Loop and double loop geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14495
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• 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 Final Research Achievements

Alex Weekes and I have published two papers about the geometry of Coulomb branches. First, we classified symplectic leaves in finite-type, and second, we constructed many symplectic leaves in general using fundamental monopole operators. I have posted a paper defining double affine Kazhdan-Lusztig R-polynomials generalizing techniques of masures and point-counting for p-adic Kac-Moody groups. Alex Weekes, Oded Yacobi and I have published a paper solving a twenty-year-old conjecture of Pappas and Rapoport using techniques we developed for studying affine Grassmannians.

Additionally, I have begun and progressed several projects during the grant. Hiraku Nakajima and I are working on understanding Intersection Cohomology stalks for affine type A Coulomb branches. August Hebert and I have constructed a completion of Kac-Moody affine Hecke algebras in a work in-progress. Finally, Anna Puskas and I are about to post a preprint about pursuing Coxeter theory of Kac-Moody affine Hecke algebras.

Free Research Field

Mathematics

Academic Significance and Societal Importance of the Research Achievements

This is about Langlands duality for loop groops via the geometry of double loop groups. This involves Coulomb branches, which come from quantum physics, and p-adic Kac-Moody groups, which come from arithmetic and number theory. The goal is to advance and connect these rich and different areas.