Introduction to Conformal Field Theory: With Applications to String ...

Chapter 2
Basics in Conformal Field Theory
The approach for studying conformal field theories is somewhat different from the
usual approach for quantum field theories. Because, instead of starting with a classical
action for the fields and quantising them via the canonical quantisation or the
path integral method, one employs the symmetries of the theory. In the spirit of
the so-called boot-strap approach, for CFTs one defines and for certain cases even
solves the theory just by exploiting the consequences of the symmetries. Such a procedure
is possible in two dimensions because the algebra of infinitesimal conformal
transformations in this case is very special: it is infinite dimensional.
In this chapter, we will introduce the basic notions of two-dimensional conformal
field theory from a rather abstract point of view. However, in Sect. 2.9, we will study
in detail three simple examples important for string theory which are given by a
Lagrangian action.
2.1 The Conformal Group
We start by introducing conformal transformations and determining the condition
for conformal invariance. Next, we are going to consider flat space in d ≥ 3dimensions
and identify the conformal group. Finally, we study in detail the case
of Euclidean two-dimensional flat space R 2,0 and determine the conformal group
and the algebra of infinitesimal conformal transformations. We also comment on
two-dimensional Minkowski space R 1,1 in the end.
2.1.1 Conformal Invariance
Conformal Transformations
Let us consider a flat space in d dimensions and transformations thereof which locally
preserve the angle between any two lines. Such transformations are illustrated
in Fig. 2.1 and are called conformal transformations.
In more mathematical terms, a conformal transformation is defined as follows.
Let us consider differentiable maps ϕ : U → V , where U ⊂ M and V ⊂ M ′ are
Blumenhagen, R., Plauschinn, E.: Basics in Conformal Field Theory. Lect. Notes Phys. 779, 5–86
(2009)
DOI 10.1007/978-3-642-00450-6 2 c○ Springer-Verlag Berlin Heidelberg 2009