Spacetime singularity resolution by M-theory fivebranes: calibrated geometry, Anti-de Sitter solutions and special holonomy metrics

The supergravity description of various configurations of supersymmetric M-fivebranes wrapped on calibrated cycles of special holonomy manifolds is studied. The description is provided by solutions of eleven-dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with Anti-de Sitter geometry. For known examples of Anti-de Sitter solutions, the associated special holonomy metric is derived. One explicit Anti-de Sitter solution of M-theory is so treated for fivebranes wrapping each of the following cycles: K\"{a}hler cycles in Calabi-Yau two-, three- and four-folds; special lagrangian cycles in three- and four-folds; associative three- and co-associative four-cycles in $G_2$ manifolds; complex lagrangian four-cycles in $Sp(2)$ manifolds; and Cayley four-cycles in $Spin(7)$ manifolds. In each case, the associated special holonomy metric is singular, and is a hyperbolic analogue of a known metric. The analogous known metrics are respectively: Eguchi-Hanson, the resolved conifold and the four-fold resolved conifold; the deformed conifold, and the Stenzel four-fold metric; the Bryant-Salamon-Gibbons-Page-Pope $G_2$ metrics on an $\mathbb{R}^4$ bundle over $S^3$, and an $\mathbb{R}^3$ bundle over $S^4$ or $\mathbb{CP}^2$; the Calabi hyper-K\"{a}hler metric on $T^*\mathbb{CP}^2$; and the Bryant-Salamon-Gibbons-Page-Pope $Spin(7)$ metric on an $\mathbb{R}^4$ bundle over $S^4$. By the AdS/CFT correspondence, a conformal field theory is associated to each of the new singular special holonomy metrics, and defines the quantum gravitational physics of the resolution of their singularities.