{\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
  {\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b\ ,$$ the dual Euclidean universal hypermultiplet Lagrangian has two sign flips in the kinetic terms, due to the fact that we now have $**=(-)^{(D-1)p}$ when acting on a $p$-form in Euclidean space. The dualization procedure yields $$\label{EUHM-action}
  {\mathcal{L}}_\mathrm{UH}^E = {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
  {\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 - |{\mathrm{d}}\varphi|^2 \big) - {\tfrac{1}{2}}\, {\mathrm{e}}^{-2
  \phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ together with the boundary term $$\label{E-BT}
  {\mathcal{L}}_\mathrm{bnd}^E = - (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
  {\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ By setting $\varphi=\chi=0$, this boundary term is the same as for the $D$-instanton of type IIB in ten dimensions, obtained by dualizing the nine-form field strength into the $R$-$R$ scalar $\sigma$ [@GGP; @GG]. In four dimensions, we generate more terms due to the fact that we dualize two tensors. The sign flips of the kinetic terms of the two dual fields $\lambda_a$ are compatible with the prescription of Wick rotating pseudoscalars $\lambda_a\rightarrow{\mathrm{i}}\lambda_a$ [@vNW]. This is consistent with the duality relation .

A Euclidean version of the universal hypermultiplet action was also proposed in [@GS1]. Both their bulk 