2023-09-04

It is useful to recap what I did today, or these days.

I have been thinking about Koszul duality a lot these days.

• where does the mysterious grading come from. What is mixed category? The terminology is weird that, mixed constructible sheaf does not form a mixed category. you need to take a (not full) subcategory.
• why the two spaces, Coulomb branch and Higgs branch, are related? I don't see the relation at all. You can say, categorically, one is about $Map(S^1, V/G)_{dR}$ the other is $Map(S^1_{dR}, V/G)$. But concretely, on a low brow way, what are we talking about?

You want to say, for the same theory, the Coulomb branch and Higgs branch, are just two shadows of the same object, the theory. Therefore, the category of line operators in the two branches are the same. Just some equivalence of categories (but where are the generators?) The BDGH paper. let me shut-up and read.

specify some N=(2,2) boundary condition in UV, and get some module over quantized Coulomb or Higgs branch algebra. Some interface (1/2 BPS).

we always says, consider some boudnary condition, that partially breaks supersymmetry.

when we say 2d sigma-model to some target space, we don't have to say the twist. it is just physics.

The UV to IR flow, the destination, depends on parameters. (so far, no twist)

Boundary condition is just boundary condition. Shall I say boundary condition forms a category? (is that true before we turn on twist?)

Anyway, we can have either $B_H \In M_H$ or $B_C \In M_C$, holomorphic Lagrangian in some hyperKahler space (wait, for which complex structure of $M_C, M_H$? for all? or does the boundary condition prefers one?)