Geometry from the viewpoint of quantization and duality

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K20877
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 11:Algebra, geometry, and related fields
• Research Institution: The 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): 2020-07-30 – 2024-03-31
• Keywords: クラスター代数 / 箙変異 / 分配級数 / 量子不変量 / ペンタゴン関係式 / 指標公式 / 量子ダイログ / 保型性

Outline of Final Research Achievements

Recently quivers and their mutations play pivotal role in mathematics and mathematical physics. In a recent joint work with Yuji Terashima (Tohoku University), we introduced a partition q-series Z(γ) for a quiver mutation loop γ. Z(γ)'s enjoy following remarkable properties: (1) Invariance under inversion and cyclic shift of γ; may be regarded as a monodromy invariant. (2) Pentagon identities similar to those for quantum dilogarithms. (3) (For Dynkin type quivers) Reproduce fermionic characters of coset CFTs, and thus have nice modularity. (4) (For reddening mutation sequences) Can be expressed as an ordered product of quantum-dilogarithms and coincide with the combinatorial Donaldson-Thomas invariants. I am now working on extending these ideas to obtain quantum invariants of triangulated three manifolds.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

学問が専門化し細分化が進むと，分野間の共通構造が見えづらくなる．我々が提唱する分配級数は，有向グラフや貼り合わせのような組合せ論的データのみから定義され、具体的対象の設定やモデルの詳細には依らない．ちょうど遺伝子が生物種を比較するときに役立つように，分配級数も，数学や物理学の諸分野にまたがる双対性（共通の性質）を追究する上で役立つことが期待される。