Quantum cohomology of infinite dimensional manifolds

Research Project

Project/Area Number	16K13759
Research Category	Grant-in-Aid for Challenging Exploratory Research
Allocation Type	Multi-year Fund
Research Field	Geometry
Research Institution	Nihon University
Principal Investigator	Otofuji Takashi 日本大学, 工学部, 准教授 (70339266)
Project Period (FY)	2016-04-01 – 2020-03-31
Project Status	Completed (Fiscal Year 2019)
Budget Amount *help	¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000) 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: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords	可積分系 / 無限次元多様体 / 無限次元旗多様体 / ループ群
Outline of Final Research Achievements	We started our research project of finding new aspects of integrable systems in quantum cohomology, expecting that the case of infinite-dimensional flag manifolds would be an interesting example. We hoped that the following known theories would be keys to our research: structures of integrable systems known for quantum cohomology of finite-dimensional manifolds (Frobenius manifolds, D-modules), some generating functions coming from instanton moduli spaces, several known theories for Toda lattices, and a two-dimensional topological field thoery with positive boundary. With any way with one of the keys above we have not succeeded to get essential new discoveries.
Academic Significance and Societal Importance of the Research Achievements	無限次元旗多様体の量子コホモロジー環を出発点として、新しい可積分系の構造を求めるという目標は研究期間内に達成することは叶わなかった。当初期待した具体的な数学的結果を得ることは出来なかった一方、可積分系に関連する様々な数学理論の理解を深める必要性から、関連分野の研究者との交流を深めるために、全4年間の研究期間のうち3か年において年一回、2日ないし3日にわたる研究集会を開催した。毎回20名前後の参加者を得て、当研究課題の広い枠組みでの意義を一定数の研究者に認知して頂く活動を行うことが出来た。