Localizing punctures in M-theory

We use equivariant localization and holography to study four-dimensional $\mathcal{N}=1$ superconformal field theories arising from M5-branes wrapped on a punctured Riemann surface. We explain how, given a Riemann surface with marked points, one can glue in a ``puncture geometry'' locally around each point. Using equivariant localization we show that the central charge consists of a bulk contribution plus localized puncture contributions. We recover and generalize the known results for locally $\mathcal{N}=2$ preserving punctures, and derive new results for genuinely locally $\mathcal{N}=1$ preserving punctures.