Symmetries in conformal field theory, operator algebras
and noncommutative geometry
We will present various symmetries appearing in the
operator algebraic approach to conformal field theory.
(1) The Moonshine conjecture connects elliptic modular functions
and the Monster group and is usually studied with theory of vertex
operator algebras. We present how to study this structure using
(2) We present classification theory of (super) conformal field
theories using representation theory of operator algebras. A
certain quantum group type symmetry plays an important role here.
A braided tensor category in the setting of the Jones theory of
subfactors appears here.
(3) Based on an analogy between the conformal Hamiltonian in
chiral conformal field theory and the Laplacian in classical
geometry, we study conformal field theory in the framework of
noncommutative geometry. We show how the entire cyclic cohomology
of "infinite dimensional noncommutative manifolds" enters the
framework of (2).