A geometric perspective on local nets in Algebraic Quantum Field Theory

In Algebraic Quantum Field Theory (AQFT) one studies nets of von Neumann algebras M(O) assigned to open subsets O of a spacetime manifold. We are interested in obtaining such nets from unitary representations of a connected Lie group and understanding the relevant structure of these nets in terms of data related to the group and its representations. The connection between AQFT and unitary representations is established by the modular theory of operator algebra that provides for so-called wedge regions a modular one-parameter group. This puts a focus on groups generated by modular one-parameter groups. A key result is that these one-parameter groups are generated by Euler elements (defining a 3-grading of the Lie algebra). Typical examples are generators of Lorentz boosts on Minkowski space. On the geometric side, they are closely related to wedge regions in causal manifolds. We shall outline the cornerstones of the theory and how the geometry of Euler elements and causal homogeneous spaces provides an interesting perspective on phenomena in AQFT.