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A<sc>bstract</sc> In the standard$$ \mathcal{N} $$ $N$= (4, 4) AdS₃/CFT₂with sym^N(T⁴), as well as the$$ \mathcal{N} $$ $N$= (2, 2) Datta-Eberhardt-Gaberdiel variant with sym^N(T⁴/ℤ₂), supersymmetric index techniques have not been applied so far to the CFT states with target-space momentum or winding. We clarify that the difficulty lies in a central extension of the SUSY algebra in the momentum and winding sectors, analogous to the central extension on the Coulomb branch of 4d$$ \mathcal{N} $$ $N$= 2 gauge theories. We define modified helicity-trace indices tailored to the momentum and winding sectors, and use them for microstate counting of the corresponding bulk black holes. In the$$ \mathcal{N} $$ $N$= (4, 4) case we reproduce the microstate matching of Larsen and Martinec. In the$$ \mathcal{N} $$ $N$= (2, 2) case we resolve a previous mismatch with the Bekenstein-Hawking formula encountered in the topologically trivial sector by going to certain winding sectors. 

A<sc>bstract</sc> We consider the 𝒩 = (2, 2) AdS₃/CFT₂dualities proposed by Eberhardt, where the bulk geometry is AdS₃× (S³×T⁴)/ℤ_k, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma modelT⁴/ℤ_k(withk= 2, 3, 4, 6). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard 𝒩 = (4, 4) case ofT⁴, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively$$ \frac{1}{2} $$ $\frac{1}{2}$,$$ \frac{2}{3} $$ $\frac{2}{3}$,$$ \frac{3}{4} $$ $\frac{3}{4}$,$$ \frac{5}{6} $$ $\frac{5}{6}$) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.