146
7 CFT Systems
factor of 2π comes from the normalization of the spatial integral. We have inserted
in order to interpret the conserved charge p as a component of the momentum
contravariant vector in string theory.
To make contact with string theory, consider D scalar fields X a (x μ ). Then, the
current becomes
J
μ
a =
i
2ππ 2 η ab ∂
μ X
b ,
(7.21)
where the position of the indices is in agreement with the standard form of Noether’s
formula (A.25a) (a current has indices in opposite locations as the parameters and
fields). Since we have η 00 = −1 = X 0 , we find that J 0μ = X 0 J
μ
0 has no epsilon
after replacing the expression (7.17) of J
μ
0 .
The transformation X a → X a + c a is a global translation in target spacetime:
the charge p a is identified with the spacetime momentum. The factor of i indicates
that p a is the Euclidean contravariant momentum vector by comparison with (A.7).
The convention of this section is to always work with quantities which will
become contravariant vector to avoid ambiguity.
7.1.2 Action on the Complex Plane
In complex coordinates, the action on flat space reads
S =
2ππ 2
dzd¯ z ∂ z X∂ ¯
z X,
(7.22)
giving the equation of motion:
∂ z ∂ ¯
z X = 0.
(7.23)
This indicates that ∂ z X and ∂ ¯
z X are, respectively, holomorphic and antiholomorphic such that
X(z, ¯
z) = X L (z) + X R (¯ z),
(7.24)
and we will remove the subscripts when there is no ambiguity (for example, when
the position dependence is written):
X(z) := X L (z),
X(¯ z) := X R (¯ z).
(7.25)
It looks like X L (z) and X R (¯ z) are unrelated, but this is not the case because of the
zero-mode, as we will see below.
7 CFT Systems
factor of 2π comes from the normalization of the spatial integral. We have inserted
in order to interpret the conserved charge p as a component of the momentum
contravariant vector in string theory.
To make contact with string theory, consider D scalar fields X a (x μ ). Then, the
current becomes
J
μ
a =
i
2ππ 2 η ab ∂
μ X
b ,
(7.21)
where the position of the indices is in agreement with the standard form of Noether’s
formula (A.25a) (a current has indices in opposite locations as the parameters and
fields). Since we have η 00 = −1 = X 0 , we find that J 0μ = X 0 J
μ
0 has no epsilon
after replacing the expression (7.17) of J
μ
0 .
The transformation X a → X a + c a is a global translation in target spacetime:
the charge p a is identified with the spacetime momentum. The factor of i indicates
that p a is the Euclidean contravariant momentum vector by comparison with (A.7).
The convention of this section is to always work with quantities which will
become contravariant vector to avoid ambiguity.
7.1.2 Action on the Complex Plane
In complex coordinates, the action on flat space reads
S =
2ππ 2
dzd¯ z ∂ z X∂ ¯
z X,
(7.22)
giving the equation of motion:
∂ z ∂ ¯
z X = 0.
(7.23)
This indicates that ∂ z X and ∂ ¯
z X are, respectively, holomorphic and antiholomorphic such that
X(z, ¯
z) = X L (z) + X R (¯ z),
(7.24)
and we will remove the subscripts when there is no ambiguity (for example, when
the position dependence is written):
X(z) := X L (z),
X(¯ z) := X R (¯ z).
(7.25)
It looks like X L (z) and X R (¯ z) are unrelated, but this is not the case because of the
zero-mode, as we will see below.