156
7 CFT Systems
7.1.5 Commutators
The commutators can be computed from (6.75a) knowing the OPE (7.37). The
modes of ∂X and ¯
∂X satisfy
[α m , α n ] = m δ m+n,0 ,
[ ¯
α m , ¯
α n ] = m δ m+n,0 ,
[α m , ¯
α n ] = 0
(7.71)
for all m, n ∈ Z (including the zero-modes). The appearance of the factor m in the
RHS explains the normalization of the number operator (7.63).
From the commutators of the zero-modes, we directly find the ones for the
momentum and winding:
[p, w] = [p, p] = [w, w] = 0,
[p, α n ] = [p, ¯
α n ] = [w, α n ] = [w, ¯
α n ] = 0.
(7.72)
The OPE (7.36) yields
[x L , p L ] = i,
[x R , p R ] = i,
(7.73)
which can be used to determine the commutators of x and q:
[x, p] = [q, w] = i,
[x, w] = [q, p] = 0.
(7.74)
This shows that (x, p) and (q, w) are pairs of conjugate variables. Interestingly, the
winding number w commutes all other modes except q, but the latter disappears
from the description. Hence, it can be interpreted as a number which labels different
representations: if no other principle (like periodicity) forbids w = 0, then one can
expect to have states with all possible w in the spectrum, each value of w forming a
different sector. There are other interpretations from the point of view of T -duality
and double field theory [3, 5, 8, 11].
The commutator of the modes with the Virasoro operators is
[L m , α n ] = −n α m+n .
(7.75)
as expected from (6.115). For m = 0, this reduces to
[L 0 , α −n ] = n α −n ,
(7.76)
which shows that negative modes increase the energy. The commutator of the
creation modes α −n with the number operators is
[N m , α −n ] = α −m δ m,n .
(7.77)
7 CFT Systems
7.1.5 Commutators
The commutators can be computed from (6.75a) knowing the OPE (7.37). The
modes of ∂X and ¯
∂X satisfy
[α m , α n ] = m δ m+n,0 ,
[ ¯
α m , ¯
α n ] = m δ m+n,0 ,
[α m , ¯
α n ] = 0
(7.71)
for all m, n ∈ Z (including the zero-modes). The appearance of the factor m in the
RHS explains the normalization of the number operator (7.63).
From the commutators of the zero-modes, we directly find the ones for the
momentum and winding:
[p, w] = [p, p] = [w, w] = 0,
[p, α n ] = [p, ¯
α n ] = [w, α n ] = [w, ¯
α n ] = 0.
(7.72)
The OPE (7.36) yields
[x L , p L ] = i,
[x R , p R ] = i,
(7.73)
which can be used to determine the commutators of x and q:
[x, p] = [q, w] = i,
[x, w] = [q, p] = 0.
(7.74)
This shows that (x, p) and (q, w) are pairs of conjugate variables. Interestingly, the
winding number w commutes all other modes except q, but the latter disappears
from the description. Hence, it can be interpreted as a number which labels different
representations: if no other principle (like periodicity) forbids w = 0, then one can
expect to have states with all possible w in the spectrum, each value of w forming a
different sector. There are other interpretations from the point of view of T -duality
and double field theory [3, 5, 8, 11].
The commutator of the modes with the Virasoro operators is
[L m , α n ] = −n α m+n .
(7.75)
as expected from (6.115). For m = 0, this reduces to
[L 0 , α −n ] = n α −n ,
(7.76)
which shows that negative modes increase the energy. The commutator of the
creation modes α −n with the number operators is
[N m , α −n ] = α −m δ m,n .
(7.77)