String Theory Demystified

198 String Theory Demystifi ed
This suggests that these operators are related to the Gamma matrices using
Γ
µ µ
= i 2d0 (11.11)
This tells us that the states in the R sector are space-time spinors. We can write the
ground state as
a
0
R
where a is a spinor index that ranges over a = 1, …, 32. As we have seen earlier, this
is because a general Dirac spinor has 2D/2 components where D is the number of
space-time dimensions. Since D = 10 for superstring theories, there are 32
a
components. The state 0 is a 32-component Majorana spinor.
R
±
Now recall that the chirality operator Γ = Γ Γ …Γ acts on states 0
11 0 1 9
R of
defi nite chirality according to
Γ
Γ
11
11
+ +
R R
0 =+ 0
− −
0 =− 0
R R
(11.12)
States with defi nite chirality are Majorana-Weyl spinors, which have half the
a
number of components, (16 in this case). We can write the state 0 as a direct sum
R
of positive and negative chirality states:
a
R
+ −
R R
0 = 0 ⊕ 0
(11.13)
This gives the state [Eq. (11.13)] 16 ⊕ 16 = 32 components. The states 0 are
+ R
space-time fermions. However, they have the bizarre property that 0 is bosonic
−
R
and 0 is fermionic on the worldsheet.
R
THE NS SECTOR
In the NS sector, we have the modal expansions of the left- and right-moving
fermionic states given by
ψ ( σ, τ)
=
µ µ − 2ir(
τ+ σ )
+
r
µ
−
r∈ Z+
12 /
µ −2ir( τ−σ )
∑ br
e
r∈ Z+
12 /
ψ ( σ, τ)
=
∑
b e
±
(11.14)