When migrating the server, I forgot to backup the yiye website, lost a lot of memories, sad. I also dumped a lot of my phd notes to dumpster anyway when I move away from Chicago, it is inevitable to let go.

If I consider coherent sheaf category as the category of B-branes, then how does it depends on the complexified Kahler parameters? One can either do it in the physical way, reading Witten, Hori etc; or one can do it in the math way, thinking about deformation of the Coh category by gerbes.

It turns out Andrei Caldararu's thesis deals exactly with B-field and twisted sheaves. Let's find out the definition.

Roughly speaking, twisted sheaf is a bunch of locally defined sheaves (hmm, aren't they already locally defined?), whose gluing function fails the tricycle condition.

Take an element $\alpha \in H^2(X, \mathcal O^*_X)$. What does this mean? I probably need to resolve this sheaf $\mathcal O_X^*$, here is a short exact sequence $$ 0 \to 2\pi i \Z \to \mathcal O \to \mathcal O^* \to 0 $$ So, if I do $X=\P^1$, then I don't think I have $H^3(\P^1, \Z)$ or $H^2(\P^1, O)$, so I do not have any non-trivial element in $H^2(X, \mathcal O^*_X)$.

It still might make sense, maybe the trivialness of $\alpha$ corresponds to a locally constant deformation. Just like a flat connection gives zero curvature, but still flat connection is useful.

So, what is the connection? We can try to do resolution, but no.

So, what is connection? Locally it is one form, but it is an affine space over the one-form space, so that $d$ makes sense.