M MATRIX Research Papers - Academia.edu (original) (raw)

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 \MA closed under transposition. The associative algebra \MA is irreducible iff its commutant \MA^{\prime} 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 {sp}(\infty,{\bb R}) corresponding to the field {{\bb R}} of reals, of u(∞, ∞) associated with the field {{\bb C}} of complex numbers, and of so*(4∞) related to the algebra {{\bb H}} of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and U(N,{{\bb H}})=Sp(2N) , respectively. 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.