The Calculus of Many Instantons

TL;DR

The Calculus of Many Instantons develops a complete, geometrically grounded framework for instanton calculations in gauge theories with arbitrary instanton number $k$, tying the ADHM moduli space to a hyper-Kähler quotient and deriving a SUSY-compatible collective coordinate measure. By integrating out fluctuations and exploiting SUSY cancellations, the authors connect nonperturbative effects to higher-dimensional sigma-models and D-brane constructions, and apply the formalism to central results such as the N=1 gluino condensate, Seiberg–Witten theory, and AdS/CFT. They also address noncommutative deformations, constrained/quasi-instanton configurations on various branches, and mass effects, illuminating when semi-classical methods are reliable. The work clarifies the role of clustering and large-N limits, showing the weakly coupled instanton calculus yields consistent, clusterable results, while strong-coupling treatments may fail to cluster, underscoring the semi-classical nature of instantons and their range of applicability in SUSY gauge theories.

Abstract

We describe the modern formalism, ideas and applications of the instanton calculus for gauge theories with, and without, supersymmetry. Particular emphasis is put on developing a formalism that can deal with any number of instantons. This necessitates a thorough review of the ADHM construction of instantons with arbitrary charge and an in-depth analysis of the resulting moduli space of solutions. We review the construction of the ADHM moduli space as a hyper-Kahler quotient. We show how the functional integral in the semi-classical approximation reduces to an integral over the instanton moduli space in each instanton sector and how the resulting matrix partition function involves various geometrical quantities on the instanton moduli space: volume form, connection, curvature, isometries, etc. One important conclusion is that this partition function is the dimensional reduction of a higher-dimensional gauged linear sigma model which naturally leads us to describe the relation of the instanton calculus to D-branes in string theory. Along the way we describe powerful applications of the calculus of many instantons to supersymmetric gauge theories including (i) the gluino condensate puzzle in N=1 theories (ii) Seiberg-Witten theory in N=2 theories; and (iii) the AdS/CFT correspondence in N=2 and N=4 theories. Finally, we briefly review the modifications of the instanton calculus for a gauge theory defined on a non-commutative spacetime and we also describe a new method for calculating instanton processes using a form of localization on the instanton moduli space.