Lectures on String Theory

– 22 –
Analogously,
{¯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 . (3.37)
To summarize
In particular,
{L m , X µ } = −e imσ− ∂ − X µ , {¯L m , X µ } = −e imσ+ ∂ + X µ . (3.38)
{¯L 0 − L 0 , X µ } = ∂ σ X µ (3.39)
i.e. ¯L 0 −L 0 generates rigid σ-translations. The transformations we consider transform
a solution
□X µ = 0
into another solution of this equation. Indeed, we have for instance
∂ + ∂ −
(
e imσ− ∂ − X µ )
= imσ − e imσ− ∂ + ∂ − X µ + e imσ− ∂ − ∂ + ∂ − X µ = 0
as the consequence of ∂ + ∂ − X µ = 0.
(P , X)
Lm
_
L m
_
L m =0
L m =0
Fig. 1. The phase space of string. The Virasoro constraints L m = 0 = ¯L m
define a time-independent hypersurface on which the dynamics of string
takes place. This hypersurface remains invariant under the action of L m ’s
and ¯L m ’s.
Even more generally, for any periodic function f we define
L f =
∫ 2π
0
dσf(σ + )T ++ .