Hey, I made some progress today. About Koszul duality, at least, I have a concrete conjecture. Let's state it as 'The symplectic Koszul duality for category O'. It goes as following: take a 3d N=4 gauge theory, some compact group $G$ acting on some representation $T^*N$. You can form two LG A-models, let's call that $(M_{H, \alpha}, W_{H,\beta})$ and $(M_{C, \beta}, W_{C, \alpha})$, where $\alpha$ and $\beta$ are some parameters. Then you prove that the two wrapped Fukaya categories are equivalent (after passing to $\Z/2$ grading? or invent a mixed version of the Fukaya category, where the hom space has a second grading)

What does mixed mean for toric variety constructible sheaf? This is quite useful, since Fukaya category for $T^*X$, $X$ toric is the same as constructible sheaf on $X$. First of all, we only consider unipotent monodromy, second of all, only an affine toric variety. Unipotent is somehow essential, because it create fiiltration on the nearby cycle. Jordan decomposition. Then, you can do fake Frobenius twist. Somehow, the fake Frob twist can be made isomorphic with the old one. Example, on $\C^2$, with basis $e_1, e_2$, we had unipotent monodromy $e_1 \mapsto e_1, e_2 \mapsto e_2+e_1$.There is nothing holy about $e_2$, if we change it to $e_2 + x e_1$, it still works. The Frob pullback monodromy is like, the old monodromy, but $p$ times. We need to find a matrix $A$, such that $$ A \begin{pmatrix} 1 & 1 \cr 0 & 1 \end{pmatrix}^p A^{-1} = \begin{pmatrix} 1 & 1 \cr 0 & 1 \end{pmatrix} $$ The trouble is that, $A$ is not unique, but only well-defined up to a stabilizers of the two matrices. But, once you fix $A$, you can diagonalize the matrix, and find eigenspaces. That eigenspace splitting is the weight decomposition.

So, the data of a Frobenius eigensheaf for the toric variety, is somehow and upgrade of the weight filtration to a weight $\Z$ 'grading'. like find a basis adapt to a flag.

What's the analog here? First of all, it is nothing about toric variety. We will have $T^*(Gr(k,n))$. If anything, it should be about a hyperKahle rotation. Namely, we have an $S^1$ worth of family for symplectic structure (together with Kahler structure).

Let's do an example. Consider $T^*\C$, with base coord $z = x + i y$, fiber coord $w = u + i v$, $x,y,u,v$ are real coordiante. Then, we have $$\lambda_\C = w dz = (u d x - v dy) + i (v dx + u dy)$$ If we take $$ \lambda_\theta = Re(e^{i\theta} \lambda_\C) = \cos(\theta) (u d x - v dy) - \sin(\theta) (v dx + u dy) $$

So, what's going on? It is like, we apply $e^{i\theta}$ rotation of the fiber (only the fiber, not the base, not $(1,1)$ weight rotation, not $(1,-1)$ weight Ham rotation, just one factor). I don't think it is going to be invariant.

The monodromy of $S^1$-family of symplectic structure on a fixed space. What is 'winding'?

In the space of symplectic structure, we have a loop.