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Membrane Instantons from Toda Field Theory

Four-dimensional quaternion-Kaehler metrics with an isometry are determined by solutions to the SU($\infty$) Toda equation. We set up a perturbation theory to construct approximate solutions to the latter which can be interpreted as membrane instanton corrections to the moduli space metric of the universal hypermultiplet. We compute one such solution exactly up to the five-instanton level, including all perturbative fluctuations about the instantons. The result shows a pattern that allows us to determine the asymptotic behaviour of all higher instanton corrections within this solution. We find that the generating function for the leading terms of the latter is given by the Lambert W-function.