Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor. Our action is shown to produce the Ricci-flat Einstein equation in arbitrary Einstein equation D = (2n − 1, 1) space-time dimensions. Even space-time dimensions © 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. 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 ψμ in four-dimensions (4D) [1]. Its action has only two fundamental fields: a vector-spinor ψμ and a Lorentz rs connection ωμ . The ψμ is a Majorana vector-spinor similar to the spin-3/2 field used in supergravity [2]. In other words, neither the m fundamental metric gμν nor the vierbein eμ is present in the basic lagrangian. Interestingly enough, upon using reasonable ansätze, the ψμ-field equation yields the conventional Einstein field equation, as desired. The basic philosophy behind the result in [1]is that a m vector-spinor ψμ is more fundamental than the metric gμν or vierbein eμ , that controls the ‘geometry of space-time’. (0) ≡ 4 L(0) 2 The action of BM-theory I4 d x 4 has the very simple lagrangian [1] L(0) =−i μνρσ R(0) R(0) =−∇ i μνρσ (0)rs 4 4 μν γ5 ρσ 8 ψμ γ5γrsψν Rρσ , (1.1) rs rs where Rμν and Rρσ are the field-strengths of ψμ and ωμ : (0) rs R ≡ − 1 −( ↔ ) ≡ − μν ∂μψν 4 ωμ γrsψν μ ν Dμψν Dνψμ , (1.2a) rs rs rs rt s rt s Rμν ≡∂μων − ∂νωμ + ωμ ωνt − ων ωμt . (1.2b) m L(0) Most importantly, there is neither metric gμν nor vierbein eμ introduced to define the lagrangian 4 [1]. * Corresponding author. E-mail addresses: [email protected] (H. Nishino), [email protected] (S. Rajpoot). 1 Also at California State University, Long Beach. ∇ 2 We use the symbol = for an equality up to a total divergence. https://doi.org/10.1016/j.physletb.2021.136288 0370-2693/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. H. Nishino and S. Rajpoot Physics Letters B 817 (2021) 136288 rs As will be explained in section 2, the field-equations of ψμ and ωμ in (2.1)are satisfied by the peculiar ansätze (2.2), based on the rs covariantly-constant Majorana-spinor θ. Under these ansätze, the field equations of ψμ and ωμ are reduced to the Ricci-flat equation . 3 Rμν(e) = 0, 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 R(0) ≥ field-strength μν . One possible generalization is to increase the rank of the vector-spinor ψμ to a tensor-spinor ψμ1···μn (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 ψμ, 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 BF-theories in diverse dimensions [8], we consider systems with higher-rank tensor- spinor ψμ1···μn 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. (0) In the original BM-theory [1], there is also a technical weak point. It is related to the invariance of the action I4 under the ‘gradient’ symmetry: δψμ = Dμ. 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 δψμ = Dμ 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. In our present paper, we will also accomplish the generalization of the original BM-theory [1]into general even dimensions D = 2n 2n−1 − = + + ··· + − = with the signature (2n 1, 1) ( , , , , ), with the manifest action-invariance under the tensor-spinor symmetry: δ ψμ1···μn−1 − = rs (n 1)D[μ1 μ2···μn−1]. Our lagrangian in D 2n has only three fields ψμ1···μn−1 , ωμ and χμ1···μn−2 , where the 1st and the last fields are Majorana-spinors, with a possible additional index for the 2 of Sp(2), depending on D = (2n − 1, 1). 2. An improved action with invariance rs In the BM-formulation with the lagrangian (1.1)[1], there are only two fundamental fields ψμ and ωμ , whose field equations are [1] δL(0) 4 =− i μνρσ rs =. 4 (γ5γrsψν)Rρσ 0 , (2.1a) δψμ δL(0) 4 =− i μνρσ R(0) =. 4 (ψνγ5γrs ρσ ) 0 . (2.1b) δωμrs m Note that neither the metric gμν nor the vierbein eμ is involved in these field equations. 4 These field equations are solved under the following ansätze [1] ∗ m ∗ ψμ = (γμθ)≡ eμ (γmθ) , Dμθ = 0 , (2.2) m where eμ 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) ∗ iμνρσ (ψ γ γ R ) =+iμνρσ (θγ γ γ γ θ)T t ν 5 rs ρσ 5 ν rs t ρσ ∗ μ μ tu μ μ ∗ = 2e(θθ) T rs + 2e[r T s] + ie(θγ5θ)rs T tu + 2e[t Tu] = 0 μ μ ∗ r ∗ =⇒ T rs + 2e[r T s] = 0 =⇒ T μν = 0 , (2.3) r . where Tm ≡ Tmr . The last equation in (2.3)is obtained by taking the trace of the 1st equation in (2.3) via Tm = 0. We also regard two sectors (θθ) and i(θγ5θ) as two independent sectors, but they are mutually consistent. This implies that ωmrs (e) ≡ (1/2)(Cmrs − Cmsr − ∗ r r r Crsm) with Cμν ≡ 2∂[μeν] . In our present paper, we do not regard the torsion-freedom Tμν = 0as one of the ansätze, but it is implied by the ω-field equation under the other ansätze (2.2). rs rs t Second, the Riemann tensor in (2.1a)is now Rμν (ω(e)) ≡ Rμν (e), satisfying the Bianchi identity R[mrs] (e) ≡ 0 and Rmnrs(e) ≡ Rrsmn(e). Using this identity and the ansätze (2.2), we see that the Ricci-flatness (or Einstein equation in empty space-time): ∗ νρσ rs =+ ν − 1 =. iμ (γ5γrsψν)Rρσ (e) 4e(γ θ) Rμν(e) 2 gμν R(e) 0 . (2.4) m In the ansätze (2.2), the vierbein eμ is introduced, not as the basic fundamental field in the theory, but is defined by the no-fermion β m β background-state |B and one-fermion state |B, α as B|ψμ |B, α = eμ (γm)α [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- L(0) = R (0) = rs = grangian 4 in (1.1)is not invariant under the proper vector-spinor transformation δψμ Dμ, because δ μν (1/4)(γrs)Rμν 0.