Beyond Generalized Moonshine

2019 Fiscal Year Annual Research Report

• Project/Area Number: 17K14152
• Research Institution: University of Tsukuba
• Principal Investigator: CARNAHAN Scott 筑波大学, 数理物質系, 准教授 (10600538)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: 代数 / 頂点作用素代数 / モジュラー形式 / ムーンシャイン / 群スケーム

Outline of Annual Research Achievements

(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.

Research Products

• [Journal Article] Building Vertex Algebras from Parts (2020)
  • Author(s): Scott Carnahan
  • Journal Title: Communications in Mathematical Physics Volume: 373 Pages: 1-36
  • DOI: 10.1007/s00220-019-03607-0
  • Peer Reviewed
• [Journal Article] A Self-Dual Integral Form of the Moonshine Module (2019)
  • Author(s): Scott Carnahan
  • Journal Title: SIGMA Volume: 15, no 030 Pages: 1-36
  • DOI: 10.3842/SIGMA.2019.030
  • Peer Reviewed / Open Access
• [Presentation] Monstrous Moonshine over the Integers (2019)
  • Author(s): Scott Carnahan
  • Organizer: JMS Algebra Symposium, Tohoku University
  • Invited
• [Presentation] Moonshine over Z (2019)
  • Author(s): Scott Carnahan
  • Organizer: The Mathematical Foundations of Conformal Field Theory and Related Topics, Nankai University
  • Int'l Joint Research / Invited