Infinite-dimensional Lie algebras in 4D conformal quantum field theory. (English) Zbl 1139.81052
Summary: 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, , where the 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 associated with the field of complex numbers, and of related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups and , respectively.
|81T40||Two-dimensional field theories, conformal field theories, etc.|
|81R10||Infinite-dimensional groups and algebras motivated by physics|
|22E70||Applications of Lie groups to physics; explicit representations|
|81R12||Relations of groups and algebras in quantum theory with integrable systems|