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 ${\mathcal{M}}^{\text{'}}$ 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 $\u2102$ of complex numbers, and of $s{o}^{*}\left(4\infty \right)$ 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 $O\left(N\right),U\left(N\right)$ and $U(N,\mathbb{H})=\text{Sp}\left(2N\right)$, respectively.

##### MSC:

81T40 | Two-dimensional field theories, conformal field theories, etc. |

81R10 | Infinite-dimensional groups and algebras motivated by physics |

22E70 | Applications of Lie groups to physics; explicit representations |

81R12 | Relations of groups and algebras in quantum theory with integrable systems |