New homotopy approach to the analysis of nonlinear higher-spin equations is developed. It is shown to directly reproduce the previously obtained local vertices. Simplest cubic (quartic in Lagrangian nomenclature) higher-spin interaction vertices in four dimensional theory are examined from locality perspective by the new approach and shown to be local. The results are obtained in a background independent fashion.
A new class of shifted homotopy operators in higher-spin gauge theory is introduced. A sufficient condition for locality of dynamical equations is formulated and Pfaffian Locality Theorem identifying a subclass of shifted homotopies that decrease the
degree of non-locality in higher orders of the perturbative expansion is proven.
In this work we classify homogeneous solutions to the Noether procedure in (A)dS for an arbitrary number of external legs and in general dimensions. We also give a review of the corresponding flat space classification and its relation with the (A)dS
result presented here. The role of dimensional dependent identities is also investigated.
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 $betato-infty$--shifted contracting homotopy introduced in th
e 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 $etabar eta$. 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.
Higher-spin theory contains a complex coupling parameter $eta$. Different higher-spin vertices are associated with different powers of $eta$ and its complex conjugate $bar eta$. Using $Z$-dominance Lemma, that controls spin-locality of the higher-spi
n equations, we show that the third-order contribution to the zero-form $B(Z;Y;K)$ admits a $Z$-dominated form that leads to spin-local vertices in the $eta^2$ and $bar eta^2$ sectors of the higher-spin equations. These vertices include, in particular, the $eta^2$ and $bar eta^2$ parts of the $phi^4$ scalar field vertex.
Vasilievs higher-spin theories in various dimensions are uniformly represented as a simple system of equations. These equations and their gauge invariances are based on two superalgebras and have a transparent algebraic meaning. For a given higher-sp
in theory these algebras can be inferred from the vacuum higher-spin symmetries. The proposed system of equations admits a concise AKSZ formulation. We also discuss novel higher-spin systems including partially-massless and massive fields in AdS, as well as conformal and massless off-shell fields.