Branes wrapped on orbifolds and their gravitational blocks

We construct new supersymmetric $\mathrm{AdS}_2\times \mathbb{M}_4$ solutions of $D=6$ gauged supergravity, where $\mathbb{M}_4$ are certain four-dimensional orbifolds. After uplifting to massive type IIA supergravity these correspond to the near-horizon limit of a system of $N$ D4-branes and $N_f$ D8-branes wrapped on $\mathbb{M}_4$. In one class of solutions $\mathbb{M}_4 = \Sigma_{\mathrm{g}}\ltimes\Sigma$ is a spindle fibred over a smooth Riemann surface of genus $\mathrm{g}>1$, while in another class $\mathbb{M}_4 = \Sigma\ltimes\Sigma$ is a spindle fibred over another spindle. Both classes can be thought of as orbifold generalizations of Hirzebruch surfaces and, in the second case, we describe the solutions in terms of toric geometry. We show that the entropy associated with these solutions is reproduced by extremizing an entropy function obtained by gluing gravitational blocks, using a general recipe for orbifolds that we propose. We also discuss how our prescription can be used to define an off-shell central charge whose extremization reproduces the gravitational central charge of analogous $\mathrm{AdS}_3\times \mathbb{M}_4$ solutions of $D=7$ gauged supergravity, arising from wrapping M5-branes on $\mathbb{M}_4$.