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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.

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There are indeed ways to view D-branes as properties of the worldsheet (s)CFT instead of as geometric objects in target space/spacetime. See, for instance, "D-branes from conformal field theory" by Gaberdiel or the references in the nLab article on D-branes in rational (s)CFTs viewed through the formalism of functorial QFT.

The basic idea is that D-branes represent information about how to extend the world-sheet (s)CFT on closed Riemann surfaces to a theory on "Riemann surfaces with boundary", i.e. the worldsheets of open strings rather than closed strings. These boundaries correspond to so-called "Ishibashi states" in the (s)CFT obeying a "Cardy condition" (hence also called "Cardy states") named after Ishibashi's "The Boundary and Crosscap States in Conformal Field Theories". Since the space of states is an intrinsic property of the (s)CFT, this is a view of D-branes that does not consider the target space or the idea of D-branes floating around in it at all.

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Although $D$-branes (and other non perturbative stringy objects) can be identified as some objects or characteristics of the string worldsheet CFT; there is not a precise map between non-perturbative objects and worldsheet properties.

Sometimes $D$-branes have CFT imprints such as restricting the vertex operator spectrum (as $D9$-branes with $O$-planes do in the type I theory) or providing the physical interpretation of a particular CFT partition function (as the Ishibashi states in the @ACuriousMind answer or the $D5-D1$ CFT as the partition function of a BPS black hole) but it's not true that we can understand general non-perturbative objects as qualities of the worldsheet CFT; at least we haven't achieved a precise understanding of how this works systematically or in full generality.

A very interesting example were is not clear how $D$-branes can be understood from the worldsheet perspective concerns how to use the RNS formalism in RR-flux backgrounds. $D$-branes are the source of RR-fluxes and sometimes they dualize to RR-flux (as in the Gopakumar-Vafa transition) but its is actually far from obvious how to use the RNS formalism itself in this context, see Superstring Perturbation Theory and Ramond-Ramond Backgrounds . It would be that the intrinsic problems of the RNS formalism could be circumvented in the GS or pure spinor formalism, but my point is just that it is generically a highly non-trivial task to understand $D$-branes from the CFT worldsheet, not just because there are no perturbative states charged under RR $p$-form but because $D$-branes have several different effects on the CFT when are "comprehensible".

References:

Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism

Xi Yin - String spectrum in RR flux background from NSR closed string field theory

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