Consistent equations for interacting massless fields of all spins in the first order in curvatures (Q1113118)

A new form of equations of motions is suggested for \(d=4\) massless fields of all spins interacting with gravity: equations of all massless fields, including the gravitational field itself, are described in terms of a free differential algebra constructed from 1-forms and 0-forms belonging both to the adjoint representation of the superalgebra of higher-spin and auxiliary fields proposed previously by E. S. Fradkin and the author. In this construction, 1-forms describe gauge massless and auxiliary fields, while 0-forms describe lower-spin fields and Weyl tensors corresponding to gauge 1-forms. The equations of motion are constructed explicitly in the first order in the Weyl 0-forms (and in all orders in 1-forms) that exceeds significantly the results of \textit{E. S. Fradkin} and the author [Phys. Lett. B 189, 89-95 (1987), Nucl. Phys. B 291, 141-171 (1987)] on the cubic higher-spin-gravitational interaction. The equations obtained are shown to remain consistent when all quantities take on their values in an arbitrary associative algebra. This enables us to describe simultaneously a class of extended-type theories with Yang-Mills gauge groups \(U(n)\times U(n)\) corresponding to massless spin-1 fields (n is arbitrary). Various consistent truncations of these extended theories are also discussed including those with Yang-Mills gauge groups \(SO(n)\times SO(n).\)