D-Brane Actions and N =2 Supergravity Solutions

Abstract

Among the most remarkable recent developments in string theory are the AdS/CFT duality, as proposed by Maldacena, and the emergence of noncommutative geometry. It has been known for some time that for a system of almost coincident D-branes the transverse displacements that represent the collective coordinates of the system become matrix-valued transforming in the adjoint representation of U(N). From a geometrical point of view this is rather surprising but, as we will see in Chapter 2, it is closely related to the noncommutative descriptions of D-branes. A consequence of the collective coordinates becoming matrix-valued is the appearance of a dielectric effect in which D-branes can become polarized into higher-dimensional fuzzy D-branes. This last aspect has inspired Polchinski and Strassler to find a nonsingular string dual of a confining four-dimensional gauge theory. The nonsingular geometry is sourced by an extended brane arising from Myers' dielectric effect. Following the spirit of the Polchinski-Strassler paper, we find N = 2 supergravity solutions with polarized branes and a field-theory dual. In our case we are able to present exact supergravity solutions by using M-theory reductions to type IIA supergravity.