# Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields

## TL;DR

This paper establishes a connection between Vasiliev's higher-spin theory and noncommutative Yang--Mills theory, demonstrating that certain charges are twisted Wilson lines, and relates these to correlation functions in free conformal field theories.

## Contribution

It proves that zero-form charges in Vasiliev's theory are twisted Wilson lines and links higher-spin invariants to correlation functions in free CFTs, including explicit computations in arbitrary dimensions.

## Key findings

- Zero-form charges are twisted open Wilson lines in noncommutative Z space.
- Higher-spin invariants match cyclically-invariant building blocks of CFT correlation functions.
- Explicit polarization vector computations of conserved current correlators in arbitrary dimensions.

## Abstract

We first prove that, in Vasiliev's theory, the zero-form charges studied in 1103.2360 and 1208.3880 are twisted open Wilson lines in the noncommutative $Z$ space. This is shown by mapping Vasiliev's higher-spin model on noncommutative Yang--Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higher-spin invariants given by the leading order of these $n$-point zero-form charges are equal to corresponding cyclically-invariant building blocks of $n$-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the higher spin gravity side, our computation reproduces the results of 1210.7963 using an alternative method amenable to the computation of subleading corrections obtained by perturbation theory in normal order. On the free CFT side, our proof involves the explicit computation of the separate cyclic building blocks of the correlation functions of $n$ conserved currents in arbitrary dimension $d>2$, using polarization vectors, which is an original result. It is shown to agree, for $d=3$, with the results obtained in 1301.3123 in various dimensions and where polarization spinors were used.