58 Chapter 2: Motivation sometimes Dp branes (for a review of D branes see [323]). It is thought that p branes and Dp branes are different descriptions of the same object, with the p brane description of the backreaction on the geometry valid for small curvature in Planck units ( r$_{−}$/l$_{p}$ ≡ R/l$_{p}$ ≫ 1). The endpoints of open strings can be charged with a non-dynamical degree of freedom. Each end is indexed by i or j running from 1 to N , where N is the number of D branes (only Type I string theory has open strings not ending on D branes, but we are not interested in this string theory); see Fig. 2.13 (a). The N × N matrices λ a ij form a basis for this part of the string wavefunction, and in a scattering amplitude each vertex carries one of these matrices. Consider a scattering amplitude in which the N D branes are coincident. In this case one must sum over all possible Chan-Paton indices and, in addition, since the right endpoint of a string is the same as the left endpoint of another string—see Fig. 2.13 (b)—the total scattering amplitude results in a trace over these matrices Tr ( λ a$_{1}$ · · · λ $^{a$_{n}$}$). Therefore the amplitude is invariant under U ( N ) transformations of these matrices, λ → UλU − $^{1}$. One may then elevate this global symmetry to a local one in spacetime by defining the vertex operator V aµ ij = λ a $_{ij}$∂$_{t}$ X $^{µ}$exp( ik · X ) that transforms under the adjoint of U ( N ). Since U ( N ) = U (1) × SU ( N )— U ∈ U ( N ) is an N × N unitary matrix. Unitarity implies that its determinant is a pure phase: UU † = 1 ⇒ | det ( U ) | = 1. Therefore any U ( N ) matrix may be written as U = exp( iφ ) SU , where SU ∈ SU ( N )—we now have a p + 1 dimensional SU ( N ) field theory defined on the stack of Dp branes. Specifically, the low energy effective action for a D 3 brane in AdS space can