Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: The present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of a sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise. *

We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT’s. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the ’t Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.

We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current j(x) = ∂φ(x) corresponding to a chiral free scalar φ in d = 2. We also compute explicitly the mutual information between the intervals. This model shows a failure of Haag duality for two intervals that translates into a loss of a symmetry property for the mutual information usually associated with modular invariance. Contrary to the case of a free massless fermion, the modular Hamiltonian turns out to be completely non local. The calculation is done diagonalizing the density matrix by computing the eigensystem of a correlator kernel operator. These eigenvectors are obtained by a novel method that involves solving an equivalent problem for an holomorphic function in the complex plane where multiplicative boundary conditions are imposed on the intervals. Using the same technique we also re-derive the free fermion modular Hamiltonian in a more transparent way. arXiv:1809.00026v2 [hep-th]