On completeness and generalized symmetries in quantum field theory

TL;DR

This paper discusses the concept of completeness in quantum field theory, linking it to the structure of operator algebras, the existence of generalized symmetries, and implications for holography and modular invariance.

Contribution

It clarifies the relationship between completeness, generalized symmetries, and algebraic structures in QFT, providing new insights into their interplay and implications.

Findings

Complete theories have maximal local operator algebras compatible with causality.

Non-complete theories necessarily feature dual pairs of generalized symmetries.

Broken symmetries occur simultaneously, either explicitly or effectively.

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.