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-Kähler 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 higherdimensional 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 brielfy 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.

Centre d'études et d'expertise sur les risques, l'environnement, la mobilité et l'aménagement

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The Seiberg-Witten solution of Nϭ2 supersymmetric SU͑2͒ gauge theory may be viewed as a prediction for the infinite family of constants ͕F n ͖ measuring the n-instanton contribution to the prepotential F. Here we examine the instanton physics directly, in particular the contribution of the general self-dual solution of topological charge n constructed by Atiyah, Drinfeld, Hitchin, and Manin ͑ADHM͒. In both the bosonic and supersymmetric cases, we determine both the large-and short-distance behavior of all the fields in this background. This allows us to construct the exact classical interaction between n ADHM ͑super͒instantons mediated by the adjoint Higgs bosons. We calculate the one-and two-instanton contributions to the low-energy Seiberg-Witten effective action and find precise agreement with their predicted values of F 1 and F 2 .

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