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

Chapter 7A First Example“Le temps n’apprécie rien qui se fait sans lui...”Lambic Brewery, BrusselsWritten on a piece <strong>of</strong> wood patientlyhanging in the fermentation chamber.As a way <strong>of</strong> getting acquainted with the problem at hand, a good place to start is toretrace the steps needed in order to obtain the results reported in [6]. In this article,a calculation is carried out in section 4 ressembling the one we will ultimately want tomake. The problem at hand is to find the spectrum <strong>of</strong> a D1-brane living on the cigartarget space using the small fluctuation <strong>DBI</strong> analysis. We will also perform the samecomputation for a D1-brane living on the trumpet, a case <strong>of</strong> interest since the D2-braneon the cigar we want to analyse came about by T-dualizing a D1-brane on the trumpet.As was mentioned earlier, we should be careful when interpreting the results <strong>of</strong> thiscomputation, since the exploding <strong>of</strong> the Φ field near the trumpet origin alerts us thatwe should also take into account extra perturbative terms. Nevertheless, despite this,going from the cigar to the trumpet is so easy, that this result will constitute a firstindication whether or not we are on the right track (<strong>of</strong> finding bound states, that is).In all the computations in this and the next chapter, we set α ′ = 1.7.1 D1-brane on the cigar7.1.1 Metric and actionThe cigar metric is given byds 2 = (dρ) 2 + (tanh ρ) 2 dτ 2 ; e −Φ = e −Φ 0chρ. (7.1)85