Conformal Field Theory as a Nuclear Functor

We consider Segal’s categorical approach to conformal field theory (CFT). Segal constructed a category whose objects are finite families of circles, and whose morphisms are Riemann surfaces with boundary compatible with the families of circles in the domain and codomain. A CFT is then defined to be a functor to the category of Hilbert spaces, preserving the appropriate structure. In particular, morphisms in the geometric category must be sent to trace class maps. However, Segal’s approach is not quite categorical, as the geometric structure he considers has an associative composition, but lacks identities. We begin by demonstrating that an appropriate method of dealing with the lack of identities in this situation is the notion of nuclear ideal, as defined by Abramsky and the first two authors. More precisely, we show that Segal’s structure is contained in a larger category as a nuclear ideal. While it is straightforward to axiomatize categories without identities, the theory of nuclear ideals further captures the idea of the identity as a singular object. An excellent example of a singular identity to keep in mind is the Dirac delta “function.” We argue that this sort of singularity is precisely what is occurring in conformal field theory. We then show that Segal’s definition of CFT can be defined as a nuclear functor to the category of Hilbert spaces and bounded linear maps, equipped with its nuclear structure of Hilbert-Schmidt maps. As a further example, we examine Neretin’s notion of correct linear relation (CLR), and show that it also contains a nuclear ideal. We then present Neretin’s construction of a functor from the geometric category to CLR, the category of Hilbert spaces and correct linear relations, and show that it is a nuclear functor and hence a generalized CFT. We conclude by noting that composition in Neretin’s category can also be seen in Girard’s geometry of interaction construction.