AdS5 solutions from M5-branes on Riemann surface and D6-branes sources

TL;DR

We address gravity duals of four-dimensional ${\mathcal{N}}=1$ SCFTs arising from M5-branes wrapped on punctured Riemann surfaces. The method reduces the problem to $AdS_5$ solutions in M-theory where the internal six-manifold has a ${U(1)^2}$ isometry, encoded by a single potential ${D_0}$ that satisfies a generalized Monge-Ampère equation; punctures appear as supersymmetric sources and include D6-brane and, in certain regimes, M9-brane sources. The results connect to known ${\mathcal{N}}=2$ GM/LLM setups in appropriate limits (e.g., ${p' or q'\to 0}$) and extend the B^3W family via an ${S}$-duality, while highlighting the role of puncture profiles in shaping flavor and R-symmetries. The framework opens avenues to classify ${\mathcal{N}}=1$ punctures, study their central charges, and explore generalizations to six-dimensional $(1,0)$ theories via richer Monge-Ampère boundary data and potential interpolating geometries.

Abstract

We describe the gravity duals of four-dimensional N=1 superconformal field theories obtained by wrapping M5-branes on a punctured Riemann surface. The internal geometry, normal to the AdS5 factor, generically preserves two U(1)s, with generators (J+,J-), that are fibered over the Riemann surface. The metric is governed by a single potential that satisfies a version of the Monge-Ampere equation. The spectrum of N=1 punctures is given by the set of supersymmetric sources of the potential that are localized on the Riemann surface and lead to regular metrics near a puncture. We use this system to study a class of punctures where the geometry near the sources corresponds to M-theory description of D6-branes. These carry a natural (p,q) label associated to the circle dual to the killing vector (p J+ + q J-) which shrinks near the source. In the generic case the world volume of the D6-branes is AdS5 X S^2 and they locally preserve N=2 supersymmetry. When p=-q, the shrinking circle is dual to a flavor U(1). The metric in this case is non-degenerate only when there are co-dimension one sources, M9-branes, obtained by smearing M5-branes that wrap the AdS5 factor and the circle dual the superconformal R-symmetry. In the IIA limit, they can interpreted as D8-branes. The D6-branes are extended along the AdS5 and on cups that end on the co-dimension one branes. In the special case when the shrinking circle is dual to the R-symmetry, the D6-branes are extended along the AdS5 and wrap an auxiliary Riemann surface with an arbitrary genus. When the Riemann surface is compact with constant curvature, the system is governed by a Monge-Ampere equation.