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

CHAPTER 5. BRANES 61conditions can end on them; i.e. that the endpoints <strong>of</strong> an open string satifsying Dirichletboundary conditions are located on a D-brane. 1What makes D-branes so extremely interesting is that they provide a way to introducegauge symmetries in string theory, which in turn hints at a possible way to createstandard model-like models from a stringy perspective.The easiest way to motivate the necessity for D-branes is by considering T-duality.Hence, this section will start with a short exposition <strong>of</strong> this subject. Then we will have alook at the influence <strong>of</strong> the gauge fields generated by the closed strings in the two TypeII superstring theories on the possible D-branes in these theories. We will conclude byhighlighting a possible low-energy action that describes the world volume <strong>of</strong> a D-brane.5.1.1 T-dualityTo make our lives easier, we will again go back for a second to the bosonic string theoryand its corresponding 26-dimensional space-time. We will investigate what happenswhen we compactify one spatial dimension. Compactifying in essence comes down totaking a dimension, and identifying (at least) two points in this dimension with eachother. A natural way <strong>of</strong> doing so, is to consider this dimension to be curled up into acircle.We will start by considering what happens to closed strings, and then have a lookat what changes for the case <strong>of</strong> open strings.Closed stringsThe first thing to do, is simply to choose one <strong>of</strong> the 25 spatial dimensions to compactify.We will not try to be original, but instead choose to do as everybody else, and hencechoose the 25 th dimension. As such, compactifying this dimension amounts to statingthat this dimension satisfies the topology <strong>of</strong> a circle with some arbitrary (but fixed, <strong>of</strong>course) radius R. Now, a closed string with a component that lives in this dimensionobviously has to go around the entire circle at least once. But nothing forbids it <strong>of</strong>winding around it multiple times. Just think <strong>of</strong> an elastic that you wind around acurled up poster or something: you can wrap it around just once, but nothing forbidsyou from making a figure-<strong>of</strong>-eight with your elastic and hence wrapping it around twice.Because <strong>of</strong> this newly acquired freedom, our original closed string boundary condition,Eq. 2.16, gets modified with an extra term as follows:X 25 (σ + π, τ) = X 25 (σ, τ) + 2πRW. (5.1)Herein, W ∈is called the winding number, and expresses how many times the stringwinds around the circular dimension. The direction in which the string is winded isencoded in the sign <strong>of</strong> W. This extra term is explained by the simple observation 2that the closed string boundary condition relates one and the same point on our string,1 This immediatly implies that a string and a D1-brane are two separate things.2 The simpler an observation, the longer it usually takes to figure it out.