trace of the Einstein equation sets the bulk Lagrangian to zero, hence nontrivial field configurations would require a nonvanishing curvature scalar if the sigma model part of the Lagrangian were positive definite. The negative signs in allow for cancellations that are compatible with $R=0$. Note also that since on the hypermultiplet side the bulk action vanishes for any solution, the instanton action comes entirely from the boundary term discussed above. As already stated, the boundary term is different from the one proposed in [@GS1]. For this reason, we get different results for the instanton action, and eventually for the instanton corrected hypermultiplet moduli space.

What is more important from the point of view of instanton calculations, is that the Euclidean Lagrangian is no longer positive definite. In a path integral formulation, this makes the finite action configurations irrelevant, since the action is not bounded from below. Moreover, perturbative fluctuations around the instanton yield diverging non-Gaussian integrals, and the semiclassical approximation would break down. Similar considerations apply to the $N=2$ tensor multiplet, whose Euclidean action is not positive definite.

On the other hand, the Euclidean double-tensor multiplet Lagrangian is bounded from below, since the matrix $M_{ab}$ is positive definite. This leads 