

Abstract

We verify the Generalised Moonshine conjectures for some irrational modular functions for the Monster centralisers related to the Harada-Norton, Held, M12 and L3(3) simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case.

Moonshine

“…Theorem 5 was used by Dong-Li-Mason in [79], following an observation of Tuite (cf. [76], and [223][224][225] for broader context), to prove Norton's conjecture for the case that g and h generate a cyclic subgroup of M, and this approach, via twisted modules for V , has been extended by Ivanov-Tuite in [141,142]. Höhn obtained a generalization of Borcherds' method by using a particular twisted module for V to construct a BKM algebra adapted to the case that g is in the class named 2A in [56] -the smaller of the two conjugacy classes of involutions in M -and in so doing established [135] generalized moonshine for the functions T (g,h) with g ∈ 2A.…”

Generalized Mathieu Moonshine

“…Although this conjecture has been proven in special cases [15][16][17], the general case is still open. 2 The Generalized Moonshine conjecture was given a physical interpretation by Dixon, Ginsparg and Harvey [24] that was later elaborated upon by Ivanov and Tuite [17][18][19]. They showed that the Norton series f (g, h; τ ) arises naturally as the character in the twisted sector V g of an orbifold of the Monster CFT V by the element g ∈ M, 'twined' by the group element h; in standard CFT language, they can therefore be interpreted as Many of the properties conjectured by Norton can be proved from holomorphic orbifold considerations [15].…”

“…In the remainder of this subsection, we illustrate how one of these non-zero twisted twining genera, namely the one associated to g ∈ 2B and h ∈ 8A, can be determined explicitly; the other five cases are completely analogous and are dealt with in Appendix A. Let us work with the conventions that the three permutations (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23) (24), (3,17,10,7,9) …”

Monstrous and Generalized Moonshine and Permutation Orbifolds

“…Norton's Generalized Moonshine Conjectures [N2] concerning centralizers of the Monster group has yet to be generally proven using either Borcherds' approach or orbifold partition function methods although some progress has recently been made in refs. [H] and [T3], [IT1], [IT2] respectively.…”