to and would now give different boundary terms. However, due to the isometries of the scalar manifold, they are related to each other by a field redefinition of the multipliers, $\tilde{\sigma}=\sigma+\varphi
\chi$, $\tilde{\varphi}=-\chi$, $\tilde{\chi}=\varphi$. Substituting into , we get $${\mathcal{L}}_\mathrm{bnd} = (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
  {\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ The total action for the universal hypermultiplet is then $${\mathcal{L}}= {\mathcal{L}}_\mathrm{UH} + {\mathcal{L}}_\mathrm{bnd}\ .$$

The fermions have been suppressed here. For hypermultiplets, the supersymmetry transformation rules and the fermion-terms in the Lagrangian are known in general. For the double-tensor multiplet Lagrangian , the fermion-terms and susy rules can be determined by dualization. However, the most general self-interacting supersymmetric double-tensor multiplet Lagrangian has not been worked out. For a discussion on this in the context of rigid $N=2$ supersymmetry, we refer to [@B].

Euclidean formulation {#euclidean-formulation .unnumbered}
---------------------

To find instanton solutions, we need the Euclidean formulation of the universal hypermultiplet, or, equivalently, the Euclidean double-tensor multiplet Lagrangian. For the latter, apart from the usual complications with the Euclidean Einstein-Hilbert term, the Wick rotation acts in the standard way on the scalars and tensors. While the double-tensor multiplet Lagrangian formally stays the same, $$\label{E-DTM}
  