a similar strategy as in [@BB; @GS1], or as in [@R] for matter coupled to $N=1$ supergravity. Although being an approximation of the exact result, the hope is that the leading supergravity corrections, combined with the constraints from quaternion-Kähler geometry, and together with some knowledge from string theory on the isometries and singularity structure of ${\cal M}_H$, should fix the answer uniquely. Such a program has worked succesfully in the context of supersymmetric field theories in three dimensions with eight supercharges, where the hypermultiplet moduli space is hyperkähler [@SW; @DKMTV]. See [@OV; @K] for related issues.

In this paper, we carry out the first steps of the supergravity instanton calculation. In section 2, we explain how the Euclidean theory is best understood in terms of the double-tensor multiplet, since then the action is manifestly positive definite, a requirement needed for a semiclassical approximation. In section 3, we derive a Bogomol’nyi bound and show that the instanton action is purely topological and given by a surface term. We then solve the BPS equation explicitly and compute the instanton action for the solutions. A similar approach was followed in [@BB] and [@GS1]. Compared to these papers, we propose a different Euclidean version 