The D-instanton partition function

The D-instanton partition function is a fascinating quantity because in the presence of N D3-branes, and in a certain decoupling limit, it reduces to the functional integral of N = 4 U(N) supersymmetric gauge theory for multi -instanton solutions. We study this quantity as a function of non-commutati vity in the D3-brane theory, VEVs corresponding to separating the D3-branes and alpha'. Explicit calculations are presented in the one-instanton secto r with arbitrary N, and in the large-N limit for all instanton charge. We f ind that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is Q-exact. Consequently the partit ion function localizes on the minima of the matrix theory action. This allo ws us to prove some general properties of these integrals. In the non-commu tative theory, the contributions come from the "Higgs Branch" and are equal to the Gauss-Bonnet-Chern integral of the resolved instanton moduli space. Separating the D3-branes leads to additional localizations on products of abelian instanton moduli spaces. In the commutative theory, there are addit ional contributions from the "Coulomb Branch" associated to the small insta nton singularities of the instanton moduli space. We also argue that both n on-commutativity and alpha'-corrections play a similar role in suppressing the contributions from these singularities. Finally we elucidate the relati on between the partition function and the Euler characteristic of the insta nton moduli space.