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?

Free bosons and Virasoro null vectors

In a recent article, Manabe and Sulkowski have proposed a method for deriving Virasoro null vectors, starting with certain deformed matrix integrals. In this blog post I will look for a conformal field theory interpretation of this method.

 

Quick reminders on Virasoro null vectors.

 

A null vector of the Virasoro algebra is labelled by two integers \(r,s\geq 1\), whose product is the level of the null vector. This null vector occurs in the Verma module with a specific conformal dimension \(\Delta_{r,s}\), and it can be written as
\[|\chi_{r,s}\rangle = L_{r,s} |\Delta_{r,s}\rangle\] where \(|\Delta_{r,s}\rangle\) is the primary state of our Verma module, and \(L_{r,s}\) is a level \(rs\) creation operator.

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: