Quarter-BPS AdS5 solutions in M-theory with a T2 bundle over a Riemann surface

TL;DR

The paper develops a unifying framework for quarter-BPS AdS$_5$ solutions in M-theory with internal geometry given by a $T^2$ bundle over a Riemann surface and two interval directions. It reduces the problem to a canonical system governed by two key functions, the conformal factor of the Riemann surface and the $T^2$ fibration, and organizes solutions into Class I (warpings leading to a warped $SU( obreak\infty)$ Toda equation) and Class II (Liouville-type on the Riemann surface). The analysis recovers and contextualizes known geometries—the GMSW family, LLM, Maldacena–Nuñez, the uplift of $Y_{p,q}$, and the Beem–Bobev–Wecht–Bah B$^3$W family—within a single coherent scheme, while outlining explicit metric forms and reduction strategies. This framework provides a structured route to new AdS$_5$ backgrounds and their holographic duals, with potential extensions to punctured Riemann surfaces and SCFT classifications. The work thus links geometric classifications to dual 4D $\ ext{N}=1$ SCFTs arising from M5-branes wrapped on Riemann surfaces, advancing the understanding of M-theory AdS$_5$/CFT$_4$ holography in this regime.

Abstract

We study and classify quarter-BPS AdS5 systems in M-theory, whose internal six-dimensional geometry is a T2 bundle over a Riemann surface and two interval directions. The general system presented, provides a unified description of all known AdS5 solutions in M-theory. These systems are governed by two functions, one that corresponds to the conformal factor of the Riemann surface and another that describes the T2 fibration. We find solutions that can be organized into two classes. In the first one, solutions are specified by the conformal factor of the Riemann surface which satisfies a warped generalization of the SU(infinity) Toda equation. The system in the second class requires the Riemann surface to be S2, H2 or T2. Class one contains the M-theory AdS5 solutions of Lin, Lunin and Maldacena; the solutions of Maldacena and Nunez; the solutions of Gauntlett, Martelli, Sparks and Waldram; and the eleven-dimensional uplift of the Y(p,q) metrics. The second includes the recently found solutions of Beem, Bobev, Wecht and the author. Within each class there are new solutions that will be studied in a companion paper.