tion of this effect. The surface c µ is indeed special; it represents the surface of another object, hitherto undiscovered in string theory. These objects are called D-branes , the ‘D’ standing for ‘Dirichlet’, and the ‘brane’ as in membrane, or surface. By the T-duality, all open strings must start and finish on a D-brane. Just as closed strings can be considered fluctuations of the background geometry, open strings can be considered as fluctuations of the D-branes. The D-branes are solitonic objects, and thus exist in the low energy limit of string theory - supergravity. It was shown [47] in 1989 that the action for some arbitrary D-brane gives the equations of motion for all the back- ground fields associated with any D-brane. This form of an action was first written down by Born and Infeld when considering a particular non- linear generalization of electromagnetism [48], and was expanded upon by Dirac [49]. All this occurred a long time before the advent of string theory and D-branes, but for historical reasons (1.52) is still called the Dirac-Born- Infeld (DBI) action. Let us briefly explain each field in turn: • Φ is the usual dilaton. • G ′ is the induced metric on the brane’s surface. The precise definition of this concept is given in section 3.2.1, but it is the equivalent of the √ − det g term in the standard Einstein-Hilbert action for general relativity. 32 1.4.4 D-brane action S ∼ ∫ M e Φ / 2 √ det( G ′ + B ′ + F ) (1.52)