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

Area of Science:

• Algebraic Quantum Field Theory
• Differential Geometry
• Lie Theory

Background:

• Investigates the relationship between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory (AQFT).
• Utilizes the perspective that the modular group's geometric implementation is derived from the flow generated by an Euler element of a Lie algebra, defining a 3-grading.
• Connects Euler elements of semisimple Lie algebras to non-compactly causal symmetric spaces.

Purpose of the Study:

• To analyze the geometry of the flow generated by an Euler element in the context of AQFT.
• To characterize the positivity region (wedge region) of this flow.
• To explore the connection between the positivity region, observer domains, and causal geodesics.

Main Methods:

• Geometric analysis of the flow generated by an Euler element in a semisimple Lie algebra.
• Characterization of the positivity region using geometric KMS conditions.
• Development of a polar decomposition for the positivity domain.
• Proof of a convexity theorem for G-translates of open H-orbits in flag manifolds.

Main Results:

• For Lie groups G with a trivial center, the positivity region W is connected.
• W coincides with the observer domain and is specified by a trajectory that is both a modular flow trajectory and a causal geodesic.
• The positivity region W is characterized by a geometric KMS condition and exhibits a structure of an equivariant fiber bundle, identified as a real form of the crown domain.

Conclusions:

• The study provides a detailed geometric description of the positivity region in the context of AQFT and symmetric spaces.
• The findings establish a strong link between modular flow, causal structures, and geometric properties of symmetric spaces.
• The developed mathematical tools, including polar decomposition and convexity theorems, offer new insights into the geometry of flag manifolds.