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?