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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, V M (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 sp(,) 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 U(N,)=Sp(2N), respectively.

MSC:
81T40Two-dimensional field theories, conformal field theories, etc.
81R10Infinite-dimensional groups and algebras motivated by physics
22E70Applications of Lie groups to physics; explicit representations
81R12Relations of groups and algebras in quantum theory with integrable systems