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

CHAPTER 3. SUPERSTRINGS 40In D dimensions, and for D even, there are 2 ⌊D/2⌋ <strong>of</strong> them. 2 In two dimensions, one canchoose a real, or Majorana, representation, and we choose[ ][ ]ρ 0 0 −1=; ρ 1 0 1= . (3.4)1 01 0The spinors ψ µ also are real, and hence have two (real) components, which, in anticipation<strong>of</strong> the use <strong>of</strong> light-cone coordinates, we denote[ψ µ ψµ]−=ψ µ . (3.5)+We further define¯ψ = ψ † β ; β = iρ 0 . (3.6)Note that since ψ µ is real, ψ † = ψ T . It should further be pointed out that the fields ψ µanticommute, as they are expected to, so{ψ µ , ψ ν } = 0. (3.7)This is only true in the classical theory though. Although we will not go into this, wemention that quantum mechanically this relation gets upgraded to{ψµA (σ, ( τ),ψν B σ ′ , τ )} = πη µν δ AB δ ( σ − σ ′) . (3.8)We see that at first sight, we are confronted with the same ghost problem we had for thebosonic string. In order to eliminate these, one has to introduce supersymmetry, whichresults in an extension <strong>of</strong> the Virasoro algebra (called the super-Virasoro algebra), inturn leading to altered super-Virasoro constraint equations, which will do the trick.For the sake <strong>of</strong> completeness, we mention that while we introduced, or even imposed,supersymmetry by hand, one could also define a superspace, which amounts to addinganticommuting coordinates to an arbitrary space-time. The advantage <strong>of</strong> this is that inthis formulation supersymmetry if manifest. For our purposes however, this does notmake any difference for the results we want to obtain, and would only needlessly makethings more complicated. Interested parties are referred to e.g. [9], §4.1.2.3.2 Equations <strong>of</strong> motionWe first <strong>of</strong> all note that the bosonic term in the string action is unaffected by thepresence <strong>of</strong> the fermionic term. Hence, nothing will change in the equations <strong>of</strong> motion,and subsequent spectrum <strong>of</strong> the bosonic part <strong>of</strong> the string. This allows us to single outthe fermionic part <strong>of</strong> the equation, and consider it separately. Let us denote this partbyS F = − 1 ∫2πα ′ d 2 σ ( ¯ψµ ρ α )∂ α ψ µ . (3.9)2 For D uneven there is one more, but we will not be confronted with this possibility.