Lectures on String Theory

– 108 –
One can also check that the world-sheet fermion transforms under conformal transformations
as the conformal field with weigh 1/2.
Consider closed strings. Using the mode expansion
ψ + (σ, τ) = ∑ ¯bµ r e −ir(τ+σ) ,
r∈Z+θ
ψ − (σ, τ) = ∑
r∈Z+θ
where θ = 0 in the R-sector and θ = 1 2
algebra of oscillators
b µ r e −ir(τ−σ) ,
{b µ r , b ν s} = −iη µν δ r+s ,
{¯b µ r , ¯b ν s} = −iη µν δ r+s ,
{b µ r , ¯b ν s} = 0 .
The reality of the Majorana spinor implies that
(b µ r ) † = b µ −r , (¯b µ r ) † = ¯b µ −r .
in the NS-sector, we obtain the Poisson
Introducing the modes of the stress tensor and the supercurrent
L m = 1
2π
∫ 2π
0
dσe −imσ T −− ,
G m = 1 π
∫ 2π
0
dσe −irσ G − .
Notice that the supercurrent G − satisfies the same boundary condition as the fermion
ψ − . Substituting the mode expansion we get
L m = 1 ∑
α −n α m+n + 1 ∑ (
r + m )
b −r b m+r ,
2
2 2
n∈Z
G r = ∑ n∈Z
α −n b r+n .
These generators generate the classical super-Virasoro algebra
{L m , L n } = −i(m − n)L m+n ,
( 1
)
{L m , G r } = −i
2 m − n G m+r ,
{G r , G s } = −2iL r+s .
7. Quantum fermionic string
Canonical quantization is again performed by substituting the Poisson bracket for
the (anti)-commutator: { , } pb → 1 [ , ]. Therefore, the anti-commutators for the
i
quantum fermionic fields are
{ψ µ +(σ), ψ ν +(σ ′ )} = 2πδ(σ − σ ′ )η µν
{ψ µ −(σ), ψ ν −(σ ′ )} = 2πδ(σ − σ ′ )η µν .
r