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

CHAPTER 6. SETTING 82But we can perform a T-duality on the circular ψ dimension <strong>of</strong> the trumpet, resultingin the cigar geometry (see Fig. 6.4), and the change <strong>of</strong> our D4-brane to a D5-brane.In factorized terms, our D3-“Minkowski”-brane remains unchainged, but the D1-branein the trumpet geometry now becomes a D2-brane covering the entire cigar geometry.This geometry is given by⎧⎨ds 2 = α ′ k [ dρ 2 + tanh 2 ρ dτ 2] ,⎩e 2Φ = e2Φ 0ch 2 ρ . (6.10)As far as the presence <strong>of</strong> the NS5-branes goes in these geometries, on the trumpet onecan consider them to be at infinity on the exploding plane at r = 0, and at the originon the cigar.6.4 RoadmapWe have now come to a point where we managed to isolate our test subject in a geometrywe know how to handle. So how should we proceed from here on?We cannot simply write down the <strong>DBI</strong> action for a D-brane in some geometry, aD2-brane in the cigar geometry in our specific case, and start varying around. Whatwe first need to do is to find the classical solution <strong>of</strong> the D-brane in this geometry.The classical solution is what one finds when one varies the unperturbed <strong>DBI</strong> D-braneaction with respect to the coordinates or fields one ultimately wants to vary, or moreprecisely can vary. This solution effectively extremizes the <strong>DBI</strong> action, thus constitutingthe equilibrium solution around which we will want to perturb. As a specific example,consider a D1-brane in the cigar and trumpet geometries. As will be shown in the nextchapter, the classical solutions for these configurations are given bytrumpet:cigar:sin(ψ − ψ 0 ) = Cchρ , (6.11)sin (τ − τ 0 ) = Cshρ . (6.12)Minimizing the energy <strong>of</strong> the D-brane in these configurations, these equations also tell ushow the brane will position itself in these geometries, i.e. they represent the embeddings<strong>of</strong> the D-brane in the geometry. In these embeddings, we see the appearance <strong>of</strong> twointegration constants, C and ψ 0 and τ 0 for the trumpet and cigar respectively. ψ 0 andτ 0 are angles with no physical interpretation; they simply define an <strong>of</strong>fset compared tothe origin in that direction, which amounts to rotating the brane around geometriesthat are symmetrical in that direction, hence not changing the physics. The constant Chowever has a very distinct interpretation: it can be considered as a measure <strong>of</strong> distance<strong>of</strong> the D1-brane to the NS5-ring, as is shown in Figure 6.5.Once we have found the classical solution, we can start thinking about varying theaction. For this, one simply introduces perturbations <strong>of</strong> coordinates and/or fields (e.g.the Maxwell field living on the D-brane). These perturbations will enter the action