In Chapter 2 of Polchinski, we begin our engagement with Conformal Field Theory (CFT). On first read, I found this to be one of the more interesting chapters in Volume 1. It is densely packed, with many subtleties and detailed workings.

But before we start to dig into CFTs in the context of bosonic string theory, perhaps it might be useful to first address the question of motivation. One may have asked, ‘why study CFTs?’ A more general answer would point to the fact that many CFTs are completely solvable. But an answer particular to what we have so far been developing alongside Polchinski’s textbook points in the direction of how, practically speaking, 2-dim CFTs prove very important when it comes to the study of the physical dynamics of the worldsheet (WS). More emphatically, the subject of conformal field theory provides essential tools that will enable us to go on and systematically study string theory.

In general and broad terms, CFTs allow one 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. But as emphasis here is on the stringy case, we will focus on how the WS theory that we’ve been developing is a 2-dim CFT. In what way is this true? For example we will see how, upon fixing the WS diffeomorphism plus Weyl symmetries, the result is precisely a conformal field theory.

Another benefit of working in the context of a CFT, concerns how theories which are invariant under conformal transformations are also invariant under scalings. This is really, I think, the essence of the motivation to study CFTs in a stringy context.

Conformal transformations

To give some geometric intuition, we’re talking of the preservation of angles between vectors (not distances). And so, we might add how a scaling transformation is part of the conformal group where, upon some change of coordinates X^{\mu} \rightarrow \tilde{X}^{\mu}(x) the metric also changes as (Tong, p.61),

    \[ g_{\alpha \beta}(\sigma) \rightarrow \omega^2(\sigma)g_{\alpha \beta}(\sigma) \]


This is a conformal transformation. As Tong summarises, “A conformal field theory is a field theory invariant under these transformations.”

We can state this is another way. If the conformal group is one that preserves angles, we can give this statement mathematical definition by understanding that if we have some metric g_{\alpha \beta}(x), it follows that under the coordinate change X^{\mu}(\sigma) \rightarrow X^{\prime \mu}(\sigma), the metric transforms as

    \[ g_{\mu \nu}(\sigma) \rightarrow g_{\mu \nu}^{\prime}(\sigma^{\prime}) = \frac{\partial \sigma^{\alpha}}{\partial \sigma^{\prime \mu}} \frac{\partial \sigma^{\beta}}{\sigma^{\prime \nu}}g_{\alpha \beta}(f(\sigma^{\prime})) \]


Where our metric is a 2-tensor and where we’re working in D-dim spacetime.

The key message is that the transformation preserves scale up to some point. This can also be stated more formally, noting that the conformal group is a subgroup of coordinate transformations. These particular coordinate transformations have the unique property in which the metric remains unchanged. So, in simple terms,the metric and angles are preserved; but length, of course, changes.

Thus, it is common in the literature, including in Polchinski, to read some version of the statement that: the conformal group displays the manner in which, upon coordinate transformation, the metric remains unchanged up to some scale factor. Mathematically, we thus have something of the form

    \[ g_{\mu \nu}(\sigma) \rightarrow g_{\mu \nu}^{\prime}(\sigma^{\prime}) = \omega(\sigma)g_{\mu \nu}(\sigma) \]

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. Thus, also, 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. So it can be seen why CFTs are useful when one wants to study massless excitations.

Additionally, it is perhaps contextually useful to note that all QFTs are essentially perturbations of CFTs. For QFTs that have conformal symmetry, we might provide a mathematically rigorous and certainly effective description of a system near criticality.

Now, in thinking again of the conformal transformation described above, another important and directly related point concerns a description of the metric. It 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, as Polchinski notes, can be undone by 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. It is also common in the literature that the background is flat.

In the next entry, we will look at the generators of conformal transformations.

References

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].