The seminar will be held online on Tuesdays from 10 am to 12 pm. The goal of the seminar is to understand the current progress in the study of fixed points of Coulomb branches of 3d N=4 gauge theories under certain involutions, and the associated representation theory.
The main geometric objects of our interest are:
● Affine Grassmannian (i)slices
● Slodowy slices in Lie algebras of types B, C, and D
● Coulomb branches of quiver gauge theories with symmetrizers
During this seminar we want to understand how these objects play a role in the representation theory of interesting algebras, such as shifted twisted Yangians, and twisted current algebras. The seminar organizers have a strong interest in Hikita conjecture and its various generalizations. One of the goals of the seminar is to understand whether it's reasonable to expect that some of the known results in the field (such as connection between the schematic fixed points of Coulomb branch of ADE quiver gauge theory with the weight spaces of (dual) Weyl modules of current algebras) can be transferred to the setting of fixed points under an involution.
The format suggests a number of papers that participants will read during the seminar. Each week will have a dedicated speaker presenting what they learned from one of the papers listed below and will mostly be focused on the discussion of ideas and details, as participants are expected to be familiar with the discussed parts. If you volunteer to give a talk, or want to suggest a paper on the topic you want to read (or both!), please contact Mitya at dmatvieievsk[at]umass.edu or Vasya at krylovasya[at]gmail.com.
Whenever it is possible, each talk must include an example for a small rank Lie algebra.
And the seminar is over, thank you to everyone participating!
Please find below the list of talks. You can request recordings by contacting Mitya by email.
1) Introduction to the seminar, and motivation from the Hikita conjecture. Vasya's notes. Mitya's notes.
2) Weyl modules and affine Demazure modules. Ilya's notes.
3) Affine Grassmanian and (generalized) affine Grassmanian slices. Trung's notes.
4) Proof of Hikita conjecture for ADE quiver gauge theories. First part of Pasha's notes. Second part of Pasha's notes.
5) Coulomb branches with symmetrizers and slices in affine Grassmanians of types BCFG. Daniil and Kolya volunteered to give a talk. First part and second part of Daniil's notes. Kolya's notes.
6) Isomorphism of quantized affine Grassmanian slices and truncated shifted Yangians, GKLO homomorphism. Alexandr's notes.
7) Satake diagrams and shifted twisted Yangians. Artem's notes.
8) Affine Grassmanian i-slices and truncated shifted twisted Yangians, iGKLO homomorphism. Rui's notes.
9) Slodowy slices and (quantized) Mirkovic-Vybornov isomorphism. First part of Kien's notes.
10) Affine Grassmanian i-slices and Slodowy slices in types BC. Lukas' notes.
A more detailed outline of the topics with references can be found in the pdf file.
Papers we plan on discussing (depending on time, audience interest, and potential continuation of the seminar):
● J, Kamnitzer, P. Tingley, B. Webster, A. Weekes, O. Yacobi. Highest weights for truncated shifted Yangians and product monomial crystals.
● H. Nakajima, A. Weekes. Coulomb branches of quiver gauge theories with symmetrizers.
● Z. Dong, H. Ma. Flag varieties of type B/C and twisted Yangian.
● K. Lu, W. Wang, A. Weekes. Shifted twisted Yangians and affine Grassmannian islices.
● K. Lu, Y. Peng, L. Tappeiner, L. Topley, W. Wang. Shifted twisted Yangians and finite W-algebras of classical type.
● Y. Shen, C. Su, R. Xiong. Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches.
● H. Nakajima. Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs.
● M. Varagnolo, E. Vasserot. Non symmetric quantum loop groups and K-theory.