The Conway-Miyamoto correspondences for the Fischer 3-transposition groups

TL;DR

This paper constructs 3-transposition groups as automorphism groups of vertex operator algebras, establishing correspondences between Fischer groups and specific Virasoro vectors within the moonshine VOA.

Contribution

It introduces a general method to realize 3-transposition groups as automorphism groups of VOAs and links Fischer groups to Virasoro vectors in the moonshine VOA.

Findings

Fischer groups $ ext{Fi}_{23}$ and $ ext{Fi}_{22}$ correspond to specific Virasoro vectors.

A new construction method for 3-transposition groups using VOAs.

Establishment of Conway-Miyamoto correspondences in the moonshine VOA.

Abstract

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.