Higher-spin initial data in twistor space with complex stargenvalues

Abstract

This paper is a supplement to and extension of arXiv:1903.01399. In the internal twistor space of the 4D Vasiliev's higher-spin gravity, we study the star-product eigenfunctions of number operators with generic complex eigenvalues. In particular, we focus on a set of eigenfunctions represented by formulas with generalized Laguerre functions. This set of eigenfunctions can be written as linear combinations of two subsets of eigenfunctions, one of which is closed under the star-multiplication with the creation operator to a generic complex power-and the other similarly with the annihilation operator. The two subsets intersect when the left and the right eigenvalues differ by an integer. We further investigate how star-multiplications with both the creation and annihilation operators together may change such eigenfunctions and briefly discuss some problems that we are facing in order to use these eigenfunctions as the initial data to construct solutions to Vasiliev's equations.

On boundary conditions and spacetime/fibre duality in Vasiliev's higher-spin gravity

“…The regular presentation above was used in [31,32,34,35,38] to deal with the simple structure of the rotationally-invariant particle and black-hole modes (as well as for solutions of cosmological interest with scalar profiles on (A)dS background in [36]), reproducing their orthogonality by means of higher-spin invariant quantities 31 . In fact, the simple small-contour integral presentation reviewed above needs to be modified in order to be extended to the most general non-diagonal operator P n L |n R (see [37] for more comments and [30,85] for more general integral presentations). Moreover, as we have seen, the regular presentation may have to be adapted to the solution space in exam -for instance, to the compact/non-compact nature of the principal Cartan generator, which is related to the existence of normalizable solution spaces with elements distinguished by discrete/continuous parametersin such a way that it is compatible with all the observable features of the solutions.…”