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

CHAPTER 6. SETTING 80with r ∈Ê+, 0 ≤ θ ≤ π 2and 0 ≤ φ, ψ ≤ 2π, considering the double scaling limitg s ,ρ 0√α ′ρ 0g s√α ′−→ 0, (6.4)= constant, (6.5)and performing a T-duality along the angular φ direction, one is left with the geometry⎧[ ( ) dω 2 ( ) ]dω2⎪⎨ ds 2 = dx µ dx µ + α ′ k dr 2 + coth 2 k + dψ + dθ 2 + tan 2 θ ,k(6.6)⎪⎩ e 2Φ = g2 eff 1k sh 2 r cos 2 θ ,with k the number <strong>of</strong> NS5-branes, ω the T-dual coordinate to φ, and g eff given by√kα ′g eff = e Φ 0. (6.7)ρ 0Note that the Kalb-Ramond field has vanished. Further note that although we nowfind ourselves with three compact dimensions (θ, φ and ψ), we did not compactify anydimension. All we did was introduce a change <strong>of</strong> coordinates, just like one could reachany point on the flat (x, y)-plane by specifying an angle and a radius. By introducingsuch a compact coordinate, one does not compactify a dimension.It is worth pauzing an extra second at this, since this is exactly what sets this approachapart from the Kaluza-Klein-like approach where one does compactify a dimensionin order to make it invisible (or more precisely: make the physics in that dimensiondecouple from the physics in the other dimensions). We do not compactify, but considera setting in which states appear that are fixed in some dimension, limiting their freedom<strong>of</strong> movement to the remaining dimensions, thereby from a physical point <strong>of</strong> view makingthe “static” dimension “invisible.”The bracketed part <strong>of</strong> the geometry described by Eq. 6.6 can be decomposed as thedirect product <strong>of</strong> two two-dimensional geometries. One is the bell,⎧⎨ds 2 = α ′ k [ dθ 2 + tan 2 θ dω 2] ,⎩e 2Φ = g2 effcos 2 θ , (6.8)and the other is the trumpet,⎧⎨ds 2 = α ′ k [ dr 2 + coth 2 r dψ 2] ,⎩e 2Φ = e2Φ 0sh 2 r . (6.9)We will not be concerned by the six-dimensional Minkowski and two-dimensional bellgeometries, as the physics in these geometries is already quite well understood. Weinstead focus on what happens in the trumpet.