Mathematical definition of conformal field theory

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:

Katrin Wendland
On the geometry of singularities in quantum field theory
  • Proceedings of the International Congress of Mathematicians Hyderabad, August 19-27, 2010
  • Hindustan Book Agency (2010), 2144-2170
Katrin Wendland
Mathematical Foundations of Conformal Field Theory and Applications
  • Proceedings of the COPROMAPH International School, Cotonou, Benin, October 26 - November 4, 2012, ICMPA-UNESCO Chair, 180-223
Katrin Wendland
  • Mathematical Aspects of Quantum Field Theories, D. Calaque and Th. Strobl, eds.
  • Mathematical Physics Studies, Springer 2015, pp. 89-129
  • arXiv:1404.3108 [hep-th]
Katrin Wendland
K3 en route From Geometry to Conformal Field Theory
  • lecture notes for the author's contribution to the 2013 Summer School “Geometric, Algebraic and Topological Methods for Quantum Field Theory” in Villa de Leyva, Colombia
  • arXiv:1503.08426 [math.DG]
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 preprint
Anne Taormina, Katrin Wendland
The Conway Moonshine Module is a Reflected K3 Theory
we have developed a procedure that we call reflection which 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. For a particular K3 theory, we show that it allows reflection, and that the resulting super vertex operator algebra plus admissible module is the Conway Moonshine Module. The study was motivated by the earlier results Derived Equivalences of K3 Surfaces and Twined Elliptic Genera of John Duncan and Sander Mack-Crane, who discovered that the respective spaces of states agree as Virasoro modules.