Quadratic Corrections to the Higher-Spin Equations by the Differential Homotopy Approach

TL;DR

This paper advances the differential homotopy method to analyze quadratic corrections in nonlinear higher-spin theories, deriving new formulae and establishing relations between formalisms, leading to explicit spin-local vertices.

Contribution

It extends the differential homotopy approach to second order perturbations and derives general star-multiplication formulas for higher-spin equations.

Findings

Derived second-order perturbation formalism for higher-spin equations.

Established relations between shifted and differential homotopy formalisms.

Obtained explicit quadratic spin-local vertices in the higher-spin theory.

Abstract

The recently proposed differential homotopy approach to the analysis of nonlinear higher spin theory is developed. The Ansatz is extended to the form applicable in the second order of the perturbation theory and general star-multiplication formulae are derived. The relation of the shifted homotopy and differential homotopy formalisms is worked out. Projectively-compact spin-local quadratic (anti)holomorphic vertices in the one-form sector of higher-spin equations are obtained within the differential homotopy formalism.