5
Conformal Symmetry in D Dimensions
Abstract
Starting with this chapter, we discuss general properties of conformal field
theories (CFT). The goal is not to be exhaustive, but to provide a short
introduction and to gather the concepts and formulas that are needed for string
theory. However, the subject is presented as a standalone topic such that it can be
of interest for a more general public.
The conformal group in any dimension is introduced in this chapter. The
specific case D = 2, which is the most relevant for the current book, is developed
in the following chapters.
5.1
CFT on a General Manifold
In this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime
M, independently from string theory (there is no reference to a target spacetime).
As such, we will use spacetime notations together with some simplifications:
coordinates are written as x μ with μ = 0, . . . , D − 1 and time is written as x 0 = t
(x 0 = τ ) in Lorentzian (Euclidean) signature.
Given a metric g μν on a D-dimensional manifold M, the conformal group
CISO(M) is the set of coordinate transformations (called conformal symmetries
or isometries)
x
μ
−→ x
= x
(x)
(5.1)
which leaves the metric invariant up to an overall scaling factor:
g μν (x) −→ g
μν (x
) =
∂x ρ
∂x
∂x σ
∂x g ρσ (x) =
)
2 g μν (x
).
(5.2)
© 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_5
101
Conformal Symmetry in D Dimensions
Abstract
Starting with this chapter, we discuss general properties of conformal field
theories (CFT). The goal is not to be exhaustive, but to provide a short
introduction and to gather the concepts and formulas that are needed for string
theory. However, the subject is presented as a standalone topic such that it can be
of interest for a more general public.
The conformal group in any dimension is introduced in this chapter. The
specific case D = 2, which is the most relevant for the current book, is developed
in the following chapters.
5.1
CFT on a General Manifold
In this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime
M, independently from string theory (there is no reference to a target spacetime).
As such, we will use spacetime notations together with some simplifications:
coordinates are written as x μ with μ = 0, . . . , D − 1 and time is written as x 0 = t
(x 0 = τ ) in Lorentzian (Euclidean) signature.
Given a metric g μν on a D-dimensional manifold M, the conformal group
CISO(M) is the set of coordinate transformations (called conformal symmetries
or isometries)
x
μ
−→ x
= x
(x)
(5.1)
which leaves the metric invariant up to an overall scaling factor:
g μν (x) −→ g
μν (x
) =
∂x ρ
∂x
∂x σ
∂x g ρσ (x) =
)
2 g μν (x
).
(5.2)
© 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_5
101