Spin-Locality of Higher-Spin Theories and Star-Product Functional Classes

TL;DR

This paper investigates the spin-locality properties of higher-spin gauge theories using star-product functional classes, extending previous results and proposing a space-time interpretation of spin-locality as a substitute for traditional locality.

Contribution

It introduces the class of functions ${ mf extit H}^{+0}$ for analyzing spin-locality and extends the Pfaffian Locality Theorem to the $eta$-shifted homotopy in higher-spin theories.

Findings

Identification of ${ mf extit H}^{+0}$ functions that do not contribute to field equations.

Extension of the Pfaffian Locality Theorem to $eta$-shifted homotopy.

Proposal that spin-locality equates to space-time locality in terms of constituent fields and currents.

Abstract

The analysis of spin-locality of higher-spin gauge theory is formulated in terms of star-product functional classes appropriate for the $\beta\to -\infty$ limiting shifted homotopy proposed recently in arXiv:1909.04876 where all $\omega^2 C^2$ higher-spin vertices were shown to be spin-local. For the $\beta\to -\infty$ limiting shifted contracting homotopy we identify the class of functions ${\mathcal H}^{+0}$, that do not contribute to the r.h.s. of HS field equations at a given order. A number of theorems and relations that organize analysis of the higher-spin equations are derived including extension of the Pfaffian Locality Theorem of arXiv:1805.11941 to the $\beta$-shifted contracting homotopy and the relation underlying locality of the $\omega^2 C^2$ sector of higher-spin equations. Space-time interpretation of spin-locality of theories involving infinite towers of fields is proposed as the property that the theory is space-time local in terms of original constituent fields $\Phi$ and their local currents $J(\Phi)$ of all ranks. Spin-locality is argued to be a proper substitute of locality for theories with finite sets of fields for which the two concepts are equivalent.