Supergravity and D-branes wrapping supersymmetric 3-cycles

We construct dual supergravity descriptions of D3-branes wrapping associative 3-cycles L. We analyse the conditions for having five-dimensional background solutions of the form AdS₂ × L and show that they require L to be of constant negative curvature type. This provides AdS₂ background solutions when L is the hyperbolic space H³ or its quotients by subgroups of its isometry group. We construct a regular numerical solution interpolating between AdS₅ in the UV and AdS₂ × H³ in the IR. The IR fixed point exists at the "intersection" of the Coulomb and Higgs branches. We analyse the singular supergravity solutions which correspond to moving into the Higgs and the Coulomb branches. For negative constant curvature spaces the singularity is of a "good" type in the Higgs branch and of a "bad" type in the Coulomb branch. For positive constant curvature spaces such as S³ the singularity is of a "bad" type in both the Higgs and the Coulomb branches. We discuss the meaning of these results.