I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some physics background, but not including string theory. I'm looking to understand "branes", which play a big part in the paper, and so far, in this context, here is my understanding (please feel free to correct any misconceptions):
Kapustin–Witten formulate a TQFT with gauge group $G$ on a spacetime of the form $\Sigma\times C$, where $C$ is going to play the role of the curve of the geometric Langlands correspondence. $\Sigma$ is a Riemann surface with boundary $\partial\Sigma$, and we are going to look at an effective field theory on $\Sigma$ after compactification on $C$. The resulting fields on $\Sigma$ are going to have to satisfy the Hitchin equations, whose solutions are parametrized by the Hitchin moduli space $\mathcal{M}_{H}(G,C)$. The assignment of the value of the field on $\Sigma$ is going to described by maps from $\Sigma$ into $\mathcal{M}_{H}(G,C)$. I believe this is what physicists refer to as a "sigma model".
As a side remark, $\mathcal{M}_{H}(G,C)$ has many fascinating properties, for example it is hyperkahler, and related to both $\operatorname{Bun}_{G}(C)$ and $\operatorname{Loc}_{G}(C)$, and its mirror pair concerns the Langlands dual ${}^{L}G$ in place of $G$. This makes it of interest in the Langlands program.
As far as I understand, a brane is a condition on where $\partial \Sigma$ gets sent to in $\mathcal{M}_{H}(G,C)$, i.e. a submanifold of $\mathcal{M}_{H}(G,C)$. There are actually two TQFT's in consideration here, an "A-model" and a "B-model", and for the A-model the submanifold $\partial \Sigma$ can get sent to is a Lagrangian (or more generally coisotropic) submanifold, while for the B-model it is a complex submanifold.
But for some reason it is more than just what I described — in fact, a brane in the B-model (or a B-brane) really consists of a coherent sheaf. Hence the appearance of the derived category of coherent sheaves in the formulation of homological mirror symmetry.
My first question is: Why are B-branes coherent sheaves? Apparently this is related to some sort of "boundary modification" (see 5.2 of Frenkel's survey Gauge Theory and Langlands Duality), but I don't know what these are, and I would be happy to see an explanation or elaboration.
Furthermore, branes (both A-branes and B-branes) form a category. There is supposed to be some sort of physical intuition for the morphisms between branes, but it is not clear to me what that is. For A-branes, the corresponding category is supposed to be some enlargement of the Fukaya category, whose morphisms are Floer chain groups.
This is my second question: What is the "physical intuition" behind morphisms of branes, why are they the Floer chain groups for A-branes, and why do branes form a category?
Morphisms of branes are actually pretty important in another way, as this is how Kapustin–Witten get from A-branes to D-modules on $\operatorname{Bun}_{G}(C)$ (in terms of which the geometric Langlands correspondence is stated). This is constructed by means of a "canonical coisotropic A-brane" (which is $\mathcal{M}_{H}(G,C)$ itself considered as an A-brane) whose endomorphisms are supposed to give a sheaf differential operators after some sort of localization or sheafification process. Kapustin's notes Lectures on Electric–Magnetic Duality and the Geometric Langlands Program say this is related to "insertion of vertex operators" (I don't really understand what this means). This also somehow looks reminiscent of Beilinson–Drinfeld's work producing D-modules (see 9.2 of Frenkel - Lectures on the Langlands Program and Conformal Field Theory) although as far as I know that makes no reference to branes.
Finally, my third question: Why do endomorphisms of the canonical coisotropic A-brane give us differential operators?
