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

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 AdS(5) 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 pJ(+) + qJ(-) which shrinks near the source. In the generic case the world volume of the D6-branes is AdS(5) 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 obtained by smearing M5-branes that wrap the AdS(5) factor and the circle dual the superconformal R-symmetry. The D6-branes are extended along the AdS(5) 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 AdS(5) 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.