Infinite-dimensional Lie algebras in 4D conformal quantum field theory**Lecture at the workshops ‘Lie Theory and Its Applications in Physics’, 18–24 June 2007, Varna, Bulgaria; ‘Infinite- Dimensional Algebras and Quantum Integrable Systems’, 23–27 July, 2007, Faro, Portugal; and ‘Supersymmetries and Quantum Symmetries’, 30 July–4 August, 2007, Dubna, Russia.

Abstract

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra closed under transposition. The associative algebra is irreducible iff its commutant coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of corresponding to the field of reals, of u(∞, ∞) associated with the field of complex numbers, and of so*(4∞) related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and , respectively.