In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully 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 conceived of 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/data. (Corrections welcome.)


In the hopes of tightening my grasp on the concept, here are some of the questions that have been nagging me during my reading.


1. 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?
2. What picture should I have in my head when an author talks about "the moduli space of D-branes"?
3. 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.)
4. What goes into verifying (or at least asserting/conjecturing) that the elements of twisted K-theory classify "D-brane charges"?


(Question reposted from [here](https://math.stackexchange.com/questions/7435/what-are-d-branes-in-a-topological-field-theory).)



What are D-branes, really?
There is an *abstract algebraic* formulation of QFT: this says that an $n$-dimensional QFT is a consistent assignment of spaces of states and of maps between them to $n$-dimensional *cobordisms*.


If one allows cobordisms with boundary here, one speaks of *open-closed QFT*. A **D-brane** in this context is the type of data assigned by the QFT to these boundaries.


There is also a geometric aspect to this: many abstractly defined QFTs are imagined to be "sigma-models". They are supposed to be induced by a process called "quantization" from a functional (the "action") on a space of maps $\Sigma \to X$ from a cobordism $\Sigma$ into a smooth manifold $X$ equipped with extra geometric data (such as metric, connections, etc.)


Under this correspondence, one may ask which abstract algebraic properties of the QFT derive from which geometric aspects of these "background structures". One finds that the data that the QFT assigns to boundaries comes from geometric data on $X$ that *tends* to look like submanifolds with their own geometric data on them (but may be considerably more general than that!). If so, this geometric data on $X$ is called a **D-brane** of the sigma-model.


There are many instances of this that are understood at the rough level at which quantum field theory was understood in the 20th century. One special case that is by now under fairly complete mathematical control and which serves as a good guide to the general concept of D-branes is what is called "2d *rational* CFT" .


There is a complete mathematical classification of 2d rational CFTs in their abstract algebraic form: they are given by special symmetric Frobenius algebra objects *internal* to a modular tensor category of representation of a vertex operator algebra.


Under this classification theorem, the boundary data = D-branes in the algebraic formulation can be proven to be precisely *modules* over this Frobenius algebra object.


In special nice cases, one understands where these come from geometrically. The notable example is the Wess-Zumino-Witten model, where the target space is a group manifold. Here one finds that in the simplest case the geometric data corresponding to these D-branes are submanifolds given by conjugacy classes, and carrying twisted vector bundles. More generally, though, the D-branes are given by cocycles in the twisted differential K-theory of the group. So the identification "D-brane = submanifold" is too naive, in general. The correct identification is:


geometric D-brane = geometric data on the sigma-model target space that induces boundary data of the corresponding algebraically defined worldvolume QFT.


For more see 


<http://ncatlab.org/nlab/show/brane>


