148
7 CFT Systems
The other primary operators of the theory are given by the vertex operators
V k (z): 3
V k (z, ¯
z) := : e
ikX(z,¯ z)
:.
(7.33)
Remark 7.2 In fact, it is possible to introduce more general vertex operators
V k L ,k R (z, ¯
z) := : e
2i
k L X(z)+k R X(¯ z)
:,
(7.34)
but we will not consider them in this book.
Remark 7.3 (Plane and Cylinder Coordinates) The action in w-coordinate (cylinder) takes the same form as a result of the conformal invariance of the scalar field,
which in practice results from the cancellation between the determinant and inverse
metric. As a consequence, every quantity derived from the classical action (equation
of motion, energy–momentum tensor. . . ) will have the same form in both coordinate
systems: we will focus on the z-coordinate, writing the w-coordinate expression
when it is insightful to compare. This is not anymore the case at the quantum
level: anomalies may translate into anomalous tensor transformations such that it is
necessary to keep track of the surface on which the tensor is defined. To distinguish
between the plane and cylinder quantities, an index “cyl” is added when necessary
(by convention, all quantities without further specification are on the plane).
7.1.3 OPE
The OPE between X and itself is directly found from the propagator:
X(z)X(w) ∼ −
2
2
ln(z − w).
(7.35)
By successive derivations, one finds the OPE between X and ∂X
∂X(z)X(w) ∼ −
2
2
1
z − w
,
(7.36)
and between ∂X with itself
∂X(z)∂X(w) ∼ −
2
2
1
(z − w) 2 .
(7.37)
The invariance under the permutation of z and w reflects the fact that X is bosonic
and that both operators in (7.37) are identical.
3 The in the exponential is consistent with interpreting X and k as a contravariant vector.
7 CFT Systems
The other primary operators of the theory are given by the vertex operators
V k (z): 3
V k (z, ¯
z) := : e
ikX(z,¯ z)
:.
(7.33)
Remark 7.2 In fact, it is possible to introduce more general vertex operators
V k L ,k R (z, ¯
z) := : e
2i
k L X(z)+k R X(¯ z)
:,
(7.34)
but we will not consider them in this book.
Remark 7.3 (Plane and Cylinder Coordinates) The action in w-coordinate (cylinder) takes the same form as a result of the conformal invariance of the scalar field,
which in practice results from the cancellation between the determinant and inverse
metric. As a consequence, every quantity derived from the classical action (equation
of motion, energy–momentum tensor. . . ) will have the same form in both coordinate
systems: we will focus on the z-coordinate, writing the w-coordinate expression
when it is insightful to compare. This is not anymore the case at the quantum
level: anomalies may translate into anomalous tensor transformations such that it is
necessary to keep track of the surface on which the tensor is defined. To distinguish
between the plane and cylinder quantities, an index “cyl” is added when necessary
(by convention, all quantities without further specification are on the plane).
7.1.3 OPE
The OPE between X and itself is directly found from the propagator:
X(z)X(w) ∼ −
2
2
ln(z − w).
(7.35)
By successive derivations, one finds the OPE between X and ∂X
∂X(z)X(w) ∼ −
2
2
1
z − w
,
(7.36)
and between ∂X with itself
∂X(z)∂X(w) ∼ −
2
2
1
(z − w) 2 .
(7.37)
The invariance under the permutation of z and w reflects the fact that X is bosonic
and that both operators in (7.37) are identical.
3 The in the exponential is consistent with interpreting X and k as a contravariant vector.