of the universal hypermultiplet Lagrangian, so our results, where comparable, are somehow different. Moreover, we have found new instanton solutions, which will play an important role in understanding the quantum corrected hypermultiplet moduli space, as explained in the discussion at the end of the paper.

The double-tensor multiplet
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As mentioned in the introduction, we are interested in the case of a single hypermultiplet coupled to $N=2$ supergravity. Classically, the four scalars of the universal hypermultiplet parametrize the homogeneous quaternion-Kähler manifold [@CFG; @FS] $$\label{UHM-QK}
  {\mathcal{M}}_H = {\frac{\raisebox{-2pt}{$\mathrm{SU}(1,2)$}}{\mathrm{U}(2)}}\ .$$ In a basis of real fields $\{\phi,\chi,\varphi,\sigma\}$, the bosonic Lagrangian takes the form[^2] $$\label{UHM-action}
  {\mathcal{L}}_\mathrm{UH} = - {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
  {\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 + |{\mathrm{d}}\varphi|^2 \big) + {\tfrac{1}{2}}\,
  {\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ with $D=4$ or $5$, depending on whether one is interested in type II or M-theory compactifications. The Lagrangian has a global SU(1,2) isometry group.

For our purposes, it will be convenient to discuss the dual version of ${\mathcal{L}}_\mathrm{UH}$ in terms of a double-tensor multiplet. Consider the first-order Lagrangian $$\label{DTM-action}
  {\mathcal{L}}_\mathrm{DT} = - {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
  {\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b - \lambda_a\,
  {\mathrm{d}}H^a\ ,$$ where the $H^a$ 