Abstract
It is shown that SU(∞) self-dual Yang–Mills equations on a four-dimensional flat Euclidean background are equivalent to Plebanski’s second heavenly equation for the self-dual gravitational background, which is associated with the cotangent space of a suitable Riemann surface. The Riemann surface in question is nothing but the internal two-dimensional surface required in the formulation of SU(∞) self-dual Yang–Mills theories. Such equivalence only occurs when a dimensional reduction has taken place. After a discussion of Euclidean supersymmetries the results are generalized to self-dual super-Yang–Mills theories and self-dual supergravity.
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