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

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.

Minimal model of Chiral Higher Spin Gravity

“…It well may be that there are resolutions 14 that lead to simpler results, see e.g. [74,75] for the discussion of the effects of ordering. It may also be useful for applications to fold the homological perturbation theory and rewrite it as certain simple equations, as it was done in [62,63], the danger being that the equations may admit formal solutions that lead to ill-defined vertices from the field theory point of view.…”

Unfolding, higher spins, metaplectic groups and resolution of classical singularities

“…(2.30) can be continued analytically to Eq. (2.23), and then further to , which is thus defined independently of the choice of branch cut [20].…”

“…It follows from Eq. (2.23), that limits of ( ) ∈ A [C 4 |R 4 ] in which 1 + Pr( ) degenerates are analytic delta sequences [20]; for details, see Appendix B. In particular, the center…”

“…However, as we shall briefly review, the ill behaviour of the individual spin-Weyl tensors translates to a delta-function behaviour of the corresponding master field at the singularity, and distributions in non-commutative variables can be considered smooth since they have good star product properties. Indeed, delta functions of non-commutative variables are equivalent to bounded functions up to a change in the ordering prescription [20]. It is in this same sense that the Weyl zero-form master field over the BGM background remains well-defined even where individual fluctuation fields are irregular.…”

“…The formulation of the dynamics in terms of a Cartan-integrable set of zero-curvature and covariant constancy conditions, that the Vasiliev system is based on, is called unfolded formulation [9][10][11][12] and can be thought of as a covariant analogue of hamiltonian dynamics. While it may superficially look inconvenient with respect to the standard framework of non-abelian gauge theories, unfolding has a number of powerful consequences, that in fact enabled the formulation of higher-spin gravity in closed form: two of the most important ones are that the gauge invariance of the vertices is a consequence of the integrability of their generating system in correspondence space; and that including the interactions as solutions of a differential constraints in auxiliary variables , with gauge and field-redefinition ambiguities encoded into the choice of resolution operator for the -dependence, gives some mathematical tool to control the resulting spacetime non-locality of the vertices and, possibly, to come up with a generalization of that concept adapted to higher-spin gravity (see [13][14][15][16][17][18][19][20] for recent progresses).…”