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Self-dual solutions to the complex Einstein equation with cosmological constant are characterised as SL(2, C) gauge fields satisfying an equation quadratic in the curvature. For a solution, a tetrad is constructed with respect to which the connection is the left handed spin connection. For real Euclidean signature, the connection reduces to an SU(2) connection.
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Capovilla et al. (1990) studied this question.
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