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.
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