Modular structure of chiral Fermi fields in conformal quantum field theory

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

“…After reviewing the concept of modular parallel transport [1], we present the vacuum modular Hamiltonian of a two-interval region in the 1+1-dimensional free fermion theory [4,5]. We emphasize the multi-local symmetry, which underlies the structure of that modular Hamiltonian [13,14] and which plays an important role in our analysis.…”

“…Multi-local symmetry. One fact, which enables an explicit computation of Hamiltonian (2.9), is that the algebra of observables of the chiral fermion on n > 1 intervals has a vacuum-preserving isomorphism with the observable algebra of n species of fermion on a single interval [13,14]. In other words, the fermions on distinct intervals act like replicas of one another in all correlation functions.…”

“…In other words, the fermions on distinct intervals act like replicas of one another in all correlation functions. References [13,14] refer to this fact as a multi-local symmetry and we adopt the same terminology. It is easiest to take the fiducial interval hosting the n species of fermions to be I = (0, ∞).…”

“…To establish modular inclusion, it suffices to show that the fermion remains there under modular flow (4.30) of the bigger region, for positive t. Here we use the fact that the isomorphism β intertwines modular flows in z-space and X-space [14]. By equation (2.15), σ t A sends ψ(z) to:…”