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Durham Research Online Durham Research Online Deposited in DRO: 06 May 2008 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Dorey, N. and Hollowood, T. J. and Khoze, V. V. and Mattis, M. P. (2002) 'The calculus of many instantons.', Physics reports., 371 (4-5). pp. 231-459. Further information on publisher's website: http://dx.doi.org/10.1016/S0370-1573(02)00301-0 Publisher's copyright statement: Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk hep-th/0206063 May 2002 The Calculus of Many Instantons Nick Dorey,1 Timothy J. Hollowood,1 Valentin V. Khoze2 and Michael P. Mattis3 1Department of Physics, University of Wales Swansea, Swansea, SA2 8PP, UK [email protected], [email protected] 2Department of Physics, University of Durham, Durham, DH1 3LE, UK [email protected] 3121 Fox Meadow Road, Scarsdale, NY10583, USA [email protected] ABSTRACT arXiv:hep-th/0206063v1 7 Jun 2002 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-K¨ahler 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 =1 N theories (ii) Seiberg-Witten theory in = 2 theories; and (iii) the AdS/CFT correspondence in =2 N N and = 4 theories. Finally, we brielfy review the modifications of the instanton calculus for a gauge N 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. I Introduction 7 I.1 The philosophy of instanton calculations . ............. 11 II Instantons in Pure Gauge Theory 13 II.1 Somebasicfacts ................................. ..... 13 II.2 Collective coordinates and moduli space . ............. 15 II.3 General properties of the moduli space of instantons . ................ 19 II.3.1 The moduli space as a complex manifold . ....... 19 II.4 The ADHM construction of instantons . .......... 21 II.4.1 The ADHM construction as a hyper-K¨ahler quotient . ............ 24 II.4.2 Symmetries and the moduli space . ...... 27 II.4.3 Singular gauge, one instanton, the dilute limit and asymptotics . 28 II.5 Zero modes and the metric on Mk ............................ 31 II.6 Singularities and small instantons . ............ 33 III The Collective Coordinate Integral 37 III.1 From the functional to the collective coordinate integral ................ 37 III.2 The volume form on the instanton moduli space . ............ 39 III.2.1 Clustering .................................. ... 41 III.3 Fluctuation determinants in the instanton background................. 41 IV Instantons in Supersymmetric Gauge Theories 45 IV.1 Action, supersymmetry and equations-of-motion . ............... 51 IV.2 The super-instanton at linear order . ............ 53 IV.2.1 Adjointfermionzeromodes. ...... 54 2 IV.2.2 Grassmann collective coordinates and the hyper-K¨ahler quotient construction 55 IV.2.3 Supersymmetric and superconformal zero modes . ........... 56 IV.3 Going beyond linear order: the quasi-instanton . ............... 58 IV.4 Scalar VEVs and constrained instantons . ............ 61 IV.4.1 Constrained instantons on the Coulomb branch . ........... 62 IV.5 Collective coordinate supersymmetry . ............. 64 V The Supersymmetric Collective Coordinate Integral 66 V.1 The supersymmetric collective coordinate measure . ............... 66 V.2 Theinstantoneffectiveaction. ......... 69 V.2.1 Geometricinterpretation . ...... 71 V.2.2 The size of a constrained instanton . ........ 73 V.2.3 Theliftingofzeromodes ......................... .... 74 V.3 The supersymmetric volume form on Mk ........................ 75 V.3.1 Supersymmetry ................................. 76 V.4 From =4 to =0viadecoupling .......................... 79 N N VI Generalizations and Miscellany 82 VI.1 Solving the ADHM constraints for N 2k ....................... 82 ≥ VI.2 The ADHM construction for Sp(N) and SO(N)..................... 86 VI.3 Matter fields and the ADHM construction . .......... 89 VI.3.1 =1theoriesontheHiggsbranch . 89 N VI.3.2 =2theoriesontheCoulombbranch . 93 N VI.4 Masses ......................................... .. 96 3 VI.5 Theinstantonpartitionfunction . ........... 97 VII The Gluino Condensate in =1 Theories 99 N VII.1 A supersymmetric Ward identity . .......... 101 VII.2 One instanton calculations of the gluino condensate . ................. 103 VII.2.1 Strongcoupling .............................. .... 103 VII.2.2 Weakcoupling ................................ ... 104 VII.3 Multi-instanton calculations of the gluino condensate.................. 109 VII.3.1 Strongcoupling .............................. .... 110 VII.3.2 Weakcoupling ................................ ... 113 VII.4 Clustering in instanton calculations . .............. 117 VIII On the Coulomb Branch of =2 Gauge Theories 119 N VIII.1 Seiberg-Witten theory and the prepotential . ................ 120 VIII.2 Extracting the prepotential from instantons . ................. 122 VIII.3 GaugegroupSU(2) .............................. ...... 125 VIII.3.1 Oneinstanton ............................... .... 127 VIII.3.2 Twoinstantons. .. .. ..... 128 VIII.4 One instanton prepotential in SU(N) .......................... 130 IX Conformal Gauge Theories at Large N 134 IX.1 The collective coordinate integrals at large N ...................... 135 IX.1.1 The =4case .................................. 136 N IX.1.2 The =2case .................................. 143 N IX.2 Large-N correlationfunctions. 145 4 IX.3 Instantons and the AdS/CFT correspondence . ............ 148 IX.3.1 The instanton collective coordinate integral . .............. 149 IX.3.2 Correlationfunctions . ...... 150 X Instantons as Solitons in Higher Dimensions and String Theory 155 X.1 Non-supersymmetric instanton branes . ........... 156 X.1.1 The moduli space approximation . ...... 157 X.2 Supersymmetric instanton branes . .......... 158 X.2.1 Action, supersymmetry and equations-of-motion . ............ 159 X.2.2 The moduli space approximation . ...... 161 X.2.3 Theeffectiveaction ............................. ... 161 X.2.4 Supersymmetry ................................. 164 X.2.5 Relation to the instanton calculus . ........ 165 X.3 Instantonsandstringtheory . ........ 166 X.3.1 The =4instantoncalculus . .. .. .. 167 N X.3.2 Probingthestringyinstanton. ....... 177 X.3.3 The =2instantoncalculus . .. .. .. 179 N X.3.4 Mass couplings and soft supersymmetry breaking . ........... 182 XI Further Directions 185 XI.1 Non-commutative gauge theories and instantons . .............. 185 XI.1.1 ADHM construction on non-commutative R4 .................. 187 XI.1.2 The prepotential of non-commutative =2gaugetheory. 189 N XI.2 Calculating the prepotential by localization . ................ 192 5 XI.2.1 Oneinstanton ................................. 198 XI.2.2 Twoinstantons................................ ... 199 Appendix A Spinors in Diverse Dimensions 206 Appendix B Complex Geometry and the Quotient Construction 209 Appendix C ADHM Algebra 216 I Introduction Yang-Mills instantons [1] have provided an enduring interest for a generation of physicists and mathe- maticians. On the physics side, instanton configurations give the leading non-perturbative contributions to the functional integral in the semi-classical approximation [2]. After the initial disappointment that instantons were not going to provide a simple explanation of quark confinement, they have proved interesting and useful both in phenomenological models of QCD (see, for example [3–5]) and for de- scribing exact non-perturbative phenomena in supersymmetric gauge theories (as we describe in this review). On the mathematical side, instantons lie at the heart of some important recent developments in topology and, in particular, of Donaldson’s construction of topological invariants of four-manifolds (see [6] and references
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