In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be viewed as submanifolds (of the target manifold in a sigma model), possibly augmented with a vector bundle, or a sheaf of [somethings], or maybe some other kind of label. (Please correct me if I got that wrong.)
In the hopes of tightening my grasp on the concept, here are some of the questions that have been nagging me during my reading.
- What specifically is the definition of a D-brane, say in the context of a topological field theory? (Or what are the most promising provisional definitions?) What references are most accessible to a mathematical audience?
- What picture should I have in my head when an author talks about "the moduli space of D-branes"?
- What is the idea behind the "dynamics of D-branes" that researchers sometimes talk about? (Perhaps when I understand better how to think about these gadgets, it will be easier to conceive of how they should change over time.)
- What goes into verifying (or at least asserting/conjecturing) that the elements of twisted K-theory classify D-brane charges?
(I considered putting this on MO, but it feels too basic relative to what TFT people talk about there for it to be appropriate.)