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

CHAPTER 5. BRANES 67Also remark that it are only the endpoints <strong>of</strong> the open string that are fixed; all otherpoints on the string are still free to move as they please.D-branesAs we have just introduced them, Dp-branes are physical objects that extend in p spatialdimensions, and whose defining property is that the endpoints <strong>of</strong> open strings satisfyingDirichlet boundary conditions can (or even have to) end on them. We see that in essence,D-branes were present right from the start, only betraying their presence through theDirichlet boundary conditions which first got dismissed as being unphysical due to thefact that they break Poincaré invariance. As a consequence, they only grew in popularitywhen T-duality was discovered in 1989.As an intuitive argument 4 as to why these objects should behave like physical objects,consider this next scenario. An open string fixed to a D-brane moves around untilsomehow both its endpoints meet. As <strong>of</strong> that moment, the string becomes a closedstring, and hence is no longer bound to the surface <strong>of</strong> this D-brane. Instead, it can get<strong>of</strong>, and roam around in space-time, enjoying the sight. But at some point, it might hitanother D-brane, at which time it can “open up” again, and become fixed to the surface<strong>of</strong> this other D-brane. In this case, two D-branes will have exchanged a (closed) string,and hence, talked to each other. Only physical objects can interact, hence implying thatD-branes should indeed be physical.Something crucial happens to the spectrum <strong>of</strong> open strings with Dirichlet boundaryconditions. We now know that if a string has Neumann conditions in some directions, letus say 0 to p (with 1 ≤ p < 25), and Dirichlet in the others, this amounts to saying thatthis string has endpoints lying on a p-brane. We can now denote the Neumann directionswith an index n and the Dirichlet directions with an index d, i.e. X µ = { X n , X d} withn ∈ {0, . . .,p} and d ∈ {p + 1, . . .,25}. The next step is to combine X 0 and X 1 intolight-cone coordinates, so that we can use the light-cone gauge to construct the spectrum<strong>of</strong> this string. After we do this, we are left with (p − 1) transverse Neumann directions,and thus (p − 1) α−1 i operators (i ∈ {2, . . .,p}) we can let act on our groundstate. Ifwe use these operators, we create a massless state, just as we saw in Chapter 2, butthis time in (p − 1) <strong>of</strong> the dimensions in which our Dp-brane is stretched. We create aphoton that lives on the D-brane! Hence, every Dp-brane has a Maxwell field living onits world volume. The α−1 d operators generate states that from the point <strong>of</strong> view <strong>of</strong> theD-brane transform as scalars. All these fields are perpendicular to the D-brane worldvolume. They can be regarded as excitations <strong>of</strong> the D-brane itself.This last point gives rise to a nice way <strong>of</strong> eliminating tachyons from the open stringbosonic spectrum. It has been suggested that a space-time filling D25-brane is unstablebecause <strong>of</strong> the tachyon that lives on its world volume. Because <strong>of</strong> this, the D25-branewould actually decay, and the products <strong>of</strong> this decay would be closed strings. This isalso true for Dp-branes with p < 25. As a consequence, bosonic string theory couldvery well not contain any stable D-branes at all. As <strong>of</strong> yet, no way has been found toeliminate the closed string tachyon, though.4 This argument has been taken from [5].