The aim of this post is to introduce the topic of Conformal Field Theories (CFTs) in string theory. In general, CFTs allow us to describe a number of systems in different areas of physics. To list one example, conformal invariance plays an important role in condensed matter physics, particularly in the context of second order phase transitions in which the critical behaviour of systems may be described. But as we are focused on the stringy case, we may motivate the study of CFTs as follows: 2-dimensional CFTs prove very important when it comes to the study of the physical dynamics of the worldsheet.
In past posts we already observed, for instance, how the internal modes along the string relate to conformal transformations. Indeed, upon fixing the worldsheet diffeomorphism plus Weyl symmetries, the result is precisely a CFT. There are many other topics that leverage the conformal symmetry of the worldsheet theory, including how we describe string-on-string interactions and how we compute scattering amplitudes. But perhaps one of the ultimate motivational factors is that, as an essential tool in perturbative string theory, CFTs enable the study of the quantum field theory of the worldsheet. There is also the added benefit that many CFTs are completely solvable.
2. Conformal Group in d-dimensions
Before we proceed with a study of conformal field theories (beginning with Chapter 2 in Polchinski), it is useful to first think generally about the conformal group and its algebra.
Formally, a CFT is a quantum field theory that is invariant under the conformal group. To give some geometric intuition, the conformal group may be described as follows: it is the set of transformations that preserve local angles but not necessarily distances. This may also be thought of as invariance under scaling, with a conformal mapping being quite simply a biholomorphic mapping.
We may give further intuition about the conformal group by revisiting a more familiar symmetry group. Recall in previous chapters a discussion about the Poincaré group. One will remember that transformations under the D-dimensional Poincaré group combine translations and Lorentz transformations. These may be thought of as symmetries of flat spacetime, such that the flat metric is left invariant.
The conformal group includes the Poincaré group, with the addition of extra spacetime symmetries. It has already been alluded, for example, that a type of conformal transformation is a scale transformation, in which we may act by zooming in and out of some region of spacetime. This extra spacetime symmetry is an act of rescaling.
More precisely, the conformal group may be thought of as the subgroup of the group of general coordinate transformations (or diffeomorphisms). Consider the following. If one has a metric (which is a 2-tensor) in d-dimensional spacetime, it follows that under the change of coordinates , we have a transformation of the general form
Now, let us consider some function of the spacetime coordinates. If a conformal transformation is a change of coordinates such that the metric changes by an overall factor, then we may consider how the metric transforms as
For some scaling factor . This is a conformal transformation of the metric. Hence why there is preservation of angles but not lengths. As this particular subgroup of coordinate transformations preserve angles while distorting lengths, in studying how to construct conformally invariant theories we will learn that conformal systems do not possess definitions of scale with respect to intrinsic length, mass or energy. For these reasons one might say the working physics is somewhat constrained or confined, such that there is no induction of a reference scale in the purest sense of the word. This is also why, in our case, CFTs prove interesting: they lend themselves quite naturally to the study of massless excitations.
Now, in thinking again of the conformal transformation described in (4.2), another important and directly related point concerns a description of the metric. It is common in the literature that the background is flat. It also turns out – and this will become more apparent later on – the background metric can either be fixed or dynamical (Tong, p.61). In the future, as we work in the Polyakov formalism, the metric is dynamical and, in this case, the transformation is a diffeomorphism – not just a gauge symmetry, but a residual gauge symmetry which, we will learn, can be undone by a Weyl transformation. But before that, in simpler examples, the background metric will be fixed and so the transformation will be representative of a global symmetry. In this case of a fixed metric, the transformation should be thought of as a genuine physical symmetry, and this global symmetry contains corresponding conserved currents. The corresponding charges for these currents are the Virasoro generators, which is something we will study later on.
Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction”.
Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.
David Tong. (2009). “String Theory” [lecture notes].
Kevin Wray. (2009). “An Introduction to String Theory” [lecture notes].