Metaplectic representation and ordering (in)dependence in Vasiliev's higher spin gravity

TL;DR

This paper explores the operator formulation of Vasiliev's higher-spin gravity, demonstrating ordering independence, constructing exact solutions, and analyzing gauge transformations and observables.

Contribution

It introduces a family of ordering prescriptions in operator form, enabling perturbative integration and exact solutions in higher-spin gravity.

Findings

Orderings are shown to be equivalent via analytic continuation.

Exact solutions are constructed in a family of gauges.

Wilson line observables are gauge-invariant up to subleading orders.

Abstract

We investigate the formulation of Vasiliev's four-dimensional higher-spin gravity in operator form, without making reference to one specific ordering. More precisely, we make use of the one-to-one mapping between operators and symbols thereof for a family of ordering prescriptions that interpolate between and go beyond Weyl and normal orderings. This correspondence allows us to perturbatively integrate the Vasiliev system in operator form and in a variety of gauges. Expanding the master fields in inhomogenous symplectic group elements, and letting products be controlled only by the group, we specify a family of factorized gauges in which we are able to integrate the system to all orders, producing exact solutions, including but not restricted to ones presented previously in the literature; and then connect, at first order, to a family of rotated Vasiliev gauges in which the solutions can be represented in terms of Fronsdal fields. The gauge function responsible for the latter transformation is explicitly constructed at first order. The analysis of the system in various orderings is facilitated by an analytic continuation of Gaussian symbols, by means of which one can distinguish and connect the two branches of the metaplectic double cover and give a rationale to the properties of the inner Klein operators as Gaussian delta sequences defining analytic delta densities. As an application of some of the techniques here developed, we evaluate twistor space Wilson line observables on our exact solutions and show their independence from auxiliary constructs up to the few first subleading orders in perturbation theory.