Generalised symmetries and state-operator correspondence for nonlocal operators

TL;DR

This paper establishes a correspondence between line operators and states in 4D conformal field theories with 1-form symmetries, generalizes current algebras to higher dimensions, and explores non-invertible symmetries.

Contribution

It introduces a novel state-operator correspondence for nonlocal operators in higher-dimensional CFTs with p-form symmetries and constructs generalized current algebras, including non-invertible cases.

Findings

Explicit correspondence between line operators and states on S^2×S^1.

Construction of universal current algebras in (2p+2)-dimensional CFTs.

Identification of the vacuum state as a squeezing operator.

Abstract

We provide a one-to-one correspondence between line operators and states in four-dimensional CFTs with continuous 1-form symmetries. In analogy with 0-form symmetries in two dimensions, such CFTs have a free photon realisation and enjoy an infinite-dimensional current algebra that generalises the familiar Kac-Moody algebras. We construct the representation theory of this current algebra, which allows for a full description of the space of states on an arbitrary closed spatial slice. On $\mathbb{S}^2\times\mathbb{S}^1$, we rederive the spectrum by performing a path integral on $\mathbb{B}^3\times\mathbb{S}^1$ with insertions of line operators. This leads to a direct and explicit correspondence between the line operators of the theory and the states on $\mathbb{S}^2\times\mathbb{S}^1$. Interestingly, we find that the vacuum state is not prepared by the empty path integral but by a squeezing operator. Additionally, we generalise some of our results in two directions. Firstly, we construct current algebras in $(2p+2)$-dimensional CFTs, that are universal whenever the theory has a $p$-form symmetry, and secondly we provide a non-invertible generalisation of those higher-dimensional current algebras.