Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

Abstract

Given a holomorphic C 2 -cofinite vertex operator algebra V with graded dimension j − 744, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of V has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly occur as trace functions on V .

Fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$

“…It is known that the automorphism group of L sl 2 (k, 0) is P SL(2). Given a finite subgroup G of P SL(2), the orbifold vertex operator algebra L sl 2 (k, 0) G is simple, and is conjecturely rational which is shown to be true when G is solvable [CM16]. It is very natrual and desirable to study the structure and representations of the orbifold vertex operator algebra L sl 2 (k, 0) G .…”