## 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 Fall 2022:

 Aug 29 Peng Zhou Homological Mirror Symmetry for 3d Coulomb branches and Skein-Strand diagram. Video Sept 5 No Seminar Sept 12 Andrei Okounkov The Eisenstein spectrum and some other things Video Sept 19 Henry Liu Multiplicative vertex algebras and wall-crossing in equivariant K-theory Video Sept 21, 1PM, Physics South 325* Henry Liu Multiplicative vertex algebras and wall-crossing in equivariant K-theory, II Video Sept 23, 1PM, Physics South 325* Henry Liu Multiplicative vertex algebras and wall-crossing in equivariant K-theory, III Video Sept 26 Robert Lipshitz Floer homology for 3-manifolds with boundary Video Oct 3 No Seminar Oct 10 Ivan Smith TBA Oct 17 TBA Oct 24 Sergei Gukov TBA Oct 31 Peter Koroteev Branes and DAHA Representations Nov 7 Ben Webster TBA Nov 14 TBA Nov 21 TBA Nov 28 Miroslav Rapcak TBA Dec 5 TBA

*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:

#### Aug. 29: Peng Zhou (UC Berkeley)

##### Homological Mirror Symmetry for 3d Coulomb branches and Skein-Strand diagram

This talk is a report of work in progress with Mina Aganagic, Yixuan Li and Ivan Danilenko. The B-side is the (resolved) additive Coulomb branch $$X$$, and A-side is the (deformed) multiplicative Coulomb branch $$Y$$ with a superpotential $$W$$, and HMS says $$\mathrm{Coh}(X) = \mathrm{Fuk}(Y,W)$$ where the RHS is the wrapped Fukaya category. Webster has shown $$\mathrm{Coh}(X)$$ is equivalent to the category of $$A_{G,N}$$-mod, where $$A_{G,N}$$ is the cKLRW algebra. We will show that $$\mathrm{Fuk}(Y,W)$$ is also equivalent to $$A_{G,N}$$-mod. As a first step, we consider the gauge group $$G = \mathrm{GL}(k)$$ and representation $$N =(\mathbb{C}^k)^n$$, where we use Honda-Tian-Yuan's technique to turn endomorphism algebra of Lagrangian to certain subquotient of skeins on $$S^1 \times S^1 \times [0,1]$$.

#### Sept. 19: Andrei Okounkov (UC Berkeley)

##### The Eisenstein spectrum and some other things

The spectral decomposition of the Hilbert space of automorphic functions is a very old and central topic in number theory, and mathematics in general. In particular, the Eisenstein series produce automorphic functions on a group G from automorphic functions on its Levi subgroups and one is interested in spectrally decomposing them. I will review some classical as well as recent results in this area, with an emphasis on some potential points of contact with the more traditional topics discussed in this seminar.

#### Sept. 19: Henry Liu (University of Oxford)

##### Multiplicative vertex algebras and wall-crossing in equivariant K-theory

I will explain how a recent “universal wall-crossing” framework of Joyce works in equivariant K-theory, which I view as a multiplicative refinement of equivariant cohomology. Enumerative invariants, possibly of strictly semistable objects living on the walls, are controlled by a certain (multiplicative version of) vertex algebra structure on the K-homology groups of the ambient stack. In very special settings like refined Vafa-Witten theory, one can obtain some explicit formulas. For moduli stacks of quiver representations, this geometric vertex algebra should be dual in some sense to the quantum loop algebras that act on the K-theory of stable loci.

#### Sept. 26: Robert Lipshitz (University of Oregon)

##### Floer homology for 3-manifolds with boundary

Ozsváth-Szabó's Heegaard Floer homology is a holomorphic curve analogue of the Seiberg-Witten Floer homology of closed 3-manifolds. Bordered Heegaard Floer homology is an extension of (one version of) Heegaard Floer homology to 3-manifolds with boundary, developed jointly with Ozsváth and Thurston. This talk is an overview of bordered Heegaard Floer homology. We will start by describing the structure and aspects of the construction of Heegaard Floer homology and bordered Floer homology, and then talk about some reinterpretations and refinements of bordered Floer homology. Some of this is work in progress with Ozsváth and Thurston.

#### Oct. 31: Peter Koroteev

##### Branes and DAHA Representations

Using brane quantization, we study the representation theory of the spherical double affine Hecke algebra of type $$A_1$$ in terms of the topological A-model on the moduli space of flat $$\mathop{\mathrm{SL}}(2,\mathbb{C})$$-connections on a once-punctured torus. In particular, we provide an explicit match between finite-dimensional representations and A-branes with compact support; one consequence is the discovery of new finite-dimensional indecomposable representations. We proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, we identify modular tensor categories behind particular finite-dimensional representations with $$\mathop{\mathrm{PSL}}(2,\mathbb{Z})$$ action. Using a further connection to the fivebrane system for the class S construction, we go on to study the relationship of Coulomb branch geometry and algebras of line operators in 4d $$\mathcal{N}=2^*$$ theories to the double affine Hecke algebra.