Coulomb Branch Student Seminar

报告人 Speaker:贾博名(YMSC),李鹏程(YMSC),梁石易新(Qiuzhen),司徒泉(YMSC),王昊(YMSC),王文越(Qiuzhen),张展维(Qiuzhen),赵启弦(YMSC)
组织者 Organizer:贾博名(YMSC)
时间 Time:Saturdays 10:30 - 11:50 (AM)
地点 Venue:静斋218室

Upcoming talk:


Title: Basic introduction of 3d N=4 mirror symmetry

Speaker: 王昊(YMSC)

Abstract: In this talk, I will talk about some basic properties of the 3d N=4 mirror symmetry, which is an infrared duality of two ultraviolet theories with 8 supercharges flow to the same infrared fixed point. Then I will talk about some basic of 3d N=4 supersymmetry theory, including its superalgebra and symmetries, its SUSY vacua known as the Higgs branch and Coulomb branch and its brane realization. In the end we will come to the famous 3d N=4 mirror symmetry.



Past talks:



Title: Chiral algebras and factorization algebras

Speaker: 伊东木(清华)

Time:9:30-10:50 am, 2023-12-23

Abstract: I will introduce the notion of chiral and factorization algebras after Beilinson--Drinfeld. We will also introduce quasi-conformal vertex algebras that serve as universal chiral/factorization algebras. Examples include the Beilinson--Drinfeld grassmannian. 


Title: Computing the Coulomb Branch when the group is a torus

Speaker: Jianyu Ren (Qiuzhen)

Abstract: In this talk, I will introduce Braverman-Finkelberg-Nakajima‘s definition of Coulomb branch as the spectrum of equivariant BM homology (with convolution product) of the variety of triples, and explain the structure of them when the group is a torus.


Title: Equivariant K-theory of Affine Grassmannian and the Universal Centralizer

Speaker: 司徒泉(YMSC)

Abstract: I will discuss a paper by Bezrukavnikov, Finkelberg and Mirkovic, in which the equivariant K-theory of the affine Grassmannian was computed and its spectrum is isomorphic to the universal centralizer. I will also compare the Poisson structure given by loop-rotation equivariance, and the standard one on the universal centralizer. In this talk, I will only focus on the computation for sl_2.



Title: The Borel-Moore Homology of Affine Grassmanian

Speaker: Wenyue Wang (Qiuzhen)

Abstract: In this talk, I will briefly review some basic notions of algebraic topology, such as the Borel Moore homology and the convolution product. Then I will use these tools to examine affine grassmanian of an algebraic  group. If time permits, I will show some properties of Borel Moore homology of affine grassmanian, such as commutativity and some explicit computations of certain algebraic groups, based on the work of Bezrukavnikov, Finkelberg and Mirkovic.  


Title: Quantum Hamiltonian Reduction

Speaker: Pengcheng Li (YMSC)

Abstract: In this talk, I will introduce the quantum Hamiltonian reduction and its applications to geometric representation theory.


Title: Basic tools in equivariant K-theory
Speaker: 梁石易新 (Qiuzhen)
Abstract: In this talk I will first introduce some basic notions in equivariant K-theory, including functoralities, convolution product, etc. Then I want to state some basic tools, including induction, reduction, Thom isomorphism, which are useful in computations of equivariant K-theory. At last, I want to compute some examples (e.g. some equivariant K-theory on flag varieties, may restrict to A_2 case) to give some intuition on equivariant K-groups.


Title: The Kazhdan-Lusztig Map for so(8)

Speaker: 张展维 (Qiuzhen)

Abstract: We introduce the notion of Kazhdan-Lusztig map for simple Lie algebras. It assigns for each nilpotent orbit a conjugacy class of the Weyl group. We also construct Lusztig's map, for which the Kazhdan-Lusztig map is its canonical section. We focus on the case so(8). We review the nilpotent orbits and compute the conjugacy classes of the Weyl group of so(8), then we compute explicitly the Kazhdan-Lusztig map for so(8) and give necessary explanations.


Title: The Affinization of T^*(G/U) and Coulomb Branch

Abstract: Let G be a complex semisimple algebraic group. The Affinization T^*(G/U)^aff is known to have symplectic singularities in the sense of Beauville. In this talk, I will sketch another explanation of this result, based on early works of Dancer, Kirwan, and Hanany, who interpreted this T^*(G/U)^aff as a Coulomb branch. Then by a recent result of Bellamy, all Coulomb branches have symplectic singularities. The example of G=SL(3) will be explained in full detail. If time permits, I will also briefly explain the relation between T^*(G/U)^aff and the Whittaker construction of the universal centralizer, and discuss another interesting relation with Coulomb branch (based on work of Bezrukavnikov, Finkelberg and Mirković.)


Tiltle: Organizational Meeting

Abstract: This seminar is designed for senior undergraduate students and first year graduate students to learn recent results related to Coulomb branch in the sense of Braverman-Finkelberg-Nakajima. The seminar is informal, learning-based, and discussion-friendly. Speakers may also discuss about their own research related to Coulomb branch. Undergraduate students are strongly encouraged to give talks (a talk based on concrete examples which explain basic definitions/properties of Coulomb branch is more than welcomed.)  No prerequisites are required, so please join us this Saturday to sign up for a talk :D.