122
6 Conformal Field Theory on the Plane
The infinitesimal variation of a field under the symmetry generated by Q reads
δ O(z, ¯
z) = −[Q, O(z, ¯
z)] = −
C z
dw
2π i
j (w)O(z, ¯
z) +
C z
d ¯
w
2π i
¯
j( ¯
w)O(z, ¯
z).
(6.80)
The contour integrals are easily evaluated once the OPE between the current and
the operator is known. This formula gives the infinitesimal variation under the
transformation for any field, not only for primaries.
Computation: Equation (6.77)
In real coordinates, the charge is defined by integrating the time component of
the current j μ over space for fixed time (A.23):
Q =
1
2π
dσ j
0 .
The first step is to rewrite this formula covariantly. Since the time is fixed on
the slice, dτ = 0 and one can write
Q =
1
2π
(dσ j
0
− dτ j
1 ) = −
1
2π
μν j
μ dx
ν .
The last formula is valid for any contour. Moreover, it can be evaluated for
complex coordinates:
Q = −
1
2π
z¯ z
j
z d¯ z − j
¯
z dz
= −
i
4π
j
z d¯ z − j
¯
z dz
= −
1
2π i
j z dz − j ¯
z d¯ z
.
One finds a contour integral because τ = cst circles of the cylinder are mapped
to |z| = cst contours.
6.4.2 Operator Product Expansions
The operator product expansion (OPE) is a tool used frequently in CFT: it means
that when two local operators come close to each other, it is possible to replace their
product by a sum of local operators
O i (z i , ¯
z i )O j (z j , ¯
z j ) =
k
c k
ij
z
h i +h j −h k
ij
¯
z
¯
h i + ¯
h j − ¯
h k
ij
O k (z j , ¯
z j ),
(6.81)
6 Conformal Field Theory on the Plane
The infinitesimal variation of a field under the symmetry generated by Q reads
δ O(z, ¯
z) = −[Q, O(z, ¯
z)] = −
C z
dw
2π i
j (w)O(z, ¯
z) +
C z
d ¯
w
2π i
¯
j( ¯
w)O(z, ¯
z).
(6.80)
The contour integrals are easily evaluated once the OPE between the current and
the operator is known. This formula gives the infinitesimal variation under the
transformation for any field, not only for primaries.
Computation: Equation (6.77)
In real coordinates, the charge is defined by integrating the time component of
the current j μ over space for fixed time (A.23):
Q =
1
2π
dσ j
0 .
The first step is to rewrite this formula covariantly. Since the time is fixed on
the slice, dτ = 0 and one can write
Q =
1
2π
(dσ j
0
− dτ j
1 ) = −
1
2π
μν j
μ dx
ν .
The last formula is valid for any contour. Moreover, it can be evaluated for
complex coordinates:
Q = −
1
2π
z¯ z
j
z d¯ z − j
¯
z dz
= −
i
4π
j
z d¯ z − j
¯
z dz
= −
1
2π i
j z dz − j ¯
z d¯ z
.
One finds a contour integral because τ = cst circles of the cylinder are mapped
to |z| = cst contours.
6.4.2 Operator Product Expansions
The operator product expansion (OPE) is a tool used frequently in CFT: it means
that when two local operators come close to each other, it is possible to replace their
product by a sum of local operators
O i (z i , ¯
z i )O j (z j , ¯
z j ) =
k
c k
ij
z
h i +h j −h k
ij
¯
z
¯
h i + ¯
h j − ¯
h k
ij
O k (z j , ¯
z j ),
(6.81)