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

CHAPTER 5. BRANES 68The presence <strong>of</strong> a Maxwell field on the D-brane has a very important consequence.Consider the countour integral ∮⃗E · d ⃗ l. (5.29)CWhen we consider the electric field to be static, and this contour to be lying somewhere inordinary space, we can invoke Stokes’ theorem in combination with the (static) Maxwellequation ∇ × ⃗ E = 0 to obtain∮C∫⃗E · d ⃗ l =S(∇ × E ⃗ )d⃗a = 0. (5.30)When considering compact dimensions, something fundamental changes, namely thecontour no longer bounds a surface since there is no “inner space” anymore. As aconsequence, Eq. 5.29 can take on non-zero values. Generalizing to the case <strong>of</strong> a vectorpotential A µ , and considering its component along the compact dimension (say x 25 ),defineW q = e iq H c dx25 A 25. (5.31)This quantity is called a Wilson line, and it takes on a constant value in the compactinterval [0, 2π]. Now we can consider a setup with several parallel Dp-branes, say N<strong>of</strong> them. We can thus consider several different open string configurations, in generaldefined by two labels i and j, respectively labeling the Dp-brane on which the stringstarts (σ = 0) and ends (σ = π). Now, suppose one direction is compactified, andthat our Dp-branes wrap around this direction. Due to the Maxwell field living on theD-branes, they will acquire a Wilson line θ i along the compact dimension. Amazinglyenough, when we take the T-dual <strong>of</strong> this compact dimension, the resulting D(p − 1)-branes will position themselves on the dual circle at angles θ i . This means that anopen string lying on branes i and j in the original description will stretch on the dualcircle between angles θ i and θ j . The labels i, j are called “Chan-Paton charges.” Thismechanism introduces gauge groups as follows: k coincident branes give rise to a U(k)group, while l seperate branes make for a U(1) l group.5.1.2 D-branes in Type IIA and Type IIB theoriesRecall the coupling <strong>of</strong> the string to the Kalb-Ramond field shown in Eq. 4.16,S B = − 1 ∫4πα ′ d 2 σǫ αβ B µν (X) ∂ α X µ ∂ β X ν .This is in fact a generalization <strong>of</strong> the coupling <strong>of</strong> a vector field to a gauge field,∫dx µS A = q dτA µdτ , (5.32)where q represents the coupling constant to this field (e.g. electric charge). This generalizationimplies that a string couples electrically to the Kalb-Ramond field.