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 continue the program of constructing cubic interactions between matter and higher spin supermultiplets. In this work we consider a complex linear superfield and we find that it can have cubic interactions only with supermultiplets with propagating spins j = s + 1, j = s + 1/2 for any non-negative integer s (half-integer superspin super-multiplets). We construct the higher spin supercurrent and supertrace, these compose the canonical supercurrent multiplet which generates the cubic interactions. We also prove that for every s there exist an alternative minimal supercurrent multiplet, with vanishing supertrace. Furthermore, we perform a duality transformation in order to make contact with the corresponding chiral theory. An interesting result is that the dual chiral theory has the same coupling constant with the complex linear theory only for odd values of s, whereas for even values of s the coupling constants for the two theories have opposite signs. Additionally we explore the component structure of the supercurrent multiplet and derive the higher spin currents. We find two bosonic currents for spins j = s and j = s + 1 and one fermionic current for spin j = s + 1/2.

We develop the BRST approach for all massless integer and half-integer higher spins in 4D Minkowski space, using the two component spinor notation and develop the Lagrangian formulation for supersymmetric higher spin models. It is shown that the problem of second class constraints disappears and the BRST procedure becomes much more simple than in tensorial notation. Furthermore, we demonstrate that the BRST procedure automatically provides extra auxiliary components that belong in the set of supersymmetry auxiliary components. Finally, we demonstrate how supersymmetry transformations transformations are realized in such an approach. As a result, we conclude that the BRST approach to higher spin supersymmetric theories allows to derive both the Lagrangian and the supersymmetry transformations. Although most part of the work is devoted to massless component supersymmetric models, we also discuss generalization for massive component supersymmetric models and for superfield models.

We consider a four dimensional generalized Wess-Zumino model formulated in terms of an arbitrary Kähler potential $$ \mathcal{K}\left(\varPhi, \overline{\varPhi}\right) $$ K Φ Φ ¯ and an arbitrary chiral superpotential $$ \mathcal{W}\left(\varPhi \right) $$ W Φ . A general analysis is given to describe the possible interactions of this theory with external higher spin gauge superfields of the (s + 1, s + 1/2) supermultiplet via higher spin supercurrents. It is shown that such interactions do not exist beyond supergravity (s ≥ 2) for any $$ \mathcal{K} $$ K and $$ \mathcal{W} $$ W . However, we find three exceptions, the theory of a free massless chiral, the theory of a free massive chiral and the theory of a free chiral with linear superpotential. For the first two, the higher spin supercurrents are known and for the third one we provide the explicit expressions. We also discuss the lower spin supercurrents. As expected, a coupling to (non-minimal) supergravity (s = 1) can always be found and we give the generating supercurrent and supertrace for arbitrary $$ \mathcal{K} $$ K and $$ \mathcal{W} $$ W . On the other hand, coupling to the vector supermultiplet (s = 0) is possible only if $$ \mathcal{K} = \mathcal{K}\left(\overline{\varPhi}\varPhi \right) $$ K = K Φ ¯ Φ and $$ \mathcal{W}=0 $$ W = 0 .

We present a new method of deriving the off-shell spectrum of supergravity and massless 4D, \( \mathcal{N} \) = 1 higher spin multiplets without the need of an action and based on a set of natural requirements: (a.) existence of an underlying superspace description, (b.) an economical description of free, massless, higher spins and (c.) equal numbers of bosonic and fermionic degrees of freedom. We prove that for any theory that respects the above, the fermionic auxiliary components come in pairs and are gauge invariant and there are two types of bosonic auxiliary components. Type (1) are pairs of a (2, 0)-tensor with real or imaginary (1, 1)-tensor with non-trivial gauge transformations. Type (2) are singlets and gauge invariant. The outcome is a set of Diophantine equations, the solutions of which determine the off-shell spectrum of supergravity and massless higher spin multiplets. This approach provides (i ) a classification of the irreducible, supersymmetric, representations of arbitrary spin and (ii ) a very clean and intuitive explanation to why some of these theories have more than one formulations (e.g. the supergravity multiplet) and others do not.

We present expressions for the supercurrents generated by a generic 4D,\( \mathcal{N}=1 \) theory of complex linear superfield Σ. We verify that these expressions satisfy the appropriate superspace conservation equations. Furthermore, we discuss the component projection in order to derive expressions for the energy-momentum tensor, the supersymmetry current and the R-symmetry current when available. In addition, we discuss aspects of the coupling of the theory to supergravity. Specifically, we present a straightforward method to select the appropriate formulations of supergravity that one must use in order to do the coupling. This procedure is controlled by a superfield X originating from the Super-Poincaré invariance of ccthe theory. We apply these results to examples of theories with higher derivative terms.