String Theory and M-Theory

6.1 The bosonic string and Dp-branes 199
an open string connecting two separated D-branes. Only when θi = θj do
we have an integer number of windings. This is illustrated in Fig. 6.5.
Fig. 6.5. Strings with fractional and integer winding number.
The mode expansion of the dual ij open string becomes
X 25
ij = ˜x0 + θi R + 2 Rσ
K + θj − θi
2π
+ . . . , (6.40)
so that one end is at ˜x0 + θi R and the other end is at ˜x0 + θj R. This is
interpreted as an open string whose σ = 0 end is attached to the ith D-brane
and whose σ = π end is attached to the jth D-brane. Note that diagonal
strings wind an integer number of times around the circle while off-diagonal
strings generally do not.
The spectrum
The masses of the particles in the ij open-string spectrum of the bosonic
string theory compactified on a circle are3 M 2
K
ij =
R + θj
2 − θi
+
2πR
1
(N − 1). (6.41)
α ′
This formula follows from the mass-shell condition and the fact that the p 25
component of the momentum is shifted according to Eq. (6.39).
Equation (6.41) shows that if all of the θi s are different, the only massless
vector states are ones that arise from strings starting and ending on the
same D-brane without wrapping the circle. All other vector string states
are massive. Therefore, when no D-branes coincide, there are N different
massless U(1) vectors given by the diagonal strings with K = 0. As a result,
the unbroken gauge symmetry is U(1) N .
3 The number operator N should not be confused with the rank of the gauge group.