Peng Zhou

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====== 2024-06-10 ======

===== koszul duality =====

==== physics side ====
the story is about 3d mirror symmetry. 

from a 3d N=4 SYM theory, you have Coulomb branch and Higgs branch as low energy effective field theory. 

Consider boundary conditions. If we did A-twist in the bulk, we have A-twist in the boundary. 2d A-model with a target space that has a G-action. 

Do we consider a sandwich, where on the left boundary, we have Dirichlet BC, and right boundary we have Neumann condition? (What does these mean?)

Or do we consider an interface: a theory and a dual theory share a common wall? 

ok, i give up. I don't know how to tell a blackbox cartoon story on the physics side. 

==== rep side ====
We had BGG category O. Say fix a Lie algebra, and fix a central character (a block), we can talk about all the highest weight rep whose highest weight belongs to this block (why we need to consider these things together? instead of just consider one?) oh, I guess different block don't talk to each other, so it is naturally split off like this, indeed, no loss of generality.

then, koszul dualit for principal block (or call it regular block) says, endomorphisms of projectives can acquire a secret grading. 

let's just be more explicit. I remember Ben explained this to me. It has something to do with projective resolution. right, if we consider projective resolution of a simple, then we consider endomoprhism of this simple, there is something about 'diagonal grading', which I don't quite remember. 

oh, are you saying, $Ext^1$ generate all higher $Ext$? Like, for endomorphism algebra,  ext-1 are the generators, and all the relations are in ext-2? but, so what, why is this a big deal? yes, all things are in quadratic relation. so what? 

Soergel computed something, endomorphism of the tilting projective? There is one largest projective object, whose endomorphism is the 'co-invariant' algebra. Somehow, on the Koszul dual side, it is the endomorphism of the simple, which is the cohomology of the flag variety. How do you put a grading on the End(Proj)? 

other projective can also match with simples. 


Somehow, BGS explains why this is true. 


==== Ben Webster's story (following BLPW, KWWY) ====

BGG category O is a special case of Cat O of quantized additive (resolved? deformed?) Coulomb branch. 

(really? Cat O as reg. hol. D-mod on flag variety G/B, so it is the quantization of $T^*(G/B)$. I GUESS, all $T^*(G/B)$ are Coulomb branches. )

Are all Coulomb branches have mirror Higgs branches? In other words, is 3d mirror symmetry 'always holds', or like 2d mirror symmetry, sometimes holds? What is the statement? 

Webster's story is that, quantized Coulomb branch's module, pick a full subcategory of things supported somewhere, then you get DQ module on Higgs branch, supported elsewhere. So, you get two algebra (from quantized Coulomb branch and Higgs branches) are Koszul dual. 

Webster's proof is like this: he cook up some hom space, that is simultaneously interpreted as endo of simple on Higgs side; or endo of projectives on the quantized Coulomb side. The endo of simple is roughly easy to see, we have a pushforward for $V'/G'$ to $V/G$ (not sure), is $G'$ Levi or parabolic, and what is $V'$? 

Quantized Coulomb branch.