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--- blog:2022-11-07 [2022/11/08 05:51] – created pzhou
+++ blog:2022-11-07 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 4: / Line 4: @@
   * Went to the [[https://math.berkeley.edu/~danilenko/StringMath#Danilenko | seminar talk ]] by Ivan
-===== DAHA =====
+===== Peter Koroteev =====
-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? **
+** 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
@@ Line 11: / Line 11: @@
 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?
+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.
+===== Constantin Teleman =====
+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 [[https://mathoverflow.net/questions/232819/modular-tensor-categories-reasoning-behind-the-axioms | 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?
+===== Ivan Danilenko then Andrei Okounkov =====
+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. [[https://projecteuclid.org/ebooks/mathematical-society-of-japan-memoirs/Lectures-on-Knizhnik-Zamolodchikov-equations-and-Hecke-algebras/chapter/Lectures-on-Knizhnik-Zamolodchikov-equations-and-Hecke-algebras/10.2969/msjmemoirs/00101C010 | 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, [[https://arxiv.org/abs/math/0602510 | Ganter-Kapranov]].