Modular parallel transport of multiple intervals in 1+1-dimensional free fermion theory

Modular parallel transport is a generalization of Berry phases, applied to modular (entanglement) Hamiltonians. Here we initiate the study of modular parallel transport for disjoint field theory regions. We study modular parallel transport in the kinematic space of multi-interval regions in the vacuum of 1+1-dimensional free fermion theory - one of the few theories for which modular Hamiltonians on disjoint regions are known. We compute explicitly the generators of modular parallel transport, and explain why their relatively simple form follows from a half-sided modular inclusion. We also compute explicitly the curvature two-form of modular parallel transport. We contrast all calculations with the expected behavior of modular parallel transport in holographic theories, emphasizing the role of non-local terms that couple distinct intervals.

Automated summary

Modular parallel transport for disjoint field theory regions is studied for the first time. The study focuses on the kinematic space of multi-interval regions in the vacuum of 1+1-dimensional free fermion theory, where modular Hamiltonians on disjoint regions are known. The generators and curvature two-form of modular parallel transport are explicitly computed, and the simplicity of their forms is explained through the concept of half-sided modular inclusion. The calculations are compared with the behavior of modular parallel transport in holographic theories, highlighting the role of non-local terms that couple distinct intervals.