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

CHAPTER 2. BOSONIC STRINGS 32This gives us two equations relating the mode expansions <strong>of</strong> X − to those <strong>of</strong> the X i ’s, arelation that can be solved explicity, and for the case <strong>of</strong> the open string the result is(αn − = 1 1l s p + 2D−2∑∞∑i=1 m=−∞: α i n−mα i m : −aδ n,0). (2.119)For the closed string, an analogous relation for ˜α n − exists. Notice the ordering constanta again. The important point to note, is that now there are no independent oscillatorsleft for X − , so here again, we can no longer excite our string in the X − direction! Thatis very good news indeed, because looking back at Eq. 2.56, and keeping in mind thatdue to our freshly introduced light-cone coordinates our metric has somehow changed abit, 8 we now see that we are no longer able to create ghost states, even if we wanted todo so. In other words, the light-cone gauge is manifestly free <strong>of</strong> ghost states.Let us now turn to the mass operator. The generic expression M 2 = −p µ p µ nowbecomesD−2∑M 2 = −p µ p µ = 2p + p − − p 2 i . (2.120)In order to work this out some more, let us focus on the case <strong>of</strong> open (Neumann) strings,and recall that in that caseα µ 0 = l sp µ .Hence, using p − = (l s ) −1 α0 − and Eq. 2.119, we compute that{ [(D−2∑D−2)2p + p − − p 2 i = 2p+ 1 ∑ ∑l s l s p + α−nα i ni + 1 2i=1i=1where we redefinedD−2∑− 1 ls2 i=1[(= 2 D−2 ∑ ∞∑ls2 i=1 n=1(αi0) 2,α i −nα i nn>0)− a],i=1D−2∑+∞∑i=1 n=−∞(αi0) 2]M 2 = 2 ls2 (N − a), (2.121)N =D−2∑i=1 n=1∞∑α−nα i n.iWe see that the mass-shell condition in light-cone gauge is identical to the one in conformalgauge (Eq. 2.104), except for the fact that we now simply have two directions lessin which we can excite our string. Hence, our groundstate still has α ′ M 2 |0〉 = −a|0〉.8 More specifically, we get g +− = g −+ = −1, and g ij = δ ij for i, j ∈ {1, . . . , D − 2}. But the twonegatively valued entries correspond to directions in which we have no oscillators, and hence we can nolonger create ghosts.− a}