\chi = \frac{2\chi_1}{g_s^2}\ .$$ The BPS equation further fixes the constant $\chi_1$ to be $$\chi_1 = - {\frac{\raisebox{-2pt}{$Q^{(2)}$}}{Q^{(1)} - \chi_0 Q^{(2)}}}\ ,$$ and the instanton action is easily computed from , $$S_3^E = \big| \tau'_\infty \big|\, \Big( \big| \hat{Q}^{(1)} \big|
  + {\tfrac{1}{2}}\big| \Delta \chi\, Q^{(2)} \big| \Big)\ .$$ We have redefined the $Q^{(1)}$ charge according to $$\hat{Q}^{(1)} \equiv Q^{(1)} - \chi_0 Q^{(2)}\ ,$$ such that it is invariant under . For $Q^{(2)}=0$, the instanton action then clearly reduces to .

The obtained results for the instanton action carry over to the hypermultiplet side, because the dualization procedure does not affect the real part of the instanton action.

Discussion
==========

In this paper, we have carried out the first steps of calculating instanton corrections to the hypermultiplet moduli space. An important ingredient was to derive a Bogomol’nyi bound for the double-tensor multiplet Lagrangian, and to solve the corresponding BPS equation. In a supersymmetric formulation, adapted to Euclidean space, we expect our instanton solutions to preserve one half of the supersymmetries. A more general formulation for the double-tensor multiplet Lagrangian, including the fermions and supersymmetry transformation rules, is presently under study. This will be important for finding the fermionic zero modes and 