Localizing punctures in M-theory

A bstract We use equivariant localization and holography to study four-dimensional $$ \mathcal{N}=1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> 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 $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> preserving punctures, and derive new results for genuinely locally $$ \mathcal{N}=1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> preserving punctures.