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

CHAPTER 2. BOSONIC STRINGS 36For the first excited level though, we now have many more possibilities, since thesestates are given byα−1˜α i j −1 |0〉, (2.138)and both excitations are essentialy independent. This results in 24 2 = 576 masslessstates. However, the fact that these excitations are indepent has a huge consequence,namely it allows us to decompose the states in three distinct parts as follows:[α−1α i j −1 |0〉 = α[i −1˜αj] −1 |0〉 + α −1˜αj) (i−1 − 1D − 2 δij α−1˜α k −1k]|0〉+ 1D − 2 δij α k −1˜α k −1|0〉, (2.139)where [...] stand for anti-symmetrization and (...) stand for symmetrization. Thus we obtainan anti-symmetric part, a traceless symmetric part, and a part that is proportionalto the unit matrix. And now comes the moment you have all been waiting for: thegraviton has entered the building! Admittedly, we are skipping a few steps here, butwe can nevertheless already identify the traceless symmetric part <strong>of</strong> Eq. 2.139 with amassless spin two particle described by a traceless symmetric rank two tensor, usuallydenoted g µν . It turns out though that if one would investigate how this particle couplesto matter and to itself, it does indeed behave like a graviton (cfr. [14], Chapter 15).Furthermore, we also find ourselves in the lovely company <strong>of</strong> a massless antisymmetricrank two tensor, which is called the Kalb-Ramond field and denoted B µν , and a masslessscalar which responds to the name dilaton field and denoted Φ, both <strong>of</strong> which will playvery important roles later on.Note that in the case <strong>of</strong> the closed string, the generating function trick describedfor the open string case does not apply. The origin for this lies in the level matchingcondition, Eq. 2.102, combined with the commutators for the modes, Eq. 2.53. The commutationrelations show that an m-level operator weights more than an (n < m)-leveloperator, resulting in the fact that one cannot simply write down an elegant generatingfunction for this situation, but has to verify the level-matching on a level-per-level basis.2.7 Oriented vs. unoriented stringsWe note in passing that one could define a projection operation that projectsσ −→ π − σ, (2.140)i.e. it flips all points on the string around the centerpoint. The consequence <strong>of</strong> thisoperation for the mode expansions isopen strings: α µ m ←→ (−1) m α µ m, (2.141)closed strings: α µ m ←→ ˜α µ m. (2.142)<strong>String</strong>s unaffected by this operation are called unoriented, in contrast to their brethrenwho do feel the operation and are called oriented.