Higher-spin initial data in twistor space with complex stargenvalues

TL;DR

This paper extends the study of higher-spin gravity in twistor space by analyzing star-product eigenfunctions with complex eigenvalues, focusing on their algebraic properties and potential as initial data for solutions.

Contribution

It introduces a new class of eigenfunctions with complex eigenvalues in twistor space and explores their algebraic structure and implications for higher-spin gravity solutions.

Findings

Eigenfunctions expressed via generalized Laguerre functions.

Two subsets of eigenfunctions with specific star-multiplication properties.

Discussion of challenges in using these eigenfunctions as initial data.

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.