## Berkeley Informal String-Math Seminar

Organized by Mina Aganagic, Ivan Danilenko, Andrei Okounkov, and Peng Zhou

Weekly on Mondays 2:10 PM (Pacific Time)

Meetings are in-person, at 402 Physics South

We have a lunch on the 4th floor of Physics South before the seminar

If you want to be added to the seminar mailing list, use this link https://forms.gle/Qk7Vpz4Uxv7rSyWaA

For those joining us remotely, we have a Zoom link: https://berkeley.zoom.us/j/99373923587?pwd=SUhXamdtbHhJMUJERlJ4NVJHL1Jtdz09

#### Schedule of talks for Spring 2023:

 Jan 17, 12:30PM* Roman Bezrukavnikov Microlocal sheaves on homogeneous affine Springer fibers and quantum groups Video Jan 23 No Seminar Jan 30 Wenjun Niu Quantum-supergroup-extensions of $$U_q^H(\mathfrak{sl}(2))$$ at the fourth root of unity Feb 6 Constantin Teleman TBA Feb 13 Feb 20 No Seminar Feb 27 No Seminar Mar 6 Pavel Putrov TBA Mar 13 Lev Rozansky TBA Mar 20 Jonathan Heckman TBA Mar 27 No Seminar Apr 3 Apr 10 Apr 17 Andrei Negut TBA Apr 24 Sergey Cherkis TBA May 1

*Special Day/Time/Location.

#### A note to the speakers:

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

### Abstracts:

#### Jan. 17: Roman Bezrukavnikov (MIT)

##### Microlocal sheaves on homogeneous affine Springer fibers and quantum groups

I will report on a joint work in progress with Pablo Boixeda Alvarez, Michael McBreen and Zhiwei Yun where categories of microlocal sheaves on some affine Springer fibers are described in terms of the Langlands dual group. In particular, in the slope 1 case we recover the regular block in the category of (graded) modules over the small quantum groups. Assuming a general formalism connecting microlocal sheaves to Fukaya categories, this yields new examples of homological mirror symmetry, including the mirror dual to $$T^*(G/B)$$.

#### Feb. 6: Wenjun Niu (UC Davis)

##### Quantum-supergroup-extensions of $$U_q^H(\mathfrak{sl}(2))$$ at the fourth root of unity

In work to appear with Ballin-Creutzig-Dimofte, we constructed vertex operator algebras associated to A and B twists of 3d N=4 abelian gauge theories. These are boundary VOAs supported on holomorphic boundary conditions of Costello-Gaiotto. For the B twist, the vertex algebra $$V_B$$ is a simple current extension of an affine Lie superalgebra, and using the work of Creutzig-Kanade-McRae, we can study its representation theory using this simple current extension. An analogous extension procedure for quantum groups was developed by Creutzig-Rupert. I will explain how to apply their strategy to $$U_q^H(\mathfrak{sl}(2))$$, the unrolled restricted quantum group at 4-th root of unity, and obtain a quantum supergroup whose category of representations is equivalent to that of $$V_B$$. This is joint work in progress with T. Creutzig and T. Dimofte.