Membrane Instantons from Toda Field Theory

TL;DR

This paper develops a perturbative approach to construct approximate solutions for the SU(∞) Toda equation, modeling membrane instanton corrections to hypermultiplet moduli space metrics, and identifies the Lambert W-function as key in describing higher instanton contributions.

Contribution

It introduces a perturbation theory for the Toda equation to compute membrane instanton corrections, including an exact five-instanton solution and the asymptotic behavior of higher instantons.

Findings

Exact five-instanton solution computed.

Pattern identified for higher instanton corrections.

Lambert W-function describes leading instanton terms.

Abstract

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.