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

CHAPTER 6. SETTING 81It was mentioned above that compact D-branes coined between two NS5-branes alsoexhibit bound states. These bound states are massless. Contrary to this, it has beenshown in [3] that the spectrum <strong>of</strong> the bound states appearing on non-compact D-branesin the close vicinity <strong>of</strong> the NS5-ring have a small mass. Recalling that these statesare bound in the x 6 direction, they can only move in the x 0,1,2,3 directions anymore,which correspond exactly to a 4D-Minkowski space-time. It has been proven, still in [3],that these massive bound states correspond to massive gauge bosons in a 4D Minkowskispace-time. This is very interesting, because it allows for yet another building block <strong>of</strong>the Standard Model to be recreated in string theory. Even though it is not yet fullyunderstood how to use this piece <strong>of</strong> the Standard Model puzzle, at least it is sure to bepresent. This result was shown using exact computations in a conformal field theoreticalapproach. The advantage <strong>of</strong> this approach is that it is exact. The drawback however isthat such a conformal field theoretical formulation is only known for a limited number<strong>of</strong> setups, and in particular it is not known in the region far away from the NS5-ring.On the other hand, the <strong>DBI</strong> approach is easily generalizable to this region, but is onlyvalid when the space-time curvature is small at the string length scale. Hence, it isinteresting to find out to which extend both methods agree in the region where they areboth applicable, so as to be able to judge if whether or not the approximate formulationis exact enough to be useful in the far-away region.Returning to the factorized space-time, remark that since we assumed our D4-braneto live in x 0,1,2,3,6 , it also gets factorized, namely into a D3-brane living in x 0,1,2,3 , and aD1-brane living in the trumpet geometry. However, we are faced with the problem thatthe trumpet geometry explodes as r −→ 0 (see Eq. 6.9 and Fig. 6.4). The problem liesin the fact that, as can be seen from Eq. 6.9, as r approaches 0, e Φ −→ ∞, indicatingthat we can no longer assume the coupling to be weak. As such, if we want to use the<strong>DBI</strong> action, we should consider extra terms in a perturbative g s expansion.Figure 6.4: Graphical representation <strong>of</strong> the trumpet (l) and cigar (r) geometries.