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

CHAPTER 7. A FIRST EXAMPLE 86The <strong>DBI</strong> action for a D1-brane thus becomes 1 (r is the coordinate on the D-brane)∫S D1 = −T 1 dre −Φ√ ∫detg αβ = −T 1 e −Φ 0drchρ √ det g αβ . (7.2)Since there is only one coordinate, we find the following for the induced metric:g αβ = g rr =( ) ∂ρ 2 ( ) ∂τ 2+ tanh 2 ρ . (7.3)∂r∂rDenoting ∂ r X = X ′ , our action takes on the form∫ √S D1 = −T 1 e −Φ 0dr ch 2 ρ (ρ ′ ) 2 + sh 2 ρ (τ ′ ) 2 . (7.4)Using reparameterization invariance to fixr = shρ, (7.5)we find thatS D1 = −T 1 e −Φ 0∫√dr 1 + r 2 (τ ′ ) 2 . (7.6)7.1.2 Classical solutionThe equations <strong>of</strong> motion for r become( ) ∂L0 = ∂ r∂ (∂ r τ)[= ∂ r − 1 2r 2 τ ′ ]√2 1 + r 2 τ ′= r2 τ ′′ + 2rτ ′√− 1 r 2 τ ′ ( 2r(τ ′ ) 2 + 2r 2 τ ′ τ ′′)1 + r 2 (τ ′ ) 2 2(1 + r 2 (τ ′ ) 2) 3/20 = rτ ′′ + 2τ ′ + r 2 ( τ ′) 3 . (7.7)This can be verified by the relation (recall that r = shρ)sin (τ − τ 0 ) = Cshρ , (7.8)with τ 0 and C constant. This gives the embedding <strong>of</strong> τ, and agrees with the D1 cig entryin Table 2, p.4 in the article.1 Note that since at the moment we do not consider a timelike coordinate on the brane, we takepdet gαβ instead <strong>of</strong> p − det g αβ .