Lectures on String Theory

– 21 –
3.3 Integrals of motion. Classical Virasoro algebra.
String has an infinite set of integrals of motion (quantities which are conserved in
time due to equations of motion) which are constructed with the help of T ±± .
Note that T ±± themselves are not the good conserved quantities as they depend
on time! However, if we define the Fourier components
then 5
dL m
dτ
= 2T
= 2T
= 2T
∫ 2π
0
∫ 2π
0
∫ 2π
0
dσ
dσ
dσ
L m = 2T
∫ 2π
0
dσ e imσ− T −− (σ, τ)
(
)
∂ τ e im(τ−σ) T −− (σ, τ) + e im(τ−σ) ∂ τ T −− (σ, τ)
(
)
ime im(τ−σ) T −− (σ, τ) − e im(τ−σ) ∂ σ T −− (σ, τ)
(
)
ime im(τ−σ) T −− (σ, τ) + ∂ σ e im(τ−σ) T −− (σ, τ) = 0
(3.34)
Thus, the Fourier components of the stress-energy tensor provide an infinite set of
the conserved constraints. Analogously, we define
¯L m = 2T
∫ 2π
0
dσ e imσ+ T ++ (σ, τ) ,
which is also an integral of motion. Note that in this derivation we never used the
constraints T ++ = 0 = T −− .
The Poisson brackets of the L m and ¯L m generators are
{L m , L n } = −i(m − n)L m+n ,
{¯L m , ¯L n } = −i(m − n)¯L m+n , (3.35)
{L m , ¯L n } = 0 .
Z
{L m , L n } = 4T 2 dσdσ ′ e −imσ−inσ′ {T −− (σ), T −− (σ ′ )}
Z
= −2T dσdσ ′ e −imσ−inσ′ ∂ σT −− (σ)δ(σ − σ ′ ) + 2T −− (σ)∂ σδ(σ − σ ′
) =
Z
= −2T dσ − T −− (σ)∂ σ e −i(m+n)σ′ + 2e −imσ T −− (σ)∂ σ e −inσ
Z
= −i(m − n) 2T dσe −i(m+n)σ T −− (σ) = −i(m − n)L m+n
This is the so-called Wit algebra. This algebra acts on X µ (σ, τ):
{L m , X µ } = 2T
∫ 2π
0
dσ ′ e imσ′− {T −− (σ ′ ), X µ (σ)} =
= − 1 2 eimσ− (Ẋµ − X ′µ ) = −e imσ− ∂ − X µ . (3.36)
5 When finding the time dynamics of L m one has to remember that the function L m has an
explicit time-dependence and, therefore dL m
dτ
= ∂ τ L m + {L m , H}.