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

Infinite Dimensional Lie Algebras in 4D Conformal Quantum Field Theory

Bojko Bakalov1, Nikolay M. Nikolov2, Karl-Henning Rehren2,3, Ivan Todorov2
1Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA
2Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria
3Institut 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, VM(x,y), where the M 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 sp(∞,R) corresponding to the field R of reals, of u(∞,∞) associated to the field C of complex numbers, and of so*(4∞) related to the algebra 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,H) = Sp(2N), respectively.

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