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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 \(\mathcal M \) closed under transposition. The associative algebra \(\mathcal M \) is irreducible iff its commutant \(\mathcal M^{\prime} \) 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:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E70 Applications of Lie groups to the sciences; explicit representations
81R12 Groups and algebras in quantum theory and relations with integrable systems
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