Limiting Shifted Homotopy in Higher-Spin Theory and Spin-Locality

TL;DR

This paper develops a novel shifted homotopy method to explicitly compute and analyze higher-spin vertices, demonstrating their spin-locality and ultra-locality properties, and clarifying their behavior on gravitational backgrounds.

Contribution

The paper introduces a $eta o-rac{1}{eta}$ shifted homotopy technique for higher-spin theory, enabling explicit calculation of spin-local vertices with new locality properties.

Findings

Vertices are spin-local with finite derivatives in spinor space.

Vertices proportional to $eta^2$ are ultra-local and background independent.

Gravitational interactions do not involve $eta^2$ vertices, ensuring positive gravitational constant.

Abstract

Higher-spin vertices containing up to quintic interactions at the Lagrangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a $\beta\to-\infty$--shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter $\eta$ in the HS equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to $\eta^2$ and $\bar \eta^2$ are in addition ultra-local, i.e. zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables $y$ or $\bar y$. Also the $\eta^2$ and $\bar \eta^2$ vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on $AdS_4$. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to $\eta\bar \eta$. It is shown that the $\beta$-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the $\beta$-dependent deformed star product.