D4-branes wrapped on a spindle

Abstract: We construct supersymmetric AdS$4\times\Sigma$ solutions of $D=6$ gauged supergravity, where $\Sigma$ is a two-dimensional orbifold known as a spindle. These uplift to solutions of massive type IIA supergravity using a general prescription, that we describe. We argue that these solutions correspond to the near-horizon limit of a system of $N_f$ D8-branes, together with $N$ D4-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. The dual field theories are $d=3$, ${\cal N }= 2$ SCFTs that arise from a twisted compactification of the $d=5$, ${\cal N}=1$ $USp(2N)$ gauge theory. We show that the holographic free energy associated to these solutions is reproduced by extremizing an off-shell free energy, that we conjecture to arise in the large $N$ limit of the localized partition function of the $d=5$ theories on $S^3\times\Sigma$. We formulate a universal proposal for a class of off-shell free energies, whose extremization reproduces all previous results for branes wrapped on spindles, as well as on genus $\mathrm{g}$ Riemann surfaces $\Sigma{\mathrm{g}}$. We further illustrate this proposal discussing D4-branes wrapped on $\Sigma\times\Sigma_{\mathrm{g}}$, for which we present a supersymmetric AdS$2\times\Sigma\times\Sigma{\mathrm{g}}$ solution of $D=6$ gauged supergravity along with the associated entropy function.