Spacetime Singularity Resolution by M-Theory Fivebranes: Calibrated Geometry, Anti-de Sitter Solutions and Special Holonomy Metrics

Abstract

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: Kähler 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-Kähler 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.