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In [1] and [2] the authors claim that the boundary states (not just the Ishibashi states) of a Kazama-Suzuki model are labelled in the same way as the primary states of the model, so that the boundary states can be seen as a subset of the set of primary states (which are related to the irreducible HW representations of $Vir$). However, they do not say where this result comes from.

Request: I would like to know of an article or textbook which discusses/proves this result. Alternatively, a clear explanation would be fine, although I would prefer if it were supported by references.

[1]: https://arxiv.org/abs/hep-th/0112221 Nozaki, M. (2002). Comments on d-branes in kazama-suzuki models and landau-ginzburg theories. Journal of High Energy Physics, 2002(03):027.

[2]: https://arxiv.org/abs/1005.2117 Behr, N. and Fredenhagen, S. (2010). D- branes and matrix factorisations in supersymmetric coset models. Journal of High Energy Physics, 2010(11):136.

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I do not know of a reference that explains this, but perhaps it is useful to make some general comments. And I have a string theory context in mind, so that when the given 2D CFT of interest is combined with the ghost and remaining matter CFT the full central charge vanishes.

A boundary state is essentially a resolution of unity subject to specified boundary conditions. In the string theory context this resolution of unity is spanned by offshell states, all of the offshell states of the given model subject to the given boundary conditions. So this includes all primaries and descendants that satisfy the given boundary conditions. Generically there are also primaries and descendants that do not satisfy the boundary conditions (which, e.g., you can expose by putting the CFT on a Riemann surface with handles and cutting open a handle), so boundary states are clearly a superposition of a subset of the full set of states.

Since the full set of states are composed of primaries and descendants, one might think that the question boils down to showing that primaries and their descendants are labelled by the same quantum numbers. This will presumably not be the case unless there is a well-defined BRST cohomology and we restrict attention to BRST cohomology classes. Because when we do restrict to a BRST cohomology class (which is in turn spanned by one primary and all of its descendants), all of the elements of that class are related by adding or subtracting BRST-exact states, so there really is a unique (once a basis has been chosen) set of labels for each cohomology class and we see that any given primary and all its descendants are described by a unique set of labels. I don't think this remains true when we go offshell (or when we depart from the BRST cohomology). Boundary states however are offshell superpositions of states, so it's not clear to me that they can have the same labelling as primaries, simply because when we go offshell we can distinguish between elements in a given class. But in a sense they should at least have the same labelling as all primaries and their descendants subjected to the boundary conditions of interest once these labels have been integrated out or summed over.

(To elaborate on that last comment, since boundary states are essentially a resolution of unity, most of the labels mentioned above are actually integrated out, and this is why I was saying that at best they correspond to superpositions of primaries and descendants as opposed to a single primary or descendant.)

At least this is my understanding. Perhaps somebody can sharpen/correct the above thoughts or provide appropriate references.

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