Modular geodesics and wedge domains in non-compactly causal symmetric spaces

TL;DR

This paper explores the geometric structure of wedge regions in non-compactly causal symmetric spaces, linking modular flows, convexity, and fiber bundle structures, with implications for algebraic quantum field theory.

Contribution

It characterizes the positivity region of modular flows as a connected wedge domain with a fiber bundle structure, relating it to crown domains and geometric KMS conditions.

Findings

Positivity region W is connected and coincides with the observer domain.

W has a natural structure as an equivariant fiber bundle over a Riemannian symmetric space.

The paper introduces a convexity theorem for G-translates of open H-orbits.

Abstract

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space M = G/H, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic, it can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open $H$-orbits in the minimal flag manifold specified by the 3-grading.