Generalized curvature and the equations of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll" class="math"><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity

It is known that, for zero fermionic sector, ψ μ α ( x ) = 0 , the bosonic equations of Cremmer–Julia–Scherk eleven-dimensional supergravity can be collected in a compact expression, R a b α γ Γ b γ β = 0 , which is a condition on the curvature R α β of the generalized connection w. In this Letter we show that the equation R b c α γ Γ a b c γ β = 4 i ( ( D ˆ ψ ) b c Γ [ a b c ) β ( ψ d Γ d ] ) α , where D ˆ is the covariant derivative for the generalized connection w, collects all the bosonic equations of D = 11 supergravity when the gravitino is nonvanishing, ψ μ α ( x ) ≠ 0 .

已知对于费米子部分为零，即\(\psi_{\mu}^{\alpha}(x) = 0\)，克莱默 - 朱利亚 - 谢尔克（Cremmer - Julia - Scherk）十一维超引力的玻色子方程可以写成一个紧凑的表达式\(R^{ab\alpha\gamma}\Gamma_{b\gamma}^{\beta}=0\)，这是对广义联络\(w\)的曲率\(R^{\alpha\beta}\)的一个条件。在这封信中我们表明，方程\(R^{bc\alpha\gamma}\Gamma^{abc\gamma\beta}=4i((\hat{D}\psi)^{bc}\Gamma_{[abc)}^{\beta}(\psi^{d}\Gamma_{d]})^{\alpha})\)，其中\(\hat{D}\)是广义联络\(w\)的协变导数，当引力微子不为零，即\(\psi_{\mu}^{\alpha}(x)\neq0\)时，它包含了\(D = 11\)超引力的所有玻色子方程。