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

CHAPTER 3. SUPERSTRINGS 423.3.1 Closed stringsWe start by considering the possibility thatψ + (σ, τ) = ±ψ + (σ + π, τ), (3.16)ψ − (σ, τ) = ±ψ − (σ + π, τ). (3.17)Note that these suppositions imply that the same relations also apply to the variationsδψ ± . Furthermore, we see that if the + sign is chosen, the functions are periodic andare said to obey Ramond boundary conditions (R), if − is chosen they are antiperiodicand obey Neveu-Schwarz boundary conditions (NS). ψ + corresponds to left-movers, ψ −to right-movers, and the periodicity condition for them can be chosen independantly,resulting in four different possible combinations, namely, choosing to write them as(left-movers,right-movers), (NS,NS), (NS,R), (R,NS) and (R,R).To realize these different boundary conditions, the following expansions can be applied:3 R left-movers: ψ + (σ, τ) = ∑ m∈˜dµ m e −2im(τ+σ) , (3.18)R right-movers: ψ − (σ, τ) = ∑ n∈d µ ne −2in(τ−σ) , (3.19)NS left-movers:ψ + (σ, τ) = ∑˜bµ r e −2ir(τ+σ) , (3.20)NS right-movers:r∈+ 1 2ψ − (σ, τ) = ∑b µ se −2is(τ−σ) . (3.21)s∈+ 12We see that the (anti)periodicity is reflected in the (half-)integer summing indices. Fromhereon, we will follow the convention <strong>of</strong> using indices (m, n) for integer numbers, and(r, s) for half-integer numbers.3.3.2 <strong>Open</strong> stringsWhen considering open strings, the names Ramond and Neveu-Schwarz boundary conditionsare still used, but they correspond to different conditions. Nevertheless, they arevery similar, explaining the double usage.Since now the positions (σ = 0) and (σ = π) correspond to two different and a prioriunrelated positions on the string, the two terms in Eq. 3.15 should vanish seperately.Hence, we see that{ψ − (0, τ)δψ − (0, τ) − ψ + (0, τ)δψ + (0, τ) = 0,(3.22)ψ − (π, τ) δψ − (π, τ) − ψ + (π, τ)δψ + (π, τ) = 0.3 Almost every reference uses b and d to denote the NS and R modes respectively, except for [14] whouse b for both, and differentiate only by means <strong>of</strong> their indices (m, n) or (r, s).