Solving two-dimensional conformal field theories

This is the text of my habilitation defense, which took place on December 21st 2018. The members of the jury were Denis Bernard, Matthias Gaberdiel, Jesper Jacobsen, Vyacheslav Rychkov, Véronique Terras, Gérard Watts and Jean-Bernard Zuber.

In this habilitation defense, I gave a subjective overview of some recent progress in solving two-dimensional conformal field theories. I discussed what solving means and which techniques can be used. I insisted that there is much to discover about Virasoro-based CFTs, i.e. CFTs that have no symmetries beyond conformal symmetry. I claimed that we should start with CFTs that exist for generic central charges, because they are simpler than CFTs at rational central charges, and can nevertheless include them as special cases or limits. Finally, I argued that in addition to writing research articles, we should use various other media, in particular Wikipedia. 

 

Introduction

Two-dimensional CFTs are defined by the presence of a Virasoro symmetry algebra. This symmetry is sometimes enough for solving CFTs, and even classifying the CFTs that obey some extra conditions. For example, we can classify CFTs whose spaces of states decompose into finitely many irreducible representations of the Virasoro algebra: they are called minimal models. In some cases, Virasoro symmetry is not enough, but the CFT can nevertheless be solved thanks to additional symmetries. In particular, we can have symmetry algebras that contain the Virasoro algebra.
Let me discuss a few CFTs that I find particularly interesting. I will classify them according to their symmetry algebras, and characterize these algebras by the spins of the corresponding chiral fields. In this notation, the Virasoro algebra is \((2)\), as its generators are the modes of the energy-momentum tensor, which has spin \(2\). The sum of the spins of the generators gives you a rough idea of the complexity of an algebra.

Differential equations from fusion rules in 2d CFT

In two-dimensional conformal field theory, correlation functions are partly (and sometimes completely) determined by the properties of the fields under symmetry transformations. In particular, correlation functions of primary fields are relatively simple, because by definition primary fields are killed by the annihilation modes of the symmetry algebra. On top of that, there exist degenerate primary fields that are killed not only by the annihilation modes, but also by some combinations of creation modes. As a result, correlation functions that involve degenerate primary fields sometimes obey nontrivial differential equations, for example BPZ equations. Usually, these equations are deduced from the relevant combinations of creation modes, called null vectors.
Determining null vectors in representations of a symmetry algebra is often complicated, as the algebraic structures of the relevant algebras and of their representations can themselves be complicated. Even in the case of the Virasoro algebra, it is not easy to explicitly determine null vectors. It is however much easier to determine which representations do have null vectors, using the fusion product. For example, if we know degenerate representations \(R_{(1,1)}\) and \(R_{(2,1)}\) with null vectors at levels \(1\) and \(2\) respectively, we can deduce that the fusion product \(R_{(2,1)}\times R_{(2,1)}\) is degenerate and contains \(R_{(1,1)}\). The remainder of \(R_{(2,1)}\times R_{(2,1)}\) must therefore be a degenerate representation, which can be identified as \(R_{(3,1)}\), and has a null vector at level \(3\). (See Section 2.3.1 of my review article for more details.)
An important idea is therefore that it is not the structures of the algebras and representations that matter, but rather the structure of the category of representations, in other words their fusion products. This idea has in particular been developed in the works of Fuchs, Runkel and Schweigert. But how does this help us compute correlation functions, and determine the differential equations that they obey? In other words, can we determine differential equations from fusion products, without computing null vectors?

The light asymptotic limit of $W$ algebra conformal blocks

\(W\) algebras are natural extensions of the Virasoro algebra, the symmetry algebra of local conformal field theories in two dimensions. Conformal field theories with \(W\) algebra symmetry include \(W\) minimal models and conformal Toda theories, which are generalizations of Virasoro minimal models and Liouville theory respectively. In particular, \(sl_N\) conformal Toda theory is based on the \(W_N\) algebra, which has \(N-1\) generators with spins \(2,3,\dots, N\), and reduces to the Virasoro algebra in the case \(N=2\).

The problem of solving conformal Toda theory

 

Solving \(sl_{N\geq 3}\) conformal Toda theory is an outstanding problem. One may think that this is due to the complexity of the \(W_N\) algebra, with its quadratic commutators. I would argue that this is rather due to the complexity of the fusion ring of \(W_{N}\) representations, with its infinite fusion multiplicities. Due to these fusion multiplicities, solving \(sl_N\) conformal Toda theory does not boil down to computing three-point function of primary fields: rather, one should also compute three-point functions of infinitely many descendent fields.

Relations between conformal field theories with affine and $W$-algebra symmetries

This is a commentary of the recent article by Creutzig, Hikida and Ronne, which I was asked to review for the Journal of High-Energy Physics. I am grateful to the authors for helpful correspondence.

It has been known for a long time that $W$-algebras can be obtained from affine Lie algebras by Drinfeld-Sokolov reduction. The reduction eliminates a number of generators of the affine Lie algebra, leaving a $W$-algebra with fewer generators (but more complicated relations). The reductions of algebras are most useful when they can be promoted into relations between correlation functions of conformal field theories. For example, the reduction from the $\widehat{\mathfrak{sl}}_2$ affine Lie algebra, to the Virasoro algebra, can be promoted into a relation between correlation functions of the $H_3^+$ model and Liouville theory, two CFTs whose symmetry algebras are $\widehat{\mathfrak{sl}}_2$ and Virasoro respectively.

Generalizing the $H_3^+$-Liouville relation to models with larger symmetry algebras could be helpful for understanding, or even solving, such models. In order to find such generalizations, there are two approaches: