We study an [Formula: see text] pure connection formulation in four dimensions for real-valued fields, inspired by the Capovilla, Dell and Jacobson complex self-dual approach. By considering the CMPR BF action, also, taking into account a more general class of the Cartan–Killing form for the Lie algebra [Formula: see text] and by refining the structure of the Lagrange multipliers, we integrate out the metric variables in order to obtain the pure connection action. Once we have obtained this action, we impose certain restrictions on the Lagrange multipliers, in such a way that the equations of motion led us to a family of torsionless conformally flat Einstein manifolds, parametrized by two numbers. Finally, we show that, by a suitable choice of parameters, self-dual spaces (Anti-) de Sitter can be obtained.

中文翻译：

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具有真实重力场的纯连接公式

受 Capovilla、Dell 和 Jacobson 复数自对偶方法的启发，我们研究了实值字段的四个维度的 [公式：见文本] 纯连接公式。通过考虑 CMPR BF 动作，还考虑到李代数的更一般的 Cartan-Killing 形式[公式：见文本]，并通过改进拉格朗日乘数的结构，我们按顺序整合度量变量获得纯连接动作。一旦我们获得了这个作用，我们就对拉格朗日乘数施加某些限制，这样运动方程就会把我们引向一个由两个数字参数化的无扭转共形平坦爱因斯坦流形族。最后，我们表明，通过选择合适的参数，可以得到自对偶空间（反）de Sitter。