DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 2. BOSONIC STRINGS 13D-dimensional space-time (or background). p usually denotes the number <strong>of</strong> spatialdimensions only, hence, D > p. When these p-dimensional objects move in space-time,they describe a (p + 1)-dimensional world volume. On this world volume, one can, anddoes, define a set <strong>of</strong> coordinates, usually denoted by σ. Now, one can imagine standingsomewhere on this world volume, surrounded by the D-dimensional external world, andthrowing out anchors, one in each dimension <strong>of</strong> the external space-time. These anchorsdefine maps that translate the position on the world volume to a position in the externalspace-time, <strong>of</strong>ten referred to as the target space, and these are usually denoted by X.With both coordinate systems, that <strong>of</strong> the world volume and <strong>of</strong> space-time, are associatedmetrics. Also associated with a brane is a brane tension which gives a notion <strong>of</strong>how resistive this object is with respect to gravitation-like forces exerted upon it fromoutside. In a sense, this is nothing else but the mass per volume <strong>of</strong> a brane. In the case<strong>of</strong> the 0-brane, which is in fact a pointlike particle, the brane tension is the mass itselfsince the 0-brane has no physical dimension, and hence no volume. Starting from the1-brane, or string if you prefer, this tension thus becomes mass per length, mass persurface, etc. The metrics, maps and brane tension are all we need to construct an actionfor the brane. Since this chapter specializes to strings, in this case p = 1.2.1 ActionOur journey starts with the action <strong>of</strong> the bosonic string. Being one-dimensional, uponmoving around in space-time this object will describe a two-dimensional sheet, calledthe world sheet, analogous to the one-dimensional world line <strong>of</strong> a point particle. Weequip this world sheet with a coordinate system, the coordinates <strong>of</strong> which are usuallylabeled (σ 0 , σ 1 ) = (τ, σ), in which τ suggests that this coordinate can be tought <strong>of</strong> asthe equivalent <strong>of</strong> the space-time time t, and σ can be interpreted as a spatial dimension.The simplest action we can associate with this object is the following:∫ √S NG = −T dσdτ (ẊX′ ) 2 − Ẋ2 X ′2 . (2.1)In this equation, T represents the string tension, Ẋ = ∂ τ X and X ′ = ∂ σ X. The subscript“NG” stands for “Nambu-Goto,” the two people who first considered this action, makingit the Nambu-Goto action. The string tension T can be interpreted as the “mass perunit length” <strong>of</strong> the string, and hence it has mass dimension equal to two. Often thestring tension is written asT = 12πα ′, (2.2)where α ′ represents the Regge slope mentioned in the introductory chapter. One couldrewrite this action as∫S NG = −T√d 2 σ − det(η µν ∂ α X µ ∂ β X ν ), (2.3)with η µν representing the D-dimensional Minkowski metric, and the subscripts (α, β)refering to derivatives with respect to the world sheet coordinates. As such, for the