On completeness and generalized symmetries in quantum field theory

TL;DR

The paper formulates a precise, model‑independent completeness criterion in quantum field theory based on regional operator algebras, showing that completeness is equivalent to the absence of generalized symmetries and the uniqueness of the net via duality. It analyzes how incompleteness generates dual generalized symmetries and uses the Jones index to quantify their size, establishing that these symmetries always appear in dual pairs and disappear together under deformations or RG flow. Through global and gauge theory examples, it clarifies the role of Wilson and 't Hooft loops, center structures, and the reconstruction of symmetry content, while linking the framework to modular invariance and holography. The work culminates with entropic order parameters and a certainty relation that track the fate of dual symmetries across scales, offering a unifying lens for understanding completeness in QFT and its implications for quantum gravity and holography.

Abstract

We review a notion of completeness in QFT arising from the analysis of basic properties of the set of operator algebras attached to regions. In words, this completeness asserts that the physical observable algebras produced by local degrees of freedom are the maximal ones compatible with causality. We elaborate on equivalent statements to this completeness principle such as the non-existence of generalized symmetries and the uniqueness of the net of algebras. We clarify that for non-complete theories, the existence of generalized symmetries is unavoidable, and further, that they always come in dual pairs with precisely the same ``size''. Moreover, the dual symmetries are always broken together, be it explicitly or effectively. Finally, we comment on several issues raised in recent literature, such as the relationship between completeness and modular invariance, dense sets of charges, and absence of generalized symmetries in the bulk of holographic theories.