We present a purely tensor-spinor theory of gravity in arbitrary even $D=2n$ space-time dimensions. This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in 4D to general even dimensions with the signature $(2n-1,1)$. In the original BM-theory in $D=(3,1)$, the conventional Einstein equation emerges from a theory based on the vector-spinor field ${\psi }_{\mu }$ from a lagrangian free of both the fundamental metric $g_{\mu \nu }$ and the vierbein $e_{\mu }{}^m$. We first improve the original BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance under the nilpotent local fermionic symmetry: ${\delta }_{ϵ}\psi =D_{\mu }ϵ$ and ${\delta }_{ϵ}\chi =-ϵ$. We next generalize it to $D=(2n-1,1)$, following the same principle based on a lagrangian free of fundamental metric or vielbein now with the field content $({\psi }_{{\mu }_1\cdots {\mu }_{n-1}},\omega _{\mu }{}^{rs},{\chi }_{{\mu }_1\cdots {\mu }_{n-2}})$, where ${\psi }_{{\mu }_1\cdots {\mu }_{n-1}}$ (or ${\chi }_{{\mu }_1\cdots {\mu }_{n-2}}$) is a $(n-1)$ (or $(n-2)$) rank tensor-spinor. Our action is shown to produce the Ricci-flat Einstein equation in arbitrary $D=(2n-1,1)$ space-time dimensions.

我们以任意偶数提出了纯张量-旋量引力理论 $d=\mathrm{2个}ñ$时空维度。这是Bars和MacDowell（BM）在4D模式下将纯矢量旋转引力理论推广为具有签名的一般偶数尺寸的方法$（\mathrm{2个}ñ-\mathrm{1个}，\mathrm{1个}）$。在最初的BM理论中$d=（3，\mathrm{1个}）$，传统的爱因斯坦方程式是从基于矢量-旋量场的理论中产生的 ${\psi }_{\mu }$没有两个基本指标的拉格朗日$G_{\mu \nu }$ 和vierbein $Ë_{\mu }{}^{米}$。我们首先通过引入补偿器χ来改善原始的BM公式，以使所得的理论在全能的局部费米子对称性下具有明显的不变性：${\delta }_{ϵ}\psi =d_{\mu }ϵ$ 和 ${\delta }_{ϵ}\chi =-ϵ$。接下来，我们将其概括为$d=（\mathrm{2个}ñ-\mathrm{1个}，\mathrm{1个}）$，遵循基于无基本度量或vielbein的拉格朗日法的相同原理，现在具有字段$（{\psi }_{{\mu }_{\mathrm{1个}}\cdots {\mu }_{ñ-\mathrm{1个}}}，\omega _{\mu }{}^{\mathrm{[R}s}，{\chi }_{{\mu }_{\mathrm{1个}}\cdots {\mu }_{ñ-\mathrm{2个}}}）$， 在哪里 ${\psi }_{{\mu }_{\mathrm{1个}}\cdots {\mu }_{ñ-\mathrm{1个}}}$ （或者 ${\chi }_{{\mu }_{\mathrm{1个}}\cdots {\mu }_{ñ-\mathrm{2个}}}$）是一个 $（ñ-\mathrm{1个}）$ （或者 $（ñ-\mathrm{2个}）$）等级张量-旋转轴。我们的行动被证明可以任意产生Ricci-flat Einstein方程$d=（\mathrm{2个}ñ-\mathrm{1个}，\mathrm{1个}）$ 时空维度。