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

CHAPTER 6. SETTING 79<strong>of</strong> the open strings living on it, we only look at low-energetic particles, meaning thelowest excitation states. In concreto, these are the scalar and vector particles.Also, the <strong>DBI</strong> action is only valid if the string coupling is weak. If this is not thecase, one should consider extra terms in a perturbative g s expansion.6.3 Geometrical settingLet us now return to our ten-dimensional Minkowski space-time, with a ring <strong>of</strong> NS5-branes extending in x µ=0,...,5 . The presence <strong>of</strong> these NS5-branes makes the flat spacetimecurved, resulting in the following metric (which is valid also if the NS5-branes arenot spread on a circle):⎧⎪⎨ ds 2 = η µν dx µ dx ν + Hδ ij dx i dx je⎪⎩2Φ = e 2Φ 0H(6.1)= ⋆ 4 dHH (3)with i, j ∈ {6, 7, 8, 9}, H defined as a function <strong>of</strong> the positions x i a <strong>of</strong> the NS5-branes asH(x i ) = 1 +k∑ α ′|x i − x i 2, (6.2)a|a=1and ⋆ 4 the Hodge dual in x i=6,...,9 , making H (3) the three-from field strength <strong>of</strong> theKalb-Ramod field. Note that the NS5-branes behave like black holes. By specializingto the case <strong>of</strong> NS5-branes spread on a circle, introducing the coordinates{(x 6 , x 7 ) = ρ 0 chr sinθ (cos ψ, sinψ),(x 8 , x 9 (6.3)) = ρ 0 shr cos θ (cos φ,sinφ) ,D1-branex 6x 7 x xboundstatefreestatex 7D1-braneCompact D1x 7 x xx 6NS5-ringNS5-ringx 6NS5Figure 6.3: Graphical illustration <strong>of</strong> bound states. When the D-brane is far enough, theexcitations are free (left). When the brane is close enough, they become bound (mid). Ifthe brane gets even closer, it gets cut in three pieces, one <strong>of</strong> which is a compact D-branefixed between two NS5-branes. Also these states are bound (right).