Loop and double loop geometry

2022 Fiscal Year Research-status Report

• Project/Area Number: 19K14495
• Research Institution: The University of Tokyo
• Principal Investigator: MUTHIAH DINAKAR 東京大学, カブリ数物連携宇宙研究機構, 客員准科学研究員 (50835410)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: Coulomb branches / KM Affine Grassmannians / KM Affine Hecke Algebras

Outline of Annual Research Achievements

One of the long term goals of this project is to better understand the Coulomb branches of quiver gauge theories, which give rise to a definition of Kac-Moody affine Grassmannian slices. Specifically the goal is to understand their geometry explicitly. In November 2022, my collaborator, Alex Weekes, and I posted a preprint titled "Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices". In this paper, we construct embeddings of Kac-Moody affine Grassmannian slices into one another using Fundamental Monopole Operators. In this way, we answer a question posed by Finkelberg in his 2018 address at the ICM (International Congress of Mathematicians). In particular, we are able to construct many Poisson subvarieties of Kac-Moody Affine Grassmannian slices in this way. Our hope is that the Fundamental Monopole Operators will be a crucial tool in further investigations of Kac-Moody Affine Grassmannian slices.

Additionally, I have made further progress in the project with Auguste Hebert on understanding completions of Kac-Moody Affine Hecke algebras. I have also made progress in the project with Anna Puskas on the T-basis of Kac-Moody Affine Hecke algebras and its relationship with the length function and Bruhat order.

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

I was able to post the preprint with Alex Weekes, and other projects are progressing smoothly.

Strategy for Future Research Activity

My goal is to work toward finishing the projects with Auguste Hebert and Anna Puskas. Additionally, I plan to make progress on the project with Hiraku Nakajima on costalks of affine Grassmannian slices in affine type A.

Causes of Carryover

Because of Covid, there were two years (2020 and 2021), where almost none of the funding was used. However, in 2022, I made good use of the funding to attend conferences. For example, I met with my collaborator Alex Weekes at a conference, and we made significant progress on our joint work. In 2023, I have many conferences that I plan to attend using the Kakenhi funds.