  This paper studies nonlinear deformations of the linear gauge theory of any
number of spin-2 and spin-3/2 fields with general formal multiplication rules
in place of standard Grassmann rules for manipulating the fields, in four
spacetime dimensions. General possibilities for multiplication rules and
coupling constants are simultaneously accommodated by regarding the set of
fields equivalently as a single algebra-valued spin-2 field and single
algebra-valued spin-3/2 field, where the underlying algebra is factorized into
a field-coupling part and an internal multiplication part. The condition that
there exist a gauge invariant Lagrangian (to within a divergence) for these
algebra-valued fields is used to derive determining equations whose solutions
give all allowed deformation terms, yielding nonlinear field equations and
nonabelian gauge symmetries, together with all allowed formal multiplication
rules as needed in the Lagrangian for demonstration of invariance under the
gauge symmetries and for derivation of the field equations. In the case of
spin-2 fields alone, the main result of this analysis is that all deformations
(without any higher derivatives than appear in the linear theory) are
equivalent to an algebra-valued Einstein gravity theory. By a systematic
examination of factorizations of the algebra, a novel type of nonlinear gauge
theory of two or more spin-2 fields is found, where the coupling for the fields
is based on structure constants of an anticommutative, anti-associative
algebra, and with formal multiplication rules that make the fields
anticommuting (while products obey anti-associativity). Supersymmetric
extensions of these results are obtained in the more general case when spin-3/2
fields are included.
