7
CFT Systems
Abstract
This chapter summarizes the properties of some CFT systems. We focus on
the free scalar field and on the first-order bc system (which generalizes the
reparametrization ghosts). For the different systems, we first provide an analysis
on a general curved background before focusing on the complex plane. This is
sufficient to describe the local properties on all Riemann surfaces g ≥ 0.
7.1
Free Scalar
7.1.1 Covariant Action
The Euclidean action of a free scalar X on a curved background g μν is
S =
4ππ 2
d
2 x
√ g g
μν ∂ μ X∂ ν X,
(7.1)
where is a length scale 1 and
:=
+1 spacelike
−1 timelike
,
√
:=
+1 spacelike
i
timelike
(7.2)
denotes the signature of the kinetic term. The field is periodic along σ
X(τ, σ ) ∼ X(τ, σ + 2π).
(7.3)
1 To be identified with the string scale, such that α = 2 .
© 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_7
143
CFT Systems
Abstract
This chapter summarizes the properties of some CFT systems. We focus on
the free scalar field and on the first-order bc system (which generalizes the
reparametrization ghosts). For the different systems, we first provide an analysis
on a general curved background before focusing on the complex plane. This is
sufficient to describe the local properties on all Riemann surfaces g ≥ 0.
7.1
Free Scalar
7.1.1 Covariant Action
The Euclidean action of a free scalar X on a curved background g μν is
S =
4ππ 2
d
2 x
√ g g
μν ∂ μ X∂ ν X,
(7.1)
where is a length scale 1 and
:=
+1 spacelike
−1 timelike
,
√
:=
+1 spacelike
i
timelike
(7.2)
denotes the signature of the kinetic term. The field is periodic along σ
X(τ, σ ) ∼ X(τ, σ + 2π).
(7.3)
1 To be identified with the string scale, such that α = 2 .
© 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_7
143