String Theory Demystified

CHAPTER 12 Heterotic String Theory 217
The usual formulas for the modal expansions apply
x p i
X
n e
i i
∞ i
i
n
( τ − σ) = + ( τ −σ)
+ ∑
2 2 2 n=
1
X
i
α −2in( τ−σ) i i
i
x p i n in
n n e
∞ α − 2 ( τ+ σ)
( ) + ∑
2 2 2 = 1
( τ + σ) = + τ + σ
x p i
X
n e
I I
∞ I
I
n
( τ + σ) = + ( τ + σ)
+ ∑
2 2 2 n=
1
∞
a −2in( τ−σ) n
n=−∞
a
S ( τ − σ)
= ∑ S e
α − 2in(
τ+ σ)
Using this approach, we write down two number operators. The number operator
for the right-moving sector is
∞
⎛ i i 1 − ⎞
N = ∑ n n + nS n Sn
⎝
⎜α−
α
− Γ
⎠
⎟
2
n=
1
For the left-moving sector we have
∞
N
i i I I
= ∑( α−nαn+ α−nαn) n=
1
The mass can be written in terms of the canonical momentum p I as
1
1
4
1
16
2
m N N p
2
I
= + − + ∑(
)
Now, let’s see how compactifi cation affects these results. It is easiest to
understand by stepping back to compactifi cation of one dimension as we did in
Chap. 8. Kaku gives a nice example which we restate here. Take a single-dimensional
theory and let
x = x+2π R
Consider a fi eld defi ned on this space φ( x ) . Since the coordinate is periodic, the
fi eld must be also
I=
1
φ( x) = φ( x+2πR) 2