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

TL;DR

This work extends the differential homotopy program to second order in higher-spin theory, deriving a generalized second-order Ansatz and new star-product formulae that keep homotopy parameters under control. It demonstrates that the shifted-homotopy construction is a special case of the differential-homotopy framework, and introduces the Ω12 construction and a β12 parameter to realize spin-local, projectively-compact holomorphic vertices in the one-form sector. The authors compute the B2 and W2 corrections, establishing projectively-compact, spin-local holomorphic vertices such as Υ(CCωω) and related sectors, and show these results converge to known ultra-local locality in the β→−∞ limit. The formalism clarifies locality control, automatically accounts for Schouten identities, and sets the stage for higher-order calculations and mixed holomorphic–antiholomorphic sectors, with potential implications for the broader locality problem in nonlinear HS theories.

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.