to a well-defined semiclassical treatment, in which the instantons dominate the Euclidean path integral. For this reason, it is important to perform all calculations on the double-tensor multiplet side, and after having computed the instanton corrections there, we can dualize to the hypermultiplet formulation.

Instanton solutions
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Asymptotics
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Before finding the explicit instanton solutions, it will be useful to discuss the asymptotic behaviour of the fields that can lead to a finite action. Since the Euclidean action consists of three positive definite terms, each term individually should integrate to a finite quantity. For simplicity we consider for the moment flat four-dimensional space. This determines the following behaviour at infinity: $$\label{large-r}
  \phi \rightarrow \phi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}} \Big)\
  ,\quad \chi \rightarrow \chi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}}
  \Big)\ ,\quad H_{\mu\nu\rho} \propto {\frac{\raisebox{-2pt}{$1$}}{r^3}}\ .$$ The asymptotic value of $\phi$ is identified with the four- (or five-) dimensional string coupling constant, $$g_s \equiv {\mathrm{e}}^{-\phi_\infty/2}\ .$$ The field strengths determine topological charges, defined by integrating the tensors $H^a$ over spheres at infinity, $$\label{HQ}
  \int_{S^{D-1}_\infty}\! H^a = Q^{(a)} \ ,\quad a = 1,2\ .$$ In the dual (hypermultiplet) formulation, topological charges become Noether charges, corresponding to the Peccei-Quinn symmetries which act as constant shifts 