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 operator algebras.

(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).