88. Belavin and Zamolodchikov on 2D quantum gravity

Both people are among inventors of conformal field theory, string theory and the chapter of field theory that is called “integrable systems” nowadays, so naturally, one cannot help taking an hour of her time and learn what each of them has new to say. But if they are co-authors of the same paper, the probability for me to try studying their paper doubles!

Let me first recall a couple of facts I have learned studying 2D quantum gravity. First of all, there are no degrees of freedom in 2D quantum gravity at the classical level (compare it to 3D QG, where there are no dynamical degrees of freedom): by a coordinate transformation one can always choose metric to be .  Nevertheless, some dynamics appears at the quantum level, as Polyakov teaches us: due to conformal anomaly the dynamical Liouville mode appears, and the latter can be in principle coupled to the matter fields living in fluctuating 2D manifold.

So, what Belavin and Zamolodchikov want to study this time is 2D minimal quantum gravity. The word “minimal” in this context means that a minimal conformal field theory lives on a 2D manifold with fluctuating metric, and we want to understand how it interacts with Liouville mode. Correlation properties of the minimal CFTs are well known, and this allows one also to better understand how 2D gravity behaves.

There is another, descrete, approach to 2D gravity known as matrix models – in this approach, a fluctuating 2D geometry is represented as an ensemble of planar graphs. The continuum limit in this case (the one we are ultimately interested in the previous approach) is achieved in the situation where graphs of very large sizes dominate (as usual, to check whether the continuum limit exists, we have to perform RG analysis of the model and make sure that irrelevant operators remain irrelevant ).

These two approaches are very different, but they are supposed describe the same physics as was explicitly checked in several examples. Still, at this point the complete mapping between two languages is lacking, and it is great to see any progress regarding this issue.

The subject of Belavin-Zamolodchikov’s study is n-point “correlation numbers”

where

are integrals of local densities of some quantitites (functions of matter fields and metric) over the manifold. While one-and two-point corr. numbers are already checked to be in one-to-one correspondence for the minimal Liouville gravity and p-critical matrix model, the authors compare tree- and four-correlation numbers and show that they are also in perfect agreement.