Abstract
We describe the gravity duals of four-dimensional \( \mathcal{N}=1 \) 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-Ampère equation. The spectrum of \( \mathcal{N}=1 \) 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 × S 2 and they locally preserve \( \mathcal{N}=2 \) 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-Ampère equation.
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Bah, I. AdS5 solutions from M5-branes on Riemann surface and D6-branes sources. J. High Energ. Phys. 2015, 163 (2015). https://doi.org/10.1007/JHEP09(2015)163
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DOI: https://doi.org/10.1007/JHEP09(2015)163