Towards Holonomy Decomposition of Process Algebras

Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak-Berger algebras

The holonomy group of an (n+2)–dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group (R × SO(n)) ⋉ R. The main ingredient of such a holonomy group is the SO(n)–projection G := prSO(n)(Holp(M,h)) and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing ...

Fock representations from U(1) holonomy algebras

We revisit the quantization of U(1) holonomy algebras using the abelian C algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of “smeared loops” and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering ...

Projective Geometry II: Holonomy Classification

The aim of this paper and its prequel is to introduce and classify the holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each p...

Berger algebras, weak-Berger algebras and Lorentzian holonomy

Representations of the holonomy algebras of gravity and non-Abelian gauge theories

Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connecti...