• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Study of the geometry suggested by dualities

Research Project

Project/Area Number 16K13752
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

Kato Akishi  東京大学, 大学院数理科学研究科, 准教授 (10211848)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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 / 分配関数 / 三角圏 / 不変量 / 弦理論 / 安定性条件 / 幾何構造
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.

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。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (8 results)

All 2019 2017 Other

All Journal Article (1 results) (of which Peer Reviewed: 1 results) Presentation (5 results) (of which Invited: 5 results) Remarks (2 results)

  • [Journal Article] Quiver Mutation Sequences and $q$-Binomial Identities2017

    • Author(s)
      Kato Akishi、Mizuno Yuma、Terashima Yuji
    • Journal Title

      International Mathematics Research Notices

      Volume: 108 Issue: 23 Pages: 1-24

    • DOI

      10.1093/imrn/rnx108

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] 力学の変遷 ―古典・量子・弦―2019

    • Author(s)
      加藤晃史
    • Organizer
      日本数学会 東京工業大学
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 力学の変遷-古典・量子・弦-2019

    • Author(s)
      加藤晃史
    • Organizer
      日本数学会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Quiver mutation and partition q-series2017

    • Author(s)
      加藤晃史
    • Organizer
      日本数学会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2017-03-26
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Quiver mutation loops and partition q-series2017

    • Author(s)
      加藤晃史
    • Organizer
      日本数学会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Quiver mutation loops and partition q-series2017

    • Author(s)
      加藤晃史
    • Organizer
      研究集会「リーマン面に関連する位相幾何学」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Remarks] 研究集会「リーマン面に関連する位相幾何学」

    • URL

      http://www.ms.u-tokyo.ac.jp/~tado/riemann_surface17.html

    • Related Report
      2017 Research-status Report
  • [Remarks] Quiver mutation sequence and q-binomial identities

    • URL

      https://arxiv.org/abs/1611.05969

    • Related Report
      2016 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2021-02-19  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi