2019 Fiscal Year Final Research Report
Study of the geometry suggested by dualities
| Project/Area Number |
16K13752
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Kato Akishi 東京大学, 大学院数理科学研究科, 准教授 (10211848)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 箙 / 変異 / クラスター代数 / 可積分系 / 低次元トポロジー / 組合せ論的データ / 分配級数 / 双対性 |
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| Outline of Final Research Achievements |
Recently quivers and their mutations play pivotal role. In a joint work with Yuji Terashima (Tohoku), we introduced "partition q-series" for quiver mutation loops. They enjoy following remarkable properties: They are invariant under inversion and cyclic shift; so may be regarded as monodromy invariants. They satisfies pentagon identities. For Dynkin quivers, they reproduce so-called fermionic character formulas, and enjoy nice modular properties as expected from the conformal field theory. For reddening sequence, they are expressed as ordered product of quantum-dilogarithms and reproduce combinatorial Donaldson-Thomas invariants of the initial quivers.
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| Free Research Field |
数理物理学
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| Academic Significance and Societal Importance of the Research Achievements |
箙(quiver)とその変異(mutation)は，クラスター代数とともに，可積分系・低次元トポロジー・表現論・代数幾何学・WKB 解析などさまざまな分野に共通して現れる構造として注目を集めている．特に，箙の変異列 (mutation sequence) とゲージ理論や３次元双曲多様体の関連が提唱され，その不変量を数学的に厳密に解析する手段の開発が必要となった．分配q級数や分配関数は組合せ論的データのみから定義され、箙が表す数学的対象の詳細には依らないので、双対性の背後にある共通の性質を追究する上で役立つと期待される。
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