Branes, Calibrations and Supergravity

These notes are based on lectures given at the Clay School on Geometry and String Theory, Isaac Newton Institute, Cambridge, 25 March - 19 April 2002. They attempt to provide an elementary and somewhat self contained discussion of the construction of supergravity solutions describing branes wrapping calibrated cycles, emphasising the geometrical aspects and focusing on D=11 supergravity. Following a discussion of the role of special holonomy backgrounds in D=11 supergravity, the basic membrane and fivebrane solutions are reviewed and the connection with the AdS/CFT correspondence is made. The world-volume description of branes is introduced and used to argue that branes wrapping calibrated cycles in special holonomy manifolds preserve supersymmetry. The corresponding supergravity solutions are constructed first in an auxiliary gauged supergravity theory which is obtained via Kaluza-Klein reduction.