ROM2F/99/19 =4 supersymmetric Yang–Mills theory N and the AdS/SCFT correspondence Stefano Kovacs Dipartimento di Fisica, Universit`adi Roma “Tor Vergata” I.N.F.N. - Sezione di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00173 Roma, ITALY Abstract This dissertation reviews various aspects of the =4 supersymmetric Yang–Mills theory in N particular in relation with the AdS/CFT correspondence. The first two chapters are introductory. The first one contains a description of the general properties of rigid supersymmetric theories in four dimensions both at the classical and at the quantum level. The second chapter is a review of the main properties of the =4 SYM theory under consideration. N Original results are reported in chapters 3, 4 and 5. A systematic re-analysis of the arXiv:hep-th/9908171v1 26 Aug 1999 perturbative properties of the theory is presented in the third chapter. Two-, three- and four-point Green functions of elementary fields are computed using the component formu- lation and/or the superfield approach and subtleties related to the gauge-fixing are pointed out. In the fourth chapter, after an introduction to instanton calculus in supersymmetric gauge theories, the computation of the one-instanton contributions to Green functions of gauge invariant composite operators in the semiclassical approximation is reported. The calculations of four-, eight- and sixteen-point Green functions of operators in the super- current multiplet are reviewed in detail. The final chapter is devoted to the AdS/SCFT correspondence. Some general aspects are discussed. Then the attention is focused on the relation between instantons in =4 SYM and D-instanton effects in type IIB string N theory. The comparison between instanton contributions to Green functions of composite operators in the boundary field theory and D-instanton generated terms in the amplitudes computed in type IIB string theory is performed and agreement between these two sources of non-perturbative effects is shown. UNIVERSITA` DEGLI STUDI DI ROMA “TOR VERGATA” FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI Dipartimento di Fisica =4 supersymmetric Yang–Mills theory N and the AdS/SCFT correspondence Tesi di dottorato di ricerca in Fisica presentata da Stefano Kovacs Relatori Professor Giancarlo Rossi Professor Massimo Bianchi Coordinatore del dottorato Professor Piergiorgio Picozza Ciclo XI Anno Accademico 1997-1998 Contents Acknowledgments v Note added v 1 Rigid supersymmetric theories in four dimensions 1 1.1 General supersymmetry algebra in four dimensions . ..... 2 1.2 Representations of the supersymmetry algebra . ..... 5 1.3 =1superspace ............................. 10 N 1.4 =1supersymmetricmodels . 17 N 1.4.1 Supersymmetric chiral theories: Wess–Zumino model . .... 18 1.4.2 Supersymmetricgaugetheories . 19 1.5 Theories of extended supersymmetry . 22 1.5.1 Extended superspace: a brief survey . 23 1.5.2 > 1supersymmetricmodels. 27 N 1.6 Quantization of supersymmetric theories . .... 29 1.6.1 Generalformalism ........................ 30 1.6.2 Superfieldpropagators . 33 1.6.3 Feynmanrules........................... 35 1.7 General properties of supersymmetric theories . ...... 36 2 =4 supersymmetric Yang–Mills theory 40 N 2.1 Diverse formulations of =4 supersymmetric Yang–Mills theory . 41 N 2.2 Classical symmetries and conserved currents . ..... 46 2.3 Finiteness of =4 supersymmetric Yang–Mills theory . 50 N 2.4 S-duality.................................. 52 2.4.1 Montonen–Olive and SL(2,Z)duality .............. 54 2.4.2 S-duality in =4 supersymmetric Yang–Mills theory . 57 N ii Table of contents iii 2.4.3 Semiclassical quantization . 60 3 Perturbative analysis of =4 supersymmetric Yang–Mills theory 66 N 3.1 Perturbation theory in components: problems with the Wess–Zumino gauge.................................... 68 3.1.1 One loop corrections to the propagator of the fermions belong- ingtothechiralmultiplet . 72 3.1.2 One loop corrections to the propagator of the scalars belonging tothechiralmultiplet . 74 3.2 Perturbation theory in =1superspace: propagators . 79 N 3.2.1 Propagatorofthechiralsuperfield. 81 3.2.2 Propagatorofthevectorsuperfield . 86 3.2.3 Discussion............................. 93 3.3 Three- and four-point functions of =1superfields . 96 N 3.3.1 Three-pointfunctions. 96 3.3.2 Four-pointfunctions . 99 4 Nonperturbative effects in =4 super Yang–Mills theory 108 N 4.1 Instanton calculus in non Abelian gauge theories . ...... 110 4.1.1 Yang–Millsinstantons . 111 4.1.2 Semiclassical quantization . 114 4.1.3 Bosoniczero-modes. 116 4.2 Instanton calculus in the presence of fermionicfields .............................. 118 4.2.1 Semiclassical approximation . 119 4.2.2 Fermioniczero-modes. 121 4.3 Instanton calculus in supersymmetric gaugetheories............................... 123 4.4 Instanton calculations in =4 supersymmetric Yang–Mills theory . 125 N 4.5 A correlation function of four scalar supercurrents . ........ 126 4.6 Eight- and sixteen-point correlation functions of current bilinears . 136 4.6.1 The correlation function of sixteen fermionic currents ..... 136 4.6.2 The correlation function of eight gaugino bilinears . ...... 137 4.7 Generalization to arbitrary N and to any K for large N ........ 138 iv Table of contents 5 AdS/SCFT correspondence 141 5.1 Type IIB superstring theory: a bird’s eye view . .... 142 5.2 D-branes.................................. 147 5.3 The AdS/SCFT correspondence conjecture . 151 5.4 Calculations of two- and three-point functions . ...... 160 5.5 Four-pointfunctions . .. .. 170 5.6 D-instanton effects in d=10 type IIB superstring theory . 177 5.6.1 Testing the AdS/CFT correspondence: type IIB D-instantons vs. Yang–Millsinstantons . 181 5.7 The D-instanton solution in AdS S5 ................. 184 5 × Conclusions 189 A Notations and spinor algebra 193 B Electric-magnetic duality 198 B.1 Electric-magnetic duality and the Dirac monopole . ...... 198 B.2 Non Abelian case: the ’t Hooft–Polyakov monopole ................................. 200 B.3 Monopolesandfermions . 204 Bibliography 207 Acknowledgments This dissertation is based on research done at the Department of Physics of the Uni- versit`adi Roma “Tor Vergata”, during the period November 1995 - October 1998, under the supervision of Prof. Massimo Bianchi and Prof. Giancarlo Rossi. I would like to thank them for introducing me to the arguments covered in this thesis, for all what they have taught me and for their willingness and constant encouragement. I also wish to thank Prof. Michael B. Green for the very enjoyable and fruitful col- laboration on the subjects reported in the last two chapters. I am grateful to Yassen Stanev for many useful and interesting discussions. Finally I thank the whole theo- retical physics group at the Department of Physics of the Universit`adi Roma “Tor Vergata” for the very stimulating environment I have found there. Note added This thesis was presented to the “Universit`adi Roma Tor Vergata” in candidacy for the degree of “Dottore di Ricerca in Fisica” and it was defended in April 1999. Hence more recent developments are not discussed here and moreover the references are not updated. However I cannot help mentioning at least the very detailed and comprehensive review by O. Arhony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz on “Large N Field Theories, String Theory and Gravity” (hep-th/9905111), that is not cited in the bibliography. Actually this paper would have made my writing of the last chapter much simpler if it came out earlier! There are some direct developments of this thesis that have been carried on during the last few months. In particular the analysis of logarithmic singularities in four- point Green functions and their relation to anomalous dimensions of un-protected operators in =4 SYM has been finalised in collaboration with Massimo Bianchi, N Giancarlo Rossi and Yassen Stanev. This issue is only briefly addressed in the con- cluding remarks. The interested reader is referred to the paper “On the logarithmic behaviour in =4 SYM theory”, hep-th/9906188. N v Chapter 1 Rigid supersymmetric theories in four dimensions Supersymmetry was introduced in [1] as a generalization of Poincar´einvariance and then implemented in the context of four dimensional quantum field theory in [2]. The idea of a symmetry transformation exchanging bosonic and fermionic degrees of freedom has proved extremely rich of consequences. Although up to now there exists no experimental evidence for supersymmetry, it plays a central rˆole in theoretical physics. At the present day the interest in the study of supersymmetric theories relies on one side on the rˆole it plays in the construction of a more fundamental model beyond the standard model of particle physics (SM) and on the other on the possibility of deriving results that can hopefully be extended to phenomenologically more relevant models. The standard model describes the strong and electro-weak interactions within the framework of a SU(3) SU(2) U(1) non-Abelian gauge theory; despite the full × × agreement with experimental data there are various motivations for believing that it cannot be a truly fundamental theory. A first problem arises in relation with the grand unification of particle interactions. The gauge couplings are believed to be unified at an energy scale of the order of 1016 GeV. The standard model contains many free parameters (masses and mixing angles) that must be “fine-tuned” in order to achieve the grand unification (hierarchy problem): the problem follows from the 1 2 Chapter 1. Rigid