We want to show that it is not possible to have only one string endpoint on a DO-brane. Intuitively, the string charge, visualized as a current on the string, has nowhere to go on the pomthke 00-brane. More quant1tat1vely, as m any D-brane, on the 00-brane there is a gauge field that couples to the string endpoint as in ( 15.54). Since the DO-brane has no spatial coordinate the gauge field is just A 0 . Show that the Maxwell action vanishes, and that, as a result, the variation of Ao imposes an inconsistent equation of motion. Note that the inconsistency is removed if the two endpoints of an open string lie on the DO-brane.

(b) Consider the covariant quantization of an open string whose endpoints Iie on a DO-brane. Describe the ground states, noting that the momentum has a single component. Con­ struct the general states with N= 0 and discuss the equation of motion for the tachyon field (t ). Construct the N = I physical states. Show that there are no relevant null states, and that we have D independent physical states of zero momentum.

(c) Give the mass m0 of  a  DO-brane  (see  Section  18.4).  In  type  IIA  superstring theory the same formula applies, and a bound state of 11 DO-branes 1s known to have a mass exactly equal to nmo. Such states can be identified with momentum states that arise from a compact eleventh dimension, assummg that the momentum p along this d1mens1on contributes to the mass of the effective ten-di mensional particle as i n m = p. What is the radius R of this eleventh dimension? How does it behave as a function of g? This result is one piece of evidence for the fact that eleven-di mensional M-theory compactified on a circle is type IIA  superstring  theory.