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
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
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
- 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]
K3 en route From Geometry to Conformal Field Theory
- Proceedings of the 2013 Summer School “Geometric, Algebraic
and Topological Methods for Quantum Field Theory” in Villa de
Leyva, Colombia, World Scientific (2017),
- 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 work
The Conway Moonshine Module is a Reflected K3 Theory
we introduce a procedure which we call
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.
was motivated by the earlier results
of K3 Surfaces and Twined Elliptic Genera
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.