String Theory and M-Theory

6.1 The bosonic string and Dp-branes 189
incorporate the boundary condition (6.1). The expansion is
X 25 (σ, τ) = x 25 + 2α ′ p 25 τ + 2RW σ + . . . , (6.2)
where the coefficient of σ is chosen to satisfy (6.1). The dots refer to the
oscillator terms, which are not modified by the compactification.
0 +1 -1
Fig. 6.1. Strings winding around a compact direction.
Since one dimension is compact, the momentum eigenvalue along that
direction, p 25 , is quantized. Remember that the quantum mechanical wave
function contains the factor exp(ip 25 x 25 ). As a result, if x 25 is increased
by 2πR, corresponding to going once around the circle, the wave function
should return to its original value. In other words, it should be single-valued
on the circle. This implies that the momentum in the 25 direction is of the
form
p 25 = K
, K ∈
R
. (6.3)
The integer K is called the Kaluza–Klein excitation number. Splitting the
expansion into left- and right-movers,
gives
X 25 (σ, τ) = X 25
L (τ + σ) + X 25
R (τ − σ), (6.4)
X 25
R (τ − σ) = 1
2 (x25 − ˜x 25 ′ K
) + (α − W R)(τ − σ) + . . . , (6.5)
R
X 25
L (τ + σ) = 1
2 (x25 + ˜x 25 ′ K
) + (α + W R)(τ + σ) + . . . , (6.6)
R
where ˜x 25 is a constant that cancels in the sum. In terms of the zero modes
, defined in Chapter 2, the mode expansion is
α 25
0
and ˜α25
0
X 25
R (τ − σ) = 1
2 (x25 − ˜x 25 ) + √ 2α ′ α 25
0 (τ − σ) + . . . , (6.7)
X 25
L (τ + σ) = 1
2 (x25 + ˜x 25 ) + √ 2α ′ ˜α 25
0 (τ + σ) + . . . , (6.8)