2022 Fiscal Year Research-status Report
New directions in vertex algebras and moonshine
| Project/Area Number |
22K03264
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2022-04-01 – 2026-03-31
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| Keywords | moonshine / vertex algebra / weak Hopf algebra / vertex operator algebra |
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| Outline of Annual Research Achievements |
Together with my student Satoru Urano, I have submitted a paper about our conjecture that unifies and generalizes Monstrous Moonshine and Modular Moonshine. Our conjecture asserts that for any subring R of the complex numbers, any subgroup G of the monster, and any ring homomorphism f from the representation ring (Green ring) of RG to the complex numbers, the "generalized McKay-Thompson series" given by applying f to the graded pieces of the monster vertex algebra is the q-expansion of a genus zero modular function.
This is known in the special cases covered by Monstrous Moonshine (where R is the complex numbers) and Modular Moonshine (where R is isomorphic to a p-adic ring and G is cyclic of order with p-valuation 1), but we proved it in some additional cases, and we have shown that all generalized McKay-Thompson series satisfy an infinite collection of relations that we call "quasi-replicability".
This paper has been accepted at IMRN.
We have additional results that have not been submitted yet. First, we have classified homomorphisms from the Green rings of all groups of order pq, where p and q are distinct primes, and we have proved our conjecture for all "totally Fricke" cyclic groups of square-free order.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Monstrous moonshine for integral group rings has gone more smoothly than planned. We have proved integral versions of the no-ghost theorem and analysis of the Laplacian on the monster Lie algebra, and we have proved a weak form of replicability for generalized McKay-Thompson series. We have also proved the Hauptmodul assertion of the conjecture for a large class of subgroups of the monster.
My project on weak Hopf algebras has gone more slowly than expected, because some parts turned out to be hard. In particular, I had hoped the internal tensor product would produce a symmetric monoidal category, but it turns out to require some laxness. We still do not have a Galois correspondence, and no Tannakian reconstruction.
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| Strategy for Future Research Activity |
In the moonshine project, I plan to do some computational experiments on classifying quasi-replicable functions. Additional directions for generalization include classifying homomorphisms from Green rings for more groups. Based on our positive results in order pq, it seems likely that all square-free order subgroups of the monster are tractable, and possibly even all groups whose Sylow subgroups are cyclic of cube-free order. After that, I intend to consider some cases with infinitely many indecomposable representations, like the 4-group.
In the project on weak Hopf algebras, I plan to work out more compatibility properties of internal intertwining operators, and construct a categorical framework that encodes them.
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| Causes of Carryover |
I plan to do more travel to conferences in the coming year, and I plan to fund more guests and speakers.
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