Lectures on String Theory

– 71 –
Equations of motion are
This equations are supplemented by
∂ − b ++ = ∂ + b −− = 0 ,
∂ + c − = ∂ − c + = 0 .
• by periodicity condition for the closed closed string case
b(σ + 2π) = b(σ) , c(σ + 2π) = c(σ) ;
• by boundary conditions for the open string case
b ++ (σ) = b −− (σ) , c + (σ) = c − (σ) for σ = 0, π ,
which follow by requiring the vanishing of the boundary terms arising upon
deriving equations of motion.
Note that for the closed string case b ++ and c + are left-moving waves, while b −− and
c − are the right-moving ones. The canonical anti-commutation relations are
{b ++ (σ, τ), c + (σ ′ , τ)} = 2πδ(σ − σ ′ ) ,
{b −− (σ, τ), c − (σ ′ , τ)} = 2πδ(σ − σ ′ ) .
For the closed string case the Fourier mode expansions look as
c + (σ, τ) =
c − (σ, τ) =
b ++ (σ, τ) =
b −− (σ, τ) =
+∞∑
n=−∞
+∞∑
n=−∞
+∞∑
n=−∞
+∞∑
n=−∞
For the anti-commutation relations this becomes
{b m , c n } = δ m+n ,
¯c n e −in(τ+σ) ,
c n e −in(τ−σ) ,
¯bn e −in(τ+σ) ,
b n e −in(τ−σ) .
{b m , b n } = {c m , c n } = 0
and the same for the barred oscillators. The Virasoro generators are
∞∑
L gh
m = (m − n) : b m+n c −n :
¯L gh
m =
n=−∞
∞∑
n=−∞
(m − n) : ¯b m+n¯c −n :