The D-instanton partition function

The D-instanton partition function is a fascinating quantity because in thepresence of N D3-branes, and in a certain decoupling limit, it reduces to thefunctional integral of N=4 U(N) supersymmetric gauge theory for multi-instantonsolutions. We study this quantity as a function of non-commutativity in theD3-brane theory, VEVs corresponding to separating the D3-branes and alpha'.Explicit calculations are presented in the one-instanton sector with arbitraryN, and in the large-N limit for all instanton charge. We find that for generalinstanton charge, the matrix theory admits a nilpotent fermionic symmetry andthat the action is Q-exact. Consequently the partition function localizes onthe minima of the matrix theory action. This allows us to prove some generalproperties of these integrals. In the non-commutative theory, the contributionscome from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integralof the resolved instanton moduli space. Separating the D3-branes leads toadditional localizations on products of abelian instanton moduli spaces. In thecommutative theory, there are additional contributions from the ``CoulombBranch'' associated to the small instanton singularities of the instantonmoduli space. We also argue that both non-commutativity and alpha'-correctionsplay a similar role in suppressing the contributions from these singularities.Finally we elucidate the relation between the partition function and the Eulercharacteristic of the instanton moduli space.Comment: 32 pages, JHEP.cls, some extra comments and typos fixe