| 研究実績の概要 |
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.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
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.
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