Michael P. Mattis

The calculus of many instantons

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 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.

Gluino condensate and magnetic monopoles in supersymmetric gluodynamics

Abstract We examine supersymmetric SU(N) gauge theories on R 3 × S 1 with a circle of circumference β. These theories interpolate between four-dimensional N = 1 pure gauge theory for β = ∞ and three-dimensional N = 2 gauge theory for β = 0. The dominant field configurations of the R 3 × S 1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R 3 × S 1 theory and hence determine its vacuum structure. to the gluino condensate in these theories and take the decompactification limit β = ∞. In this way we obtain a value for the gluino condensate in the four-dimensional N = 1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known ‘weak coupling’ expression but not with the ‘strong coupling’ expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.

Multi - instanton calculus in N=2 supersymmetric gauge theory

The Seiberg-Witten solution of

Multi-instanton calculus and the AdS / CFT correspondence in N=4 superconformal field theory

N=2

Multi-instanton calculus in N=2 supersymmetric gauge theory. II. Coupling to matter.

supersymmetric SU(2) gauge theory may be viewed as a prediction for the infinite family of constants {

Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin manifold

{\mathcal{F}}_{n}

On mass-deformed N = 4 supersymmetric Yang-Mills theory

} measuring the

Yang-Mills instantons in the large N limit and the AdS / CFT correspondence

n

On N = 2 supersymmetric QCD with four flavors

-instanton contribution to the prepotential

Multi-instanton calculus in {ital N}=2 supersymmetric gauge theory

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