Supergravity and D-branes Wrapping Special Lagrangian Cycles

We construct dual supergravity descriptions of D3-branes wrapping special Lagrangian cycles L in Calabi-Yau 3-folds. We analyse the conditions for having five-dimensional background solutions of the form AdS_2 \times L and show that they require L to be of constant negative curvature type. This provides AdS_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 AdS_5 in the UV and AdS_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.