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The calculus of many instantons

Dorey, N.; Hollowood, T.J.; Khoze, V.V.; Mattis, M.P.

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N. Dorey

T.J. Hollowood

M.P. Mattis


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 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 theories (ii) Seiberg–Witten theory in theories; and (iii) the AdS/CFT correspondence in and 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.


Dorey, N., Hollowood, T., Khoze, V., & Mattis, M. (2002). The calculus of many instantons. Physics Reports, 371(4-5), 231-459.

Journal Article Type Article
Publication Date 2002-12
Deposit Date May 6, 2008
Publicly Available Date May 6, 2008
Journal Physics Reports
Print ISSN 0370-1573
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 371
Issue 4-5
Pages 231-459
Keywords Supersymmetric gauge-theories, Yang-Mills theory, Topological field-theories, ADS-CFT correspondence, Low-energy dynamics, ADS/CFT correspondence, Multiinstanton calculus, Noncommutative R-4, Gluino condensate, Sigma-models.


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