Since conformal field theory is at the heart of most of my research projects, the apparent lack of mathematical literature on the subject is problematic. The various approaches to conformal field theory on the market are either not accepted as mathematically sound theories, or their applications in string theory have not been successful, so far. I have therefore developed an axiomatic formulation of conformal field theory, to be presented in a book, which is now close to completion. I have a preliminary contract for publication of this book in the AMS-Series Graduate Studies in Mathematics. Summaries of my approach have already been published:

An ingredient to conformal field theory, which by now has become mathematically well established, are the so-called vertex operator algebras. Although they only allow to capture a small part of the structure of a conformal field theory, it is useful to clarify the relationship between the two. In our work we introduce a procedure which we call reflection and which gives the precise relationship between the two models. Reflection transforms certain superconformal field theories into super vertex operator algebras plus their admissible modules. This builds a bridge between the two, which may be used in both directions, allowing vertex algebraists access to the realm of full-fledged superconformal field theories. The study was motivated by the earlier results Derived Equivalences of K3 Surfaces and Twined Elliptic Genera of Duncan and Mack-Crane, who discovered that the respective spaces of states agree as Virasoro modules. Our construction of reflection has since been applied in simpler examples as well, for instance by Anagiannis, Cheng, Duncan and Volpato, who however restrict their attention to toroidal superconformal field theories with c=6.