have now written an Idea-section here with some basics. Am still collecting references.

I’d like to know to which extent the proposal has been checked concretely, that the D-brane charge classification by (equivariant, twisted) topological K-theory of the target spacetime is reproduced by equivalence classes of boundary states in the BCFT under renormalization group flow. I see some special cases have been checked, also some discrepancies seem to have been reported, but I am still looking for a non-preliminary statement.

In particular I’d like to see this for fractional D-branes at $G$-orbifold singularities. How exactly does boundary state formalism here compare to the expected answer that the fractional D-branes are classified by $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$?

I see plenty of authors behave like this is true, but did anyone write anything close to a comprehensive proof and classification?

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