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On The Linearization Stability Of The Conformally (Anti-) Self-Dual Einstein Equations, Charles G. Torre

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The Einstein equations with a cosmological constant, when restricted to Euclidean space‐times with anti‐self‐dual Weyl tensor, can be replaced by a quadratic condition on the curvature of an SU(2) (spin) connection. As has been shown elsewhere, when the cosmological constant is positive and the space‐time is compact, the moduli space of gauge‐inequivalent solutions to this equation is discrete, i.e., zero dimensional; when the cosmological constant is negative, the dimension of the moduli space is essentially controlled by the Atiyah–Singer index theorem provided the field equations are linearization stable. It is shown that linearization instability occurs whenever the unperturbed geometry possesses …