Gluing two discs into a spindle

We construct AdS${}_5\times \mathrm{\Sigma }$ solutions of $D=7$ gauged supergravity where $\mathrm{\Sigma }$ is a disc, also known as a half-spindle, and uplift them to $D=11$ supergravity. By computing the anomaly polynomial of the M5-brane theory compactified on $\mathrm{\Sigma }$ in the large $N$ limit, we obtain the off-shell central charges of the dual $D=4$, $\mathcal{N}=1$ superconformal field theory to disc solutions. These off-shell central charges decompose into contributions from the center and the boundary of the disc and are referred to as the gravitational blocks for disc. By suitably gluing the gravitational blocks for two discs along their boundaries, we reproduce the off-shell central charge for a single spindle.