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Register a new userHomotopy Operators and Locality Theorems in Higher-Spin Equations
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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.
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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 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 $etabar eta$. 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.
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.
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-spin 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.
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-spin 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.
We elaborate on the spin projection operators in three dimensions and use them to derive a new representation for the linearised higher-spin Cotton tensors.
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