Tensor-spinor theory of gravitation in general even space-time dimensions

Article history: Received 18 March 2021 Accepted 9 April 2021 Available online 21 April 2021 Editor: N. Lambert


Introduction
In late 1970's, Bars and MacDowell (BM) presented an interesting re-formulation of the general relativity by A. Einstein, based on a purely vector-spinor field ψ μ in four-dimensions (4D) [1]. Its action has only two fundamental fields: a vector-spinor ψ μ and a Lorentz connection ω μ rs . The ψ μ is a Majorana vector-spinor similar to the spin-3/2 field used in supergravity [2]. In other words, neither the fundamental metric g μν nor the vierbein e μ m is present in the basic lagrangian. Interestingly enough, upon using reasonable ansätze, the ψ μ -field equation yields the conventional Einstein field equation, as desired. The basic philosophy behind the result in [1] is that a vector-spinor ψ μ is more fundamental than the metric g μν or vierbein e μ m , that controls the 'geometry of space-time'.
The action of BM-theory I  (1.1) where R μν and R ρσ rs are the field-strengths of ψ μ and ω μ rs : (1.2b) Most importantly, there is neither metric g μν nor vierbein e μ m introduced to define the lagrangian L (0) 4 [1].
As will be explained in section 2, the field-equations of ψ μ and ω μ rs in (2.1) are satisfied by the peculiar ansätze (2.2), based on the covariantly-constant Majorana-spinor θ . Under these ansätze, the field equations of ψ μ and ω μ rs are reduced to the Ricci-flat equation R μν (e) . = 0, 3 i.e., the Einstein equation in vacuum with the torsion-free Lorentz connection in general relativity.
The original BM-formulation [1] seems to work only in 4D, and therefore our 4D space-time is uniquely singled out. This is because of the combination of the totally antisymmetric 4th rank constant tensor μνρσ , multiplied by the bilinear form of the 2nd-rank gravitino field-strength R (0) μν . One possible generalization is to increase the rank of the vector-spinor ψ μ to a tensor-spinor ψ μ 1 ···μ n (n ≥ 2). But we are discouraged to do so, due to the common problem of the consistency with higher-rank spinors [3] [4]. Even for a vector-spinor ψ μ , supergravity [2] theory is supposed to be the only consistent interacting system. It is nowadays a common notion [5][6] that unless we introduce infinite tower of massive higher spins as in (super)string theory [7], no higher-spin fields can interact consistently.
However, in our recent paper on supersymmetric B F -theories in diverse dimensions [8], we consider systems with higher-rank tensorspinor ψ μ 1 ···μ n without inconsistency. Our system is also shown to embed integrable models in lower dimensions, such as super KP systems. This example indicates that such higher-rank tensor can exist consistently even with consistent non-trivial interactions.
In the original BM-theory [1], there is also a technical weak point. It is related to the invariance of the action I 4 under the 'gradient' symmetry: δ ψ μ = D μ . We will improve this point in the next section by introducing the Proca-Stëckelberg-type [9] compensator field χ , such that our improved action is invariant under the symmetry δ . Since δ ψ μ = D μ is regarded as nilpotent local fermionic symmetry [10][11] [12][13] [14], the original BM-theory [1] can be re-interpreted as nothing but an application of nilpotent local fermionic symmetry.

An improved action with invariance
In the BM-formulation with the lagrangian (1.1) [1], there are only two fundamental fields ψ μ and ω μ rs , whose field equations are [1] δL (0) Note that neither the metric g μν nor the vierbein e μ m is involved in these field equations.
These field equations are solved under the following ansätze [1] 4 where e μ m is the conventional vierbein. The Majorana-spinor θ resembles the fermionic coordinates in superspace [15], but the difference is that it is a covariantly-constant spinor [1], as (2.2) shows. First of all, (2.1b) implies that = 0. We also regard two sectors (θθ) and i(θ γ 5 θ) as two independent sectors, but they are mutually consistent. This implies that ω mrs (e) r . In our present paper, we do not regard the torsion-freedom T μν In the ansätze (2.2), the vierbein e μ m is introduced, not as the basic fundamental field in the theory, but is defined by the no-fermion background-state |B and one-fermion state |B, α as B|ψ μ β |B, α = e μ m (γ m ) α β [1].
As has been mentioned, a weak-point in this original BM-theory [1] is the lack of action-invariance principle. For example, the la- To improve this point, we replace the bare ψ μ -field in (1.1) by the field-strength of the Proca-Stückelberg field χ [9], following tensorhierarchy formulations [16] [17] or our previous work on nilpotent local fermionic symmetry [10][11] [12][13] [14]. To be more specific, we introduce a Proca-Stückelberg compensator field χ [9] in its modified field strength 3 The symbol .
= stands for a field equation. 4 We use the symbol * = for an ansatz, in order to reproduce conventional general relativity.
which is invariant: δ P μ = 0 under the infinitesimal local fermionic symmetry δ ψ μ = D μ , δ χ = − . (2.6) The P μ and the modified R μν satisfy their Bianchi identities In other words, we can simply replace ψ μ everywhere in the original BM-theory by P μ to make the improved action invariant under (2.6). Namely, the lagrangian (1.1) is improved to As usual [9], the compensator χ is gauged away by the use of (2.6), and there is no essential effect in physical degrees of freedom. For example, the new χ -field equation is a sufficient condition of the ψ -field equation. However, the important mission of χ is to make the Note that the previous procedure of getting the Ricci-flatness equation R mn (e) * = 0 is not affected by the modification of ψ μ to P μ , because we can always require χ * = 0 as an additional ansatz to (2.2).
Accordingly, the original ansätze (2.2) is modified to The BM-theory [1] has neither metric nor vierbein, but it is induced out of the vector-spinor ψ μ or P μ by the relationship [1] ( . The fermionic generators Q α for our local transformation δ in (2.6) satisfy the nilpotent anti-commutator: In other words, our modified BM-theory of [1] is noting but an example of many other previously-established nilpotent local fermionic theories [10][11] [12][13] [14]. In our generalization to D = (2n − 1, 1), we will use the modified field-strength P [n−1] of the higher-rank generalization of the compensator χ [n−2] .

The Lagrangian in general D=(2n−1,1)
As has been stated, the field content of our system is (ψ [n−1] , ω μ rs , χ [n−2] ). 5 Accordingly, we have to define the generalized γ 5 -tensor γ 2n+1 , as described in the Appendix. Our action is where the constant c n depends on D = (2n − 1, 1) as satisfying their Bianchi identities Eqs. (3.3) and (3.4) with generalized Chern-Simons terms follow the same pattern as the general tensor-hierarchy formulations [16][17], even though the latter have been developed only for bosonic fields.
As in general tensor-hierarchy formulation [16] [17], the ψ and χ -fields have their proper gauge-symmetries. In particular, χ [n−2] itself is also a tensor with its own gauge symmetry: In the previous case of 4D with n = 2, the χ -field was not a gauge field lacking its proper gauge symmetry. Relevantly, the field strengths R and P are invariant under δ and δ η : The field equations for ψ, ω and χ are δL 2n (3.8c)
Since we can regard the two groups of terms: (θγ [0] θ) and (θ γ [4] θ) as independent, if the former groups imply that T rs t * = 0, then the latter group of terms vanish, because they are all proportional to T rs t .
After substituting the ansatz (4.2) for R and ω μ rs * = ω μ rs (e) into (3.8a), the original ψ -field equation is equivalent to   As before, the constants ξ , η and ζ depend on n of D = (2n − 1, 1). Using (4.11) in (4.10), we get where G rs (e) ≡ R rs (e) − (1/2)η rs R(e) is the Einstein tensor. The terms with η do not contribute, because of the Bianchi identities u (e) ≡ 0 and R [rs] (e) ≡ 0. In the last step in (4.12), we implicitly assumed that ζ = 0. This can be explicitly confirmed for general n ≥ 2, by multiplying (4.11) by δ [m r δ t s γ u] . To be more specific, we get and therefore ζ = 0 for n ≥ 2. The proof of G rs (e) * = 0 from (4.12) is as follows: We multiply (4.12) by (θγ t ) to get (4.14) The above method is simple, but started with the multiplication of our ψ -field equation (3.1a) by γ m 2 ···m n−1 from the left. So, the final condition (4.12) may be too strong. There is a direct way of proving the satisfaction of our ψ -field equation.   Here again, we used (θ γ [2] θ) * = 0 and R m [rst] . Some readers may wonder, what is the relationship corresponding to (2.10), i.e., how to define the metric g μν in terms of tensor-spinor ψ μ 1 ···μ n−1 . To this end, consider the n = 3, D = (5, 1) case. For example, relating (P μρ P νσ ) to g μν by 6 g ρσ (P μρ P νσ ) * = − (θγ μρ γ ν ρ θ) = 5(θθ) g μν (4.19) does not work, because the most left-side needs the inverse metric g ρσ , and therefore, it does not define the metric in terms of ψ μν in a closed form. However, we can still solve (4.19) perturbatively for the metric as in quantum gravity. Defining g μν ≡ η μν + h μν and g μν = η μν − h μν + h μρ h ρ ν + O(h 3 ) into (4.19), we can get the perturbative solution for h μν : (4.20) where X μν,ρσ is defined by X μν,ρσ ≡ (P μν P ρσ ) − 2η ρ[μ η ν]σ (θθ) . (4.21) In other words, we can express h μν in terms of P μν at least perturbatively. Even though this is not a closed form, still our fundamental field ψ μν (or P μν ) defines the metric g μν . This is not limited to the special n = 3 case, but we can get similar perturbative expressions also for any arbitrary n = 2, 3, 4, · · · .

Concluding remarks
In this paper, we have accomplished two major objectives: First, we introduced the compensator field χ in D = (3, 1), so that the total action of the original BM-theory [1] becomes invariant under the nilpotent local fermionic symmetry δ ψ μ = D μ and δ χ = − . In other words, the original BM-theory [1] is re-interpreted as another example of local nilpotent fermionic symmetry [10][11] [13][12] [14].
In the past, the importance of nilpotent local fermionic symmetry never drew enough attention. The reason is that the vanishing poorlooking anti-commutator such as (2.11) does not seem to produce any significant interactions, at least, compared with supersymmetry or supergravity [22]. We keep showing that that is not the case in our past papers [10][11] [12][13] [14]. In the present paper, we have shown yet another important application of nilpotent local fermionic symmetry in terms of BM-theory [1] generalized to D = (2n − 1, 1) without basic vielbeins. We stress that the success of our formulation in this paper is based on the peculiar combination of both nilpotent fermionic symmetry formulations [10][11] [12][13] [14] and tensor-hierarchy formulations [16] [17]. Our postulate for the gamma-matrix analysis (4.5) through (4.7) and (4.11) also played a technically significant role. Without any of these formulations and techniques, our theory would not be successful in such a straightforward way as in the present paper.
The original BM-theory was presented in late 1970's [1]. There are four main reasons for the delayed development afterwards. First, the tensor-hierarchy formulation [16] [17] is the foundation of our present theory, in particular, for the invariance of our action I 2n under two distinct symmetries δ and δ η in (3.5) for ψ [n−1] and χ [n−2] . In the original BM-formulation [1], the invariance of I (0) 4 was obscure. Second, the categorization of all fermions in general even dimensional space-time is possible, thanks to the systematic analyses in [23] [24]. This is merely a technical point, but it still plays an important practical role in our formulation. Third, the non-trivial and crucial γ -matrix algebra, such as (4.5), (4.6), (4.7) and (4.11) in general D = (2n − 1, 1) was hard to handle. Fourth, the method (4.17) is very decisive for our analysis, which does not seem well-known for general D = (2n − 1, 1) dimensions, even though a similar method works in Euclidian space as (4.18).
We have not introduced a separate action for 'matter' fields in our formulation. However, our higher-dimensional Einstein field equation Gμν * = 0 7 in D = (2n − 1, 1) creates Yang-Mills field equations, or σ -model type scalar-field equations, out of simple dimensional reductions [20]. Consider a simple dimensional-reduction [20] from D = (2n − 1, 1) into D = (3, 1) with the metric-tensor reduction This is nothing but the Einstein equation with the vector-fields A μ α in D = (3, 1). Even though this mechanism covers only bosonic fields, such as the vector fields and σ -model scalar fields, we expect that fermionic fields may well be produced by a certain lagrangians that contain neither metric nor vielbeins in the future.