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

CHAPTER 2. BOSONIC STRINGS 18we see that∂ + = 1 2( ∂τ ∂∂σ + ∂τ + ∂σ )∂∂σ + ,∂σ= 1 2 (∂ τ + ∂ σ ) . (2.22)Similarly,∂ − = 1 2 (∂ τ − ∂ σ ) . (2.23)For the metric, we compute thatand analogous for η −− , andη ++ = ∂σµ∂σ + ∂σ ν∂σ +η µν,= (∂ + σ) 2 − (∂ + τ) 2 ,= 1 4 − 1 4 ,= 0, (2.24)η ±∓ = (∂ + σ)(∂ − σ) − (∂ + τ) (∂ − τ),= − 1 4 − 1 4 ,= − 1 2 , (2.25)leaving us with [ ]η++ η +−= − 1 [ 0 1η −+ η −− 2 1 0]. (2.26)The nice thing about these coordinates is that they allow us to write the equations <strong>of</strong>motion expressed in Eq. 2.12 as follows:∂ + ∂ − X µ = 0, (2.27)which immediatly suggests the general solutionX µ (σ, τ) = X µ (R σ− ) + X µ (L σ+ ) . (2.28)X µ R,Lare called the right- and left-moving modes <strong>of</strong> the string respectively.Things get a bit messy when we want to expand these modes, because we have tolook at it on a case per case basis, depending on what boundary conditions are appliedto the string under consideration.