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

CHAPTER 5. BRANES 66i.e. our original Neumann boundary conditions got transformed into Dirichlet boundaryconditions! Had we started out with Dirichlet boundary conditions, it is easy to seefrom Eq. 2.39 that the same would have happened but the other way around, i.e. wewould have found that D boundary conditions got turned into N boundary conditions.Furthermore, we note that˜X 25 (π, τ) − ˜X 25 (0, τ) = 2π ˜Rn, (5.28)meaning our string has to wrap an integer time around the T-dualized circle. This isquite an interesting observation, given that if a string has Neumann boundary conditionsin some (compact) direction, it can not have a winding number in that direction.Intuitively this is clear: the string can move freely in that direction, so nothing can forceit to keep its endpoints fixed. But since it can move, it also gains momentum. In theT-dual picture, this observation gets inversed: the string endpoints become fixed, so itcan wind (and as Eq. 5.28 shows, it is even forced to), but since it cannot move, itcannot gain any momentum.The crucial point in all this, is that as far as closed strings are concerned, thereis no physically significant difference between compactification on a circle <strong>of</strong> radius R,or a circle <strong>of</strong> radius ˜R = α ′ /R. For open strings however, this is not the same, andthat is something we do not like, because T-duality seemed like a really nice idea andvery powerful tool. So how can we fix this awkward little problem? Answer: introduceD-branes! So in order to make sense <strong>of</strong> this picture, it has been proposed (and thisproposition has since been developped in full glory) that an open string with Dirichletboundary conditions does not have its endpoints fixed to some arbitrary points in spacetimethat somehow are more special than their neighbours, but instead that its endpointslie on the surface <strong>of</strong> a physical object that lives in (p + 1)-dimensional space-time. Theendpoints <strong>of</strong> the string are free to move on this surface, but are bound to remain fixed indirections perpendicular to it. This object is called a Dp-brane, with the D referencingto Dirichlet for obvious reasons. So in the case <strong>of</strong> our original string that had Neumannboundary conditions in all 25 spatial dimensions, we can now interpret this as an openstring whose endpoints live on the surface <strong>of</strong> a space-time-filling D25-brane. Since thesurface <strong>of</strong> this D-brane comprises all <strong>of</strong> space-time, the open string is free to move as itpleases. When we compactify the X 25 direction and T-dualize it however, what happens“underneath the surface” is that we remove one dimension <strong>of</strong> our D-brane, making it aD24-brane that does no longer extend in the X 25 direction. Hence, since this brane stilllives in a 25 spatial-dimensional world, its position must be fixed in the X 25 dimension, 3which explains why suddenly our open string gains Dirichlet boundary conditions in thisdirection and is forced to wrap an integer number <strong>of</strong> times around it.This idea can easily be generalized to compactification <strong>of</strong> more than one spatialdimension. Remark however that because an open string wraps around a circular direction,and hence has both its endpoints identified in this direction, does not make ita closed string! The endpoints can still take on different values in the other directions.3 Consider e.g. a disc in the x − y plane that lives in a 3D world. If this disc cannot move in the zdirection, when we project it onto the z-axis it is bound to take on a fixed value.