The D-Instanton Partition Function

TL;DR

The paper investigates the D-instanton partition function for $k$ D-instantons in the background of $N$ D3-branes, revealing a deep link with the ADHM moduli space of $k$-instantons in $U(N)$ gauge theory. By formulating the D-instanton matrix model as a cohomological, $\mathcal{Q}$-exact theory, the authors show the partition function localizes on the zeros of the action, corresponding to the classical Higgs and Coulomb branches, with FI terms, non-commutativity, and string corrections regulating small-instanton singularities. In the one-instanton sector, explicit calculations yield the Gauss-Bonnet-Chern integral of the resolved moduli space and expose a topological interpretation as the Euler characteristic, $\chi(\widehat{\mathfrak M}_{1,N}^{(\zeta)})=N$, while in the large-$N$ limit a closed-form asymptotic is obtained for arbitrary $k$. Overall, the work shows that the D-instanton partition function provides a powerful, topologically controlled handle on multi-instanton effects in ${\cal N}=4$ SYM$_4$, with precise links between stringy regularization, non-commutativity, and instanton moduli-space geometry.

Abstract

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-commutativity in the D3-brane theory, VEVs corresponding to separating the D3-branes and alpha'. Explicit calculations are presented in the one-instanton sector with arbitrary N, and in the large-N limit for all instanton charge. We find that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is Q-exact. Consequently the partition function localizes on the minima of the matrix theory action. This allows us to prove some general properties of these integrals. In the non-commutative 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 additional contributions from the ``Coulomb Branch'' associated to the small instanton singularities of the instanton moduli space. We also argue that both non-commutativity and alpha'-corrections play a similar role in suppressing the contributions from these singularities. Finally we elucidate the relation between the partition function and the Euler characteristic of the instanton moduli space.