2022-11-07

• Watched the youtube talk by Teleman at CMSA this year, on Kapustin-Rozanski-Saulina 2-category theory.
• Reading the paper by Peter on DAHA and A-brane
• Went to the seminar talk by Ivan

Brane and Quantization One interesting thing is about this coisotropic brane. This was introduced by Kapustin-Orlov in 2001, then Aldi-Zaslow worked out an example, then Gukov-Witten, Gaiotto-Witten did some work on it. Still, there is no definition. Are these just module over the deformation quantization algebra?

DAHA has two parameters $A_{q,t}$. We suppose to have $$ A_{q,t} = O^q(X_{t}) $$ where $X_t$ is the moduli space of $SL(2,\C)$ local system on a once-punctured torus, where the monodromy over the puncture is $diag(t,1/t)$.

How is this related to the Fukaya category? Is this space exact symplectic or has non-trivial Kahler moduli? Is it true that $$ Fuk(X_t, \omega_\hbar) = A_{q,t}-Mod = \mathcal{O}^q(X_t)-mod $$ It is far from clear. Is the Fukaya wrapped Fukaya or the usual Fukaya? What's the Kahler form that we are using?

It is a paper which has many potential truth, but needs careful interpretation.

I somehow got interested in MTC, modular tensor category. It is a very fancy version of braided tensor category, with a lot of nice properties, making it into a fusion category.

An MO question about MTC axioms: where does the non-degeneracy condition come from? The answer is really cool. It is about something being not trivial but invertible. Something inspired by Kevin Walker's work.

What is an example of all these stuff?

It was an enjoyable 2 hours talk. New stuff learned

• Stable envelope for the B-side is known by Andrei Smirnov using elliptic stable envelope, then passing to K-theory.
• One need to consider the localized K-theory, losing a lot of information. Can we do better? see below.
• Affine braid group action of moving the punctures around. (Can it be realized using kernels? Why would any one want to do that? Just to compare with B-side?)
• The quantum differential equation on the B-side space, is about cohomology $H^*_T(X)$. But somehow, one need to use K-theory $K_T(X)$.

How to go further?

• Can we do elliptic version? Andrei says try the analogy of using theta function / section, instead of $e^{-W}$, and try the contour /summation formula for producing qKZ equation. lecture on KZ by Cherednik
• Can we go beyond localized K-theory? One potential trouble is convergence, for example if we try to do character of $\C[t,t^{-1}]$, we possibly have infinite contributions. And can we specialize to non-generic monodromy? Would be interesting to work out an example and see the whole picture. I hope this is not as delicate as the Bernstein-Sato polynomial.
• 2-category, 2-representation, elliptic cohomology, Ganter-Kapranov.