## 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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