2018 Fiscal Year Research-status Report
Beyond Generalized Moonshine
| Project/Area Number |
17K14152
|
|---|
| Research Institution | University of Tsukuba |
|---|
Principal Investigator |
|
|---|
| 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 |
|---|
| Outline of Annual Research Achievements |
I have the following research achievements.
1. Refinement of Moonshine integral form work - My paper on the self dual integral form of the Moonshine module has been accepted for publication, and the results have been significantly strengthened during the course of the referee process. I applied a new variant of faithfully flat descent to glue vertex algebras over different rings together, and this gave me a strong uniqueness result. Furthermore, a corollary was pointed out to me by Griess: My results imply the existence of a rank 196884 positive-definite self-dual lattice with monster symmetry, namely, an integral form for the algebra from Griess's original construction of the Monster.
2. More Modular moonshine - Together with my student S. Urano, I have results in progress concerning a natural generalization of Ryba's 1994 Modular moonshine conjecture. The original conjecture matched the characteristic zero trace of an element of the Monster on the Moonshine vertex operator algebra with the graded Brauer character of a p-regular element on a certain mod p vertex algebra. The odd prime cases were solved by Borcherds and Ryba in the late 1990s, and the p=2 case was solved by me last year as part of my work on integral forms. The generalized conjecture concerns elements of the Monster with composite order, and requires new advances in Tate cohomology and modular representation theory.
|
| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
My work on a geometrical interpretation of the moonshine cohomology class has been delayed for a few reasons.
First, I have been able to make fast progress on integrality and modular moonshine questions, so these have taken priority.
Second, there has been progress on this problem from other directions: Johnson-Freyd has recent work using modular tensor categories and conformal nets that conjecturally identifies the cohomology class that comes from vertex operator algebras. This has made me less enthusiastic about the originality of the results I will get.
Third, a solid identification of the class seems to require a substantial generalization of the orbifold regularity results given in my work with Miyamoto, and that has not been finished.
Finally, some of the foundational questions in geometry have been unexpectedly time-consuming.
|
| Strategy for Future Research Activity |
I am continuing my work on twisted conformal blocks on log-smooth curves and non-abelian fusion. I am not sure I can finish by the end of this year. I am also working on enhanced modular moonshine and other questions about vertex algebras and Tate cohomology over small rings.
|
