The D-Instanton Partition Function

TL;DR

This paper analyzes the D-instanton partition function in supersymmetric gauge theories, demonstrating localization properties, effects of non-commutativity, and connections to the topology of instanton moduli spaces.

Contribution

It provides explicit calculations of the D-instanton partition function, reveals its localization on minima, and explores the effects of non-commutativity and alpha'-corrections.

Findings

Partition function localizes on minima of matrix theory action.

Contributions from Higgs and Coulomb branches depend on non-commutativity.

Relation established between partition function and Euler characteristic of moduli space.

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.