• Reading Teleman
• chatting with Spencer

What's the input data? A compact Lie group (what's the difference between this and a complex reductive group?) and a polarisable quaternionic representation $E$ (ok, this is saying we can write $E =T^*V$, but we don't have a canonical choice of $V$, and we shouldn't fixiate on a choice.)

I don't know what's the difference between $g^{reg}/G$ and $g/G$ (adjoint action). OK, well, if we just do $g/G$, there might be too many orbits (with the same eigenvalues), and picking out $g^{reg}$ is saying, I am looking at the most generic (?) guy with this eigenvalues.

Why we call this integrable structure “Toda integrable system”. I thought it is like Hitchin integrable system, because we are taking eigenvalues of a matrix. Who is Toda, and what did he do? What's 'inverse scattering method'?

OK, let's not be distracted.

The 'reconstruction' result, you mean the gluing two copies method? , is from the 2d gauge theory interpretation. (what do you mean by that 2d gauge theory moduli space?)

The hook question: if you know Floer homology with Lagrangians is categorifying the intersection of middle dimensional homology cycles (quantum homology?)

wait, do we have a notion of product in the decategorified case? Wow, previously, you have access to each individual intersection points, so you can say who talks to who. if you only remember the intersection number, you don't have enough structure!

why do we care about chain complex, rather than just its cohomology? because it is interesting to have a local system family of chain complexes over $S^2$ (simply connected). However,if you pass to cohomology, you lose too much information, and you cannot detect the variation. Passing to fiberwise cohomology, and taking global section, gives the wrong answer.

Why does Teleman says, if $G$ is simply connected, (then $\pi_2(BG)=0$), then we don't have interesting 'local system of categories on BG'?

What does he mean, when he says representation admits a character theory? Do you mean restrict to an abelian subgroup, then the representation split as direct sum of representations of the subgroup? No, he doesn't mean that. Say $G$ is compact Lie group. Given a representation, $\rho: G \to End(V)$, and there is the trace map $End(V) \to k$, so composing them, we can get a function from the conjugacy classes of $G$ to $k$. That is the character function, living on $G/G$. (what does it look like? examples? intersting questions about it?)

So, are you suggesting, we look at coherent sheaves on $G/G$? well, what does that even mean?

Wait wait wait. So, you are saying, (categorical) representation of $G$ is related to the BFM space of the $G^v$ (tada, Langlands dual group is here). So, what do you mean by this $BFM(G^v)$? what's your convention?

This is related to $T^*(G^\vee / conj)$. Now, I am confused. why we change so much? Why we go to the Langldans dual?

Multiplicity (vector) space, of two G-representations; is replaced by the multiplicity categories.

What does $n$ points on a circle mean? Can you get $cNH_n$? There must be some flag variety in this game. How do you get the usual $NH_n$, where does the usual polynomial come in? So, I guess, just guessing, the dots are equivariant variables in the usual equivariant cohomology, like cohomology of BG, which is $G$-invariant function on $\mathfrak g$, the generators are in degree two.

But, precisely, why we use flag variety? Are we using $pt / G(K)$, and what do we push down? $pt/I$? Then, the fiber is huge, $G(K)/I$. This sounds right, since we have no matter. What does hom mean?

wait wait, why did BFN reduces to BFM in the pure gauge theory case?

How about this: conjugation action of $U(n)$ by $U(n)$, the result is $Sym^n U(1)$. We have $U(n) \to Sym^n U(1)$, the fiber is $U(n) / \prod U(n_i)$, which should be partial flag. I just don't know how to use this data.

Read Auroux, Perutz, Weirheim