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Zbl 1139.81052
Bakalov, Bojko; Nikolov, Nikolay M.; Rehren, Karl-Henning; Todorov, Ivan
Infinite-dimensional Lie algebras in 4D conformal quantum field theory. (English)
[J] J. Phys. A, Math. Theor. 41, No. 19, Article ID 194002, 12 p. (2008). ISSN 1751-8113; ISSN 1751-8121/e

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 $\cal M $ closed under transposition. The associative algebra $\cal M $ is irreducible iff its commutant $\cal M^{\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 $\text{sp}(\infty,{\mathbb R}) $ corresponding to the field ${{\mathbb R}} $ of reals, of $u(\infty , \infty )$ associated with the field ${{\mathbb C}} $ of complex numbers, and of $so^*(4\infty )$ related to the algebra ${{\mathbb 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,{{\mathbb H}})= \text{Sp}(2N) $, respectively.

MSC 2000:: *81T40 Two-dimensional field theories, etc.
81R10 Repres. of infinite-dim. groups and algebras from quantum theory
22E70 Appl. of Lie groups to physics
81R12 Relations with integrable systems

Cited in: Zbl 1202.81188 Zbl 1165.81028

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