An extremely useful relation between the Kronecker delta and the Levi-Civita symbol.
The brackets mean that we completely anti-symmetrize in the indices m,npic.twitter.com/Hp8tK8qZ3D
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Holds only in two dimensions.
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Look also for the SL(2,C) gauge theory of gravity by Moshe Carmeli. He considers gravity as an SL(2,C) Yang-Mills theory in which he recovers the Newman-Penrose (NP) equations (also used to study e.g. gravtitational waves). He formulates GTR in concise elegant 2-spinor formalism.
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Alas, these things never work. They always come to grief over soldering forms. GR is not a gauge theory in the usual sense. Its symmetries are not internal. The soldering forms, and the torsion-free requirement which inevitably alludes to them, are the canaries in the coal mine.
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SL(2, C) is a wondrous Lie group. One of my all-time favorite playgrounds.
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Abstract indices do not "run over" anything. The right-hand side here is the identity map on the space of 2-forms and the left-hand side is a projection mapping on the space of 2-forms, with 1-dimensional range. These can coincide only when the space of 2-forms is 1-dimensional.
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