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 $Ad{S}_2\times L$ and show that they require $L$ to be of constant negative curvature type. This provides $Ad{S}_2$ background solutions when $L$ is the hyperbolic space $H^3$ or its quotients by subgroups of its isometry group. We construct a regular numerical solution interpolating between $Ad{S}_5$ in the UV and $Ad{S}_2\times H^3$ 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 $S^3$ the singularity is of a bad’’ type in both the Higgs and the Coulomb branches. We discuss the meaning of these results.