D-branes are best known as hypersurfaces in spacetime on which open strings with Dirichlet boundary conditions can end. Furthermore, they have several geometric properties associated with them, such as stacking, being able to wrap around cycles, etc.
As explained in this answer however, a dual picture to the "strings in spacetime" viewpoint of string theory is its description in terms of 2D (super)conformal field theories living on Riemannian surfaces of various genera. The description of D-branes in the previous paragraph falls squarely in the first of these, which brings me to my question.
Does there exist a description of D-brane as an aspect of the worldsheet conformal field theory? If so, how do features like the worldvolume gauge fields and coincident D-branes manifest in this formalism? Naïvely I would expect such a description to exist, since although the D-branes do not intersect the worldsheet, they confer boundary conditions upon its "edges". I would additionally be interested in hearing whether there are analogous structures in other CFTs (not just critical ones) and what their interpretation is, if any.