2022 Fiscal Year Final Research Report
Operator algebras and Fukaya categories
| Project/Area Number |
19K21832
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | The University of Tokyo |
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Principal Investigator |
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| Project Period (FY) |
2019-06-28 – 2023-03-31
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| Keywords | 作用素環 / テンソル圏 / フュージョン圏 / 部分因子環 |
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| Outline of Final Research Achievements |
We studied relations between subfactor theory in operator algebras and tensor categories. We have shown that 4-tensors appearing in 2-dimensional topological order in condensed matter physics are essentitally the same as bi-uninary connections which are well-studied in subfactor theory. We have further proved that the range of projector matrix product operator in physics setting is the same as the higher relative commutant of the subfactor arising from the 4-tensor. Moreover, we gave a characterization when a general 4-tensor is of the form considered in this setting.
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| Free Research Field |
作用素環論
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| Academic Significance and Societal Importance of the Research Achievements |
最近注目を集めている2次元トポロジカル物性における数学的構造を圏論と作用素環論の立場から研究した．これによって数理物理学に現れる圏論の構造が，作用素環論における部分因子環の言葉で記述できることが分かった．特に2次元トポロジカル物性で研究されているテンソルネットワークに現れる 4-tensor と，作用素環論における部分因子環論で非退化な commuting square の記述に現れる bi-unitary connection とが実質的に同じであることを明らかにし，この関係を追究することによってこれまで知られていなかった部分因子環論との新たな関係を見出した．
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