Supergravity description of field theories on curved manifolds and a no go theorem

TL;DR

Maldacena and Nuñez develop a framework for holographically describing field theories on curved manifolds via twisted brane constructions, yielding AdS-based gravity duals for flows across dimensions such as from $d+2$ to $d$ dimensions. They compute explicit supergravity solutions for D3 and M5 branes wrapped on Riemann surfaces, revealing IR fixed points as $AdS_{d+1}$ geometries and, in certain twists, $AdS_3$ regions with calculable central charges that scale as $N^2$ or $N^3$ depending on the brane setup. A central contribution is a general criterion for allowable IR singularities and a robust no-go theorem showing the nonexistence of non-singular Randall–Sundrum or de Sitter compactifications for a broad class of theories with $V\le 0$, extended to massive IIA. The work provides concrete holographic duals for new 4d ${\cal N}=2,1$ SCFTs arising from wrapped M5-branes and clarifies when wrapped-brane geometries yield physically acceptable IR physics, outlining both UV completions and IR dynamics. The results have significant implications for AdS/CFT constructions with curved manifolds and for understanding constraints on warped compactifications in string/M-theory.

Abstract

In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form $R^d \times Σ$ where $Σ$ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside $K3$ or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to $AdS_5$. We also propose a criterion for permissible singularities in supergravity solutions. In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories.