| 研究実績の概要 |
(1) The Moonshine Module over the integers: I constructed a self-dual integral form of the Moonshine vertex operator algebra with monster symmetry. The existence of this object resolves both Ryba's 1994 Modular Moonshine conjecture, and a strong form proposed by Borcherds and Ryba in 1996. My results imply the existence of a rank 196884 positive-definite self-dual lattice with monster symmetry, conjectured by Conway and Norton in 1985.
(2) 51 constructions of the Moonshine module: I solved Tuite's 1993 conjecture on the orbifold duality correspondence between non-Fricke elements of the Monster simple group and fixed-point free automorphisms of the Leech lattice that satisfy a "no massless states" condition. From this, I showed that the ambiguous constants that appear in the Generalized Moonshine conjecture for Hauptmoduln are necessarily roots of unity.
(3) Vertex algebras and non-monstrous functions: In joint work with T. Komuro and S. Urano, I work out conditions on completely replicable functions that are necessary for such functions to come from holomorphic vertex operator algebras. Using orbifold conformal field theory, we eliminate all but 3 of the 157 candidate non-monstrous functions.
(4) Automorphism group schemes of lattice vertex operator algebras: In joint work with H. Mochizuki, I have made significant progress on determining the symmetries of the vertex operator algebra (over any commutative ring) attached to any positive definite even lattice.
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