Beyond Generalized Moonshine

• Project/Area Number: 17K14152
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• 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
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 代数 / ムーンシャイン / 頂点作用素代数 / モジュラー形式 / 群スケーム / 代数学 / 環論 / 頂点代数 / モンスター群

Outline of Final Research Achievements

In this project, I have four main results:1) I have constructed a self-dual integral form of the Moonshine vertex operator algebra, with monster symmetry. This resolves Ryba's 1994 Modular moonshine conjecture. 2) I strengthened my earlier proof of the Generalized Moonshine conjecture, by showing that a collection of ambiguous constants are roots of unity. 3) I proved Tuite's 1993 orbifold duality conjecture, connecting non-Fricke elements of the monster to certain fixed-point free automorphisms of the Leech lattice. 4) Together with T. Komuro and S. Urano, I showed that 154 of the 157 non-monstrous completely replicable functions cannot arise as traces of automorphisms of holomorphic vertex operator algebras.

Academic Significance and Societal Importance of the Research Achievements

Monstrous moonshine is a mathematical phenomenon that was discovered when someone noticed some large numbers from calculations in two different fields of mathematics were very similar. My work in this project has answered some old questions about the nature of this connection and similar phenomena.

Research Products

• [Journal Article] Building Vertex Algebras from Parts (2020)
  • Author(s): Scott Carnahan
  • Journal Title: Communications in Mathematical Physics Volume: 373 Issue: 1 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
• [Journal Article] 51 constructions of the Moonshine module (2018)
  • Author(s): Carnahan Scott
  • Journal Title: Communications in Number Theory and Physics Volume: 12 Issue: 2 Pages: 305-334
  • DOI: 10.4310/cntp.2018.v12.n2.a3
  • Peer Reviewed
• [Journal Article] Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions (2018)
  • Author(s): Carnahan Scott、University of Tsukuba, Japan、Komuro Takahiro、Urano Satoru、University of Tsukuba, Japan、University of Tsukuba, Japan
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 14 Pages: 1-8
  • DOI: 10.3842/sigma.2018.114
  • Peer Reviewed / Open Access
• [Journal Article] Moonshine over Z? (2018)
  • Author(s): Carnahan Scott
  • Journal Title: RIMS Kokyuroku Volume: 2086 Pages: 91-98
  • 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
• [Presentation] Extended Monstrous Moonshine (2018)
  • Author(s): Carnahan Scott
  • Organizer: Vertex operator algebras, number theory, and related topics, Sacramento State University
  • Int'l Joint Research / Invited
• [Presentation] Extended Monstrous Moonshine (2018)
  • Author(s): Carnahan Scott
  • Organizer: Pan-Asia Number Theory Conference 2018, IMS, National University of Singapore
  • Int'l Joint Research / Invited
• [Presentation] More Modular Moonshine (2018)
  • Author(s): Carnahan Scott
  • Organizer: RIMS Gasshuku-style seminar on Conformal Field Theory, Kansai Seminar House
  • Invited
• [Presentation] More Modular Moonshine (2018)
  • Author(s): Carnahan Scott
  • Organizer: 30th Kusatsu Group Theory Seminar, Kusatsu Seminar House
  • Invited
• [Presentation] More Modular Moonshine (2018)
  • Author(s): Carnahan Scott
  • Organizer: Workshop on Moonshine, Erwin Schrodinger Institute
  • Int'l Joint Research / Invited
• [Presentation] Generalized Monstrous Moonshine (2017)
  • Author(s): Scott Carnahan
  • Organizer: Nankai Symposium on Physics, Geometry, and Number Theory
  • Int'l Joint Research / Invited
• [Presentation] 51 constructions of the Moonshine module (2017)
  • Author(s): Scott Carnahan
  • Organizer: Workshop on Calabi-Yau Manifolds: Arithmetic, Geometry, and Physics
  • Invited
• [Presentation] Generalized Moonshine (2017)
  • Author(s): Scott Carnahan
  • Organizer: Conference on Number Theory, Geometry, Moonshine & Strings
  • Int'l Joint Research / Invited
• [Presentation] A self-dual integral form for the Moonshine Module. (2017)
  • Author(s): Scott Carnahan
  • Organizer: Research on algebraic combinatorics and representation theory of finite groups and vertex operator algebras
  • Int'l Joint Research / Invited
• [Presentation] Generalized Moonshine (2017)
  • Author(s): Scott Carnahan
  • Organizer: Trends in Modular forms
  • Int'l Joint Research / Invited
• [Presentation] Moonshine over Z (2017)
  • Author(s): Scott Carnahan
  • Organizer: Finite Groups, VOAs, and Related Topics 2018
  • Invited