We investigate the structure of equations of motion and lagrangian constraints in a general theory of massive spin-2 field interacting with external gravity. We demonstrate how consistency with the flat limit can be achieved in a number of specific spacetimes. One such example is an arbitrary static spacetime though equations of motion in this case may lack causal properties. Another example is provided by external gravity fulfilling vacuum Einstein equations with arbitrary cosmological constant. In the latter case there exists one-parameter family of theories describing causal propagation of the correct number of degrees of freedom for the massive spin-2 field in arbitrary dimension. For a specific value of the parameter a gauge invariance with a vector parameter appears, this value is interpreted as massless limit of the theory. Another specific value of the parameter produces gauge invariance with a scalar parameter and this cannot be interpreted as a consistent massive or massless theory.

We present a Lagrangian formulation for the free superspin-$\text{3}\text{2}$ massive 4D, N=1 superfield. The model is described by a dynamical real vector superfield and an auxiliary real scalar superfield. The corresponding multiplet contains spin-1, spin-2 and two spin-$\text{3}\text{2}$ propagating component fields on-shell. The auxiliary superfield is needed to ensure that the on-shell vector superfield carries only the irreducible representation of the Poincaré supergroup with a given mass and the fixed superspin of $\text{3}\text{2}$. The bosonic sector of the component level of the model is also presented.

We investigate cubic interactions between a chiral superfield and higher spin superfields corresponding to irreducible representations of the $4D,\mathcal{N}=1$ super-Poincaré algebra. We do this by demanding an invariance under the most general transformation, linear in the chiral superfield. Following Noether’s method we construct an infinite tower of higher spin supercurrent multiplets which are quadratic in the chiral superfield and include higher derivatives. The results are that a single, massless, chiral superfield can couple only to the half-integer spin supermultiplets $(s+1,s+1/2)$ and for every value of spin there is an appropriate improvement term that reduces the supercurrent multiplet to a minimal multiplet which matches that of superconformal higher spins. On the other hand a single, massive, chiral superfield can couple only to higher spin supermultiplets of type $(2l+2,2l+3/2)$ (only odd values of s, $s=2l+1$ ) and there is no minimal multiplet. Furthermore, for the massless case we discuss the component level higher spin currents and provide explicit expressions for the integer and half-integer spin conserved currents together with a R-symmetry current.

We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincaré group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case [I.L. Buchbinder, V.A. Krykhtin, A. Pashnev, Nucl. Phys. B 711 (2005) 367, hep-th/0410215], the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.

We study the effective actions in massive rank 2 and rank 3 antisymmetric tensor field models in curved space-time. These models are classically equivalent to massive vector field and massive scalar field with minimal coupling to gravity, respectively. We prove that the effective action for massive rank 2 antisymmetric tensor field is exactly equal to that for massive vector field and the effective action for massive rank 3 antisymmetric tensor field is exactly equal to that for massive scalar field. The proof is based on an identity for mass-dependent zeta functions associated with Laplacians acting on $p$ forms.