2014 Fiscal Year Annual Research Report
| Project/Area Number |
24740005
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2012-04-01 – 2015-03-31
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| Keywords | 代数 / ムーンシャイン |
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| Outline of Annual Research Achievements |
This project had three main components: (1) discovering fusion rules and higher associativity data for twisted modules via conformal blocks (2) applying these data to construct generalized vertex algebras and Lie algebras (3) using the Lie algebras to prove the Generalized Moonshine conjecture. Parts (1) and (2) are essentially complete, but part (3) is missing an important step.
In 2014, I posted the paper "Building Vertex Algebras from Parts" to the ArXiv and submitted it to a top-tier journal. The results are the following: (1) I described a homological method to construct new vertex algebras from pieces. (2) I reduced the problem of existence of simple current extensions for general abelian groups to the problem for abelian 2-groups. The 2-group problem is still open, even for the special case of regular vertex operator algebras.
My work on conformal blocks has become quite long, so I have split it into several papers. I have generalized the previous results for each of these papers, so they may be useful for researchers who do not study moonshine. Among the new results are general descent properties of Hom stacks, and a generalization of the higher frame construction to arbitrary smooth Deligne-Mumford maps between stacks.
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