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

The analysis of spin-locality of higher-spin gauge theory is formulated in terms of star-product functional classes appropriate for the $β\to -\infty$ limiting shifted homotopy proposed recently in arXiv:1909.04876 where all $ω^2 C^2$ higher-spin vertices were shown to be spin-local. For the $β\to -\infty$ 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 $β$-shifted contracting homotopy and the relation underlying locality of the $ω^2 C^2$ 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 $Φ$ and their local currents $J(Φ)$ 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.

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