6.2 Classical CFTs
115
dimension and spin s:
:= h + ¯
h,
s := h − ¯
h.
(6.46)
The conformal weights correspond to the charges of the operator under 0 and ¯
0 .
We will use “(h, ¯
h) (quasi-)primary” as a synonym of “(quasi-)primary field with
conformal weight (h, ¯
h)”.
Remark 6.2 (Complex Conformal Weights) While we consider h, ¯
h ∈ R, and more
specifically h, ¯
h ≥ 0 for a unitary theory (which is the case of string theory except
for the reparametrization ghosts), theories with h, ¯
h ∈ C make perfectly sense. One
example is the Liouville theory with complex central charge c ∈ C [31, 33] (central
charges are defined below, see (6.58)).
Primaries and quasi-primaries are hence operators which have nice transformations, respectively, under the algebra and group. Obviously, a primary is also
a quasi-primary. These transformations are similar to those of a tensor with h
holomorphic and ¯
h anti-holomorphic indices (Sect. 4.1). Another point of view is
that the object
O(z, ¯
z) dz
h d¯ z
¯
h
(6.47)
is invariant under local / global conformal transformations.
The notation f ◦ O indicates the complete change of coordinates, including
the tensor transformation law and the possible corrections if the operator is not
primary. 6 For a primary field, we have
f ◦ O(z, ¯
z) := f
(z)
h ¯
f
(¯ z)
¯
h
O
f (z), ¯
f (¯ z)
.
(6.48)
We stress that it does not correspond to function composition.
Under an infinitesimal transformation
δz = v(z),
δ ¯
z = ¯
v(¯ z),
(6.49)
a primary operator changes as
δO(z, ¯
z) = (h ∂v + v ∂)O(z, ¯
z) + ( ¯
h ¯
∂ ¯
v + ¯
v ¯
∂)O(z, ¯
z).
(6.50)
The transformation of a non-primary field contains additional terms, see, for
example, (6.89).
Remark 6.3 (Higher-Genus Riemann Surfaces) According to Remark 5.1, all Riemann surfaces g share the same conformal algebra since locally they are all subsets
6 In fact, one has f ◦ O := f ∗ O in the notations of Chap. 2.
115
dimension and spin s:
:= h + ¯
h,
s := h − ¯
h.
(6.46)
The conformal weights correspond to the charges of the operator under 0 and ¯
0 .
We will use “(h, ¯
h) (quasi-)primary” as a synonym of “(quasi-)primary field with
conformal weight (h, ¯
h)”.
Remark 6.2 (Complex Conformal Weights) While we consider h, ¯
h ∈ R, and more
specifically h, ¯
h ≥ 0 for a unitary theory (which is the case of string theory except
for the reparametrization ghosts), theories with h, ¯
h ∈ C make perfectly sense. One
example is the Liouville theory with complex central charge c ∈ C [31, 33] (central
charges are defined below, see (6.58)).
Primaries and quasi-primaries are hence operators which have nice transformations, respectively, under the algebra and group. Obviously, a primary is also
a quasi-primary. These transformations are similar to those of a tensor with h
holomorphic and ¯
h anti-holomorphic indices (Sect. 4.1). Another point of view is
that the object
O(z, ¯
z) dz
h d¯ z
¯
h
(6.47)
is invariant under local / global conformal transformations.
The notation f ◦ O indicates the complete change of coordinates, including
the tensor transformation law and the possible corrections if the operator is not
primary. 6 For a primary field, we have
f ◦ O(z, ¯
z) := f
(z)
h ¯
f
(¯ z)
¯
h
O
f (z), ¯
f (¯ z)
.
(6.48)
We stress that it does not correspond to function composition.
Under an infinitesimal transformation
δz = v(z),
δ ¯
z = ¯
v(¯ z),
(6.49)
a primary operator changes as
δO(z, ¯
z) = (h ∂v + v ∂)O(z, ¯
z) + ( ¯
h ¯
∂ ¯
v + ¯
v ¯
∂)O(z, ¯
z).
(6.50)
The transformation of a non-primary field contains additional terms, see, for
example, (6.89).
Remark 6.3 (Higher-Genus Riemann Surfaces) According to Remark 5.1, all Riemann surfaces g share the same conformal algebra since locally they are all subsets
6 In fact, one has f ◦ O := f ∗ O in the notations of Chap. 2.