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

CHAPTER 7. A FIRST EXAMPLE 927.2.1 Metric and actionThe trumpet metric is given byds 2 = (dρ) 2 + (coth ρ) 2 dψ 2 ; e −Φ = e −Φ 0shρ. (7.38)The <strong>DBI</strong> action for a D1-brane thus becomes (r is the coordinate on the D-brane)∫S D1 = −T 1 dr e −Φ√ ∫det g αβ = −T 1 e −Φ 0dr shρ √ det g αβ . (7.39)The induced metric becomesg αβ = g rr =( ) ∂ρ 2 ( ) ∂ψ 2+ coth 2 ρ . (7.40)∂r∂rDenoting ∂ r X = X ′ , our action takes on the form∫ √S D1 = −T 1 e −Φ 0dr sh 2 ρ (ρ ′ ) 2 + ch 2 ρ (ψ ′ ) 2 . (7.41)Using reparameterization invariance to fixr = chρ, (7.42)we find thatS D1 = −T 1 e −Φ 0∫√dr 1 + r 2 (ψ ′ ) 2 . (7.43)7.2.2 Classical solutionThe equations <strong>of</strong> motion for r become( ) ∂L0 = ∂ r∂∂ r ψ= r2 ψ ′′ + 2rψ ′√− 1 r 2 ψ ′ ( 2r(ψ ′ ) 2 + 2r 2 ψ ′ ψ ′′)r 2 − 1 (ψ ′ ) 2 2(r 2 − 1 (ψ ′ ) 2) 3/20 = rψ ′′ + 2ψ ′ + r 2 ( ψ ′) 3 . (7.44)This can be verified by the relation (recall that r = chρ)with ψ 0 and C constant. This gives the embedding <strong>of</strong> ψ.sin(ψ − ψ 0 ) = Cchρ , (7.45)