The search for the exotic : subfactors, conformal field theories and modular tensor categories

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In the early 1980's, Vaughan Jones introduced his subfactor theory in the analysis of von Neumann algebras of operators. This theory has since found deep connections with knot theory in topology and geometry, statistical mechanics and conformal quantum field theory. A subfactor encodes the symmetry of a statistical mechanical model or at the critical temperature a conformal quantum field theory. Dimension is ubiquitous in mathematics. The Jones index measures the relative dimension of the larger algebra over the smaller. K-theory also provides tools for understanding dimension and both of these notions are at the heart of this project.It was thought that exotic models could be produced in the subfactor framework beyond the standard models which had underlying classical group symmetries. However producing such models proved elusive. In 1993 Uffe Haagerup produced a candidate for an exotic subfactor of irrational dimension, namely the Jones index is half of 5 plus the square root of 13 - by essentially producing the Boltzmann weights at criticality by his bare hands with strong integrability properties. However Evans and Gannon have shown that one can understand this subfactor from classical symmetries of an elementary finite group, the symmetries of three objects, and the group of orthogonal transformations in 13 dimensions, producing evidence for an underlying conformal quantum field theory to be described by a conformal net or vertex operator algebra. They have also provided evidence that the Haagerup family belongs to an infinite family and most recently found another related series of quadratic systems, with again strong evidence of underlying conformal field theories based on underlying classical symmetries.This programme is to analyse and shed light on these mysterious subfactors and quadratic systems, the Haagerup systems and related series, thought to be infinite, the near group systems, their common generalizations as quadratic systems. The tools are from operator algebras, particularly subfactors, conformal nets and twisted equivariant K-theory. The objectives are the realization of the underlying modular tensor categories of their doubles or symmetric enveloping algebras and the recovery of the quadratic systems.