Anomalies, Black strings and the charged Cardy formula

We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds ${\mathcal{M}}_d$, including the contribution of the isometries of ${\mathcal{M}}_d$. We then use the result to perform a counting of microstates for electrically charged and rotating supersymmetric black strings in AdS${}_5\times S^5$ and AdS${}_7\times S^4$ with horizon topology BTZ$⋉S^2$ and BTZ$⋉S^2\times {\mathrm{\Sigma }}_{\mathfrak{g}}$, respectively, where ${\mathrm{\Sigma }}_{\mathfrak{g}}$ is a Riemann surface. We explicitly construct the latter class of solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula.