Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 114, 8 pages      arXiv:1712.10160
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

Scott Carnahan, Takahiro Komuro and Satoru Urano
Division of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan

Received May 07, 2018, in final form October 15, 2018; Published online October 25, 2018

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 $V$.

Key words: moonshine; vertex operator algebra; modular function; orbifold.

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