String Theory Demystified

202 String Theory Demystifi ed
µ
oscillators and φ is a state with an odd number of d oscillators. It so
0
happens that
Now defi ne
Then if
Γ ψ = ψ Γ φ =− φ
(11.23)
11 11
Γ = Γ −1
( )F
11
Γ ψ =+ ψ
µ
the state ψ has an even number of d excitations. On the other hand if
−n
Γ ψ =− ψ
(11.24)
µ
the state ψ has an odd number of d excitations.
−n
Type II A theory is characterized by states with opposite chiralities. So, the GSO
projection of the right movers has the opposite sign of the GSO projection of the
left movers. This means that the chirality of space-time fermions will turn out to be
opposite. Specifi cally note that
( Γ 0 Γ 0
11 ) =−( 11 )
R left R right (11.25)
in type II A theory. Now consider the spin fi eld Sa . Since Γ 0 =± 0
11 and
a
R R
0 = S 0 where 0 is bosonic, we can consider the action of Γ on the
R
NS
NS 11
spin fi eld itself. In type II A theory
a a a a
Γ S = S and Γ S =−S
(11.26)
11 11
The action of GSO projection is to take the 32-component spinor into a
16-component Majorana-Weyl spinor.