Generalised symmetries and state-operator correspondence for nonlocal operators

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.