Lagrangian and boundary term differ from ours. This has important consequences since the instanton action defines the weight in the path integral, and hence correlation functions and eventually the quantum-corrected hypermultiplet moduli space will be different.

Due to the sign changes in , the geometry of the scalar manifold is no longer SU(1,2)/U(2). Instead, it is given by the coset space $$\label{SL3}
  {\mathcal{M}}_H^E = {\frac{\raisebox{-2pt}{$\mathrm{SL}(3,\fieldR)$}}{\mathrm{SL}(2,\fieldR) \times
  \mathrm{SO}(1,1)}}\ ,$$ which is *not* a quaternion-Kähler manifold. This is not in contradiction with supersymmetry, since only *Minkowskian* supersymmtry requires the target space to be quaternionic [@BW]. A brief discussion on the geometry of the space is given in appendix \[SL3R\].

In four dimensions, the same target space can be obtained by applying the **c**-map [@CFG] to pure $N=2$ Euclidean supergravity [@TvN]. This turns the four bosonic degrees of freedom contained in the metric and graviphoton into the four scalars of the universal hypermultiplet and gives rise to the two sign flips. Moreover, the **c**-map maps Reissner-Nordstrom black hole solutions to D-instantons in the universal hypermultiplet, as was shown in [@BGLMM].

We remark that it is the inverted signs in the Euclidean hypermultiplet action that make instanton solutions in flat space possible. Indeed, the 