6
Conformal Field Theory on the Plane
Abstract
Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the
complex plane (or equivalently the sphere). We start by describing the geometry
of the sphere and the relation to the complex plane and to the cylinder, in order
to make contact with the string worldsheet. Then, we discuss classical CFTs and
the Witt algebra obtained by classifying the conformal isometries of the complex
plane. Then, we describe quantum CFTs and introduce the operator formalism.
This last section is the most important for this book as it includes information on
the operator product expansion, Hilbert space, Hermitian and BPZ conjugations.
As described at the beginning of Chap. 5, we use spacetime notations for
the coordinates, but follow otherwise the normalization for the worldsheet. In
particular, integrals are normalized by 2π . However, the spatial coordinate on
the cylinder is still written as σ to avoid confusions: x μ = (τ, σ ).
6.1
The Riemann Sphere
6.1.1 Map to the Complex Plane
The Riemann sphere 0 , which is diffeomorphic to the unit sphere S 2 , has genus
g = 0 and is thus the simplest Riemann surface. Its most straightforward description
is obtained by mapping it to the extended 1 complex plane ¯
C (also denoted ˆ
C), which
is the complex plane z ∈ C to which the point at infinity z = ∞ is added:
¯
C = C ∪ {∞}.
(6.1)
1 This qualification will often be omitted.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_6
105
Conformal Field Theory on the Plane
Abstract
Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the
complex plane (or equivalently the sphere). We start by describing the geometry
of the sphere and the relation to the complex plane and to the cylinder, in order
to make contact with the string worldsheet. Then, we discuss classical CFTs and
the Witt algebra obtained by classifying the conformal isometries of the complex
plane. Then, we describe quantum CFTs and introduce the operator formalism.
This last section is the most important for this book as it includes information on
the operator product expansion, Hilbert space, Hermitian and BPZ conjugations.
As described at the beginning of Chap. 5, we use spacetime notations for
the coordinates, but follow otherwise the normalization for the worldsheet. In
particular, integrals are normalized by 2π . However, the spatial coordinate on
the cylinder is still written as σ to avoid confusions: x μ = (τ, σ ).
6.1
The Riemann Sphere
6.1.1 Map to the Complex Plane
The Riemann sphere 0 , which is diffeomorphic to the unit sphere S 2 , has genus
g = 0 and is thus the simplest Riemann surface. Its most straightforward description
is obtained by mapping it to the extended 1 complex plane ¯
C (also denoted ˆ
C), which
is the complex plane z ∈ C to which the point at infinity z = ∞ is added:
¯
C = C ∪ {∞}.
(6.1)
1 This qualification will often be omitted.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_6
105