We present a three-dimensional metric affine theory of gravity whose field equations lead to a connection introduced by Schrödinger many decades ago. Although involving nonmetricity, the Schrödinger connection preserves the length of vectors under parallel transport, and appears thus to be more physical than the one proposed by Weyl. By considering solutions with constant scalar curvature, we obtain a self-duality relation for the nonmetricity vector which implies a Proca equation that may also be interpreted in terms of inhomogeneous Maxwell equations emerging from affine geometry.

中文翻译：

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带2 + 1维自对偶非度量向量的Schrödinger连接

我们提出了一种重力的三维度量仿射理论，其场方程导致了数十年前Schrödinger提出的联系。尽管涉及非度量，但薛定ding连接保留了并行传输下矢量的长度，因此比Weyl提出的物理连接更具物理性。通过考虑具有恒定标量曲率的解，我们获得了非度量向量的自对偶关系，这暗示了一个Proca方程，该方程也可以用仿射几何中出现的非均匀麦克斯韦方程来解释。