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Nilpotent Spinor Symmetry with Interacting Spin 3/2 Field

Consistent interactions of spin 3/2 field that realize a nilpotent spinorial symmetry are presented. Based on our previous results on purely bosonic non-Abelian tensor with consistent interactions, we present a new system for interacting spin 3/2 field that realizes the nilpotent fermionic symmetry.


INTRODUCTION
We have recently constructed a system of interacting non-Abelian tensor for purely bosonic consistent interactions [1]. Encouraged by this result, we generalized the basic structure to a fermionic system, in particular with a spin 3/2 field [2], which is different from the well-known supergravity case. We found that there are indeed consistent interactions of spin 3/2 field, realizing the nilpotent spinor algebra {Q α I , Q β J } = 0.

(I) NON-ABELIAN TENSOR (BOSONIC CASE)
The problem with constructing a system with non-Abelian tensor with consistent interactions can be described as follows: Consider the tensor B µν I with the adjoint index I. The problem is that its na1ve field strength G µνρ A clue to solving this problem lies in the dimensional reduction technique of Scherk and Schwarz [3]. In the generalized dimensional reduction in [3], the original coordinates ( xμ) in D + E space-time dimensions are reduced into (x µ , y α ), where µ, ν, ··· = 0, 1, ···, D−1 for the D -dimensions, and α, β , ··· = 1, 2, ···, E for the compact E -dimensional extra-coordinates. For example, there arises the field strength component as the field strength of the tensor B µνα with both the D -and E -dimensional coordinates. The important point is the involvement of the Chern-Simons terms at the end, different from the above-mentioned na1vely-constructed field strength.
In [1], we mimicked these structures for dimensional reductions in [3] to solve the non-Abelian tensor problem. We introduce the set of fields (B µν I ,C µ IJ , K IJK ; A µ I ) and their field strengths: Note that the total number of space-time indices and adjoint indices in the field strengths G, H and L is always four just as in [3]. These field strengths are now invariant under the tensorial Λ -gauge transformations In other words, we succeeded in defining invariant field strengths that are invariant under the Λ -tensorial transformations.

(II) FERMIONIC NILPOTENT SYMMETRY
The algebra realizing the nilpotent fermionic symmetry is dictated by the three (anti)commutators: where I, J, ··· = 1, 2, ···, g are for the adjoint representation of an arbitrary gauge group G, while α, β , ··· are for the Majorana spinors components in an arbitrary spacetime dimensions D. As the first anti-commutator in (4) shows, our spinor charges are nilpotent. The last equation in (4) is nothing but the usual non-Abelian generator commutations, while the second one implies that our spinor charges Q α I are in the adjoint representations.
The required set of fields can be fixed by mimicking the previous bosonic non-Abelian tensors. The correspondence between the bosonic and fermionic systems is Note that the total number of the space-time bosonic, fermionic and adjoint indices is always three in the present fermionic system, similarly to the previous bosonic case. The correspondence between potentials and field strengths in the present fermionic system is tabulated as All the terms accompanying the leading gradient terms are interpreted as generalized Chern-Simons terms. Our fermionic symmetry transformation rule is now We can confirm that the commutator [δ Q (ε 1 ), δ Q (ε 2 )] vanishes, consistently with the anti-commutator in (4). Amazingly and amusingly, the field strengths R and L are both invariant under the fermionic symmetry: The usual gauge transformation is δ T A µ , under which the field strengths R and L are both covariant, as desired: δ T R µν . Needless to say, these transformation rules are consistent with the algebra (4). Now that we have established the invariant field strengths R and L, it is straightforward to construct invariant actions under both δ Q and usual gauge transformations δ T . Our classical total action I T ≡ I 1 + I 2 + I 3 consists of the typical invariant actions As noted in the bosonic case, the invariance of the field strengths is closely related to the consistency of all the field equations. This can be seen from the ψ µ and χ -field equations 2 δ I T δ ψ µ = is for a field equation. Relevantly, the symbol ? = is for an equation under question.
The question now is whether the ψ µ -field equation in (10) satisfies the consistency equation D µ δ I T /δ ψ µ I ?
= 0 . Fortunately, our system has a good answer to this question, thanks to the identity: Up to this stage, no field equation has been used. Note also that this identity is nothing but the invariance of the action δ Q I T = 0. Now, the first term in (11) vanishes, as the necessary condition of the χ -field equation. This feature is also associated with the F χ -term and ψ -linear term in the R and L -field strengths.
In passing we note that we have also performed the quantization of the system [2] by confirming the BRST invariance.