#### Bulg. J. Phys. vol.35 no.s1 (2008), pp. 125-138

#### Infinite Dimensional Lie Algebras in 4D Conformal Quantum Field Theory

Bojko Bakalov

go back^{1}, Nikolay M. Nikolov^{2}, Karl-Henning Rehren^{2,3}, Ivan Todorov^{2}^{1}*Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA*^{2}*Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria*^{3}*Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany*Abstract.The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of 2-dimensional chiral conformal field theory, to a higher (even) dimensional space-time. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields,V, where the_{M}(x,y)span a finite dimensional real matrix algebra $\mathcal{M}$ closed under transposition. The associative algebra $\mathcal{M}$ is irreducible iff its commutant $\mathcal{M}'$ 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_{M}sp(∞,R)corresponding to the fieldRof reals, ofu(∞,∞)associated to the fieldCof complex numbers, and ofsorelated to the algebra^{*}(4∞)Hof quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groupsO(N), U(N), andU(N,H) = Sp(2N), respectively.