6.2 Classical CFTs
111
is a conformal transformation since the metric becomes
ds
2
= dz
d¯ z
=
df
dz
2
dzd¯ z.
(6.21)
However, only holomorphic functions which are globally defined on ¯
C are
elements of the group. At the algebra level, any holomorphic function f (z) regular
in a domain D gives a well-defined transformation in this domain D. Hence, the
algebra is infinite-dimensional. On the other hand, f (z) is only meromorphic on
C generically: it cannot be exponentiated to a group element. We first characterize
the algebra and then obtain the conditions to promote the local transformations to
global ones.
Since the transformations are defined only locally, it is sufficient to consider an
infinitesimal transformation
δz = v(z),
δ ¯
z = ¯
v(¯ z),
(6.22)
where v(z) is a meromorphic vector field on the Riemann sphere. Indeed, the
conformal Killing equation (5.6) in D = 2 is equivalent to the Cauchy–Riemann
equations:
¯
∂v = 0,
∂¯ v = 0.
(6.23)
The vector field admits a Laurent series
v(z) =
n∈Z
v n z
n+1 ,
¯
v(¯ z) =
n∈Z
¯
v n ¯
z
n+1 ,
(6.24)
and the v n and ¯
v n are to be interpreted as the parameters of the transformation. A
basis of vectors (generators) is
n = −z
n+1 ∂ z ,
¯
n = −¯ z
n+1 ∂ ¯
z ,
n∈ Z.
(6.25)
One can check that each set of generators satisfies the Witt algebra
[ m , , n ] = (m − n)) m+n ,
[ ¯
m , ¯
n ] = (m − n) ¯
m+n ,
[ m , ¯
n ] = 0. (6.26)
Since there are two commuting copies of the Witt algebra, it is natural to
extend the ranges of the coordinates from C to C 2 and to consider z and ¯
z as
independent variables. In particular, this gives a natural action of the product algebra
over C 2 . This procedure will be further motivated when studying CFTs since the
holomorphic and anti-holomorphic parts will generally split, and it makes sense to
study them separately. Ultimately, physical quantities can be extracted by imposing
the condition ¯
z = z ∗ at the end (the star is always reserved for the complex
111
is a conformal transformation since the metric becomes
ds
2
= dz
d¯ z
=
df
dz
2
dzd¯ z.
(6.21)
However, only holomorphic functions which are globally defined on ¯
C are
elements of the group. At the algebra level, any holomorphic function f (z) regular
in a domain D gives a well-defined transformation in this domain D. Hence, the
algebra is infinite-dimensional. On the other hand, f (z) is only meromorphic on
C generically: it cannot be exponentiated to a group element. We first characterize
the algebra and then obtain the conditions to promote the local transformations to
global ones.
Since the transformations are defined only locally, it is sufficient to consider an
infinitesimal transformation
δz = v(z),
δ ¯
z = ¯
v(¯ z),
(6.22)
where v(z) is a meromorphic vector field on the Riemann sphere. Indeed, the
conformal Killing equation (5.6) in D = 2 is equivalent to the Cauchy–Riemann
equations:
¯
∂v = 0,
∂¯ v = 0.
(6.23)
The vector field admits a Laurent series
v(z) =
n∈Z
v n z
n+1 ,
¯
v(¯ z) =
n∈Z
¯
v n ¯
z
n+1 ,
(6.24)
and the v n and ¯
v n are to be interpreted as the parameters of the transformation. A
basis of vectors (generators) is
n = −z
n+1 ∂ z ,
¯
n = −¯ z
n+1 ∂ ¯
z ,
n∈ Z.
(6.25)
One can check that each set of generators satisfies the Witt algebra
[ m , , n ] = (m − n)) m+n ,
[ ¯
m , ¯
n ] = (m − n) ¯
m+n ,
[ m , ¯
n ] = 0. (6.26)
Since there are two commuting copies of the Witt algebra, it is natural to
extend the ranges of the coordinates from C to C 2 and to consider z and ¯
z as
independent variables. In particular, this gives a natural action of the product algebra
over C 2 . This procedure will be further motivated when studying CFTs since the
holomorphic and anti-holomorphic parts will generally split, and it makes sense to
study them separately. Ultimately, physical quantities can be extracted by imposing
the condition ¯
z = z ∗ at the end (the star is always reserved for the complex