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A First Course on Twistors, Integrability and Gluon Scattering

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A First Course on Twistors, Integrability and Gluon Scattering DAMTP 2010–05 A First Course on Twistors, Integrability and Gluon Scattering Amplitudes Martin Wolf∗ † Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce Road, Cambridge CB3 0WA, United Kingdom Abstract These notes accompany an introductory lecture course on the twistor approach to supersymmetric gauge theories aimed at early-stage PhD students. It was held by the author at the University of Cambridge during the Michaelmas term in 2009. The lectures assume a working knowledge of differential geometry and quantum field theory. No prior knowledge of twistor theory is required. arXiv:1001.3871v2 [hep-th] 26 Aug 2010 21st January 2010 ∗Also at the Wolfson College, Barton Road, Cambridge CB3 9BB, United Kingdom. †E-mail address: [email protected] Preface The course is divided into two main parts: I) The re-formulation of gauge theory on twistor space and II) the construction of tree-level gauge theory scattering amplitudes. More specifically, the first few lectures deal with the basics of twistor geometry and its application to free field theories. We then move on and discuss the non-linear field equations of self-dual Yang–Mills theory. The subsequent lectures deal with supersymmetric self-dual Yang–Mills theories and the extension to the full non-self-dual supersymmetric Yang–Mills theory in the case of maximal = 4 supersym- N metry. Whilst studying the field equations of these theories, we shall also discuss the associated action functionals on twistor space. Having re-interpreted = 4 supersymmetric Yang–Mills N theory on twistor space, we discuss the construction of tree-level scattering amplitudes. We first transform, to twistor space, the so-called maximally-helicity-violating amplitudes. Afterwards we discuss the construction of general tree-level amplitudes by means of the Cachazo–Svrˇcek–Witten rules and the Britto–Cachazo–Feng–Witten recursion relations. Some mathematical concepts un- derlying twistor geometry are summarised in several appendices. The computation of scattering amplitudes beyond tree-level is not covered here. My main motivation for writing these lecture notes was to provide an opportunity for stu- dents and researchers in mathematical physics to get a grip of twistor geometry and its ap- plication to perturbative gauge theory without having to go through the wealth of text books and research papers but at the same time providing as detailed derivations as possible. Since the present article should be understood as notes accompanying an introductory lecture course rather than as an exhaustive review article of the field, I emphasise that even though I tried to refer to the original literature as accurately as possible, I had to make certain choices for the clarity of presentation. As a result, the list of references is by no means complete. Moreover, to keep the notes rather short in length, I had to omit various interesting topics and recent de- velopments. Therefore, the reader is urged to consult Spires HEP and arXiv.org for the latest advancements and especially the citations of Witten’s paper on twistor string theory, published in Commun. Math. Phys. 252, 189 (2004), arXiv:hep-th/0312171. Should you find any typos or mistakes in the text, please let me know by sending an email to [email protected]. For the most recent version of these lecture notes, please also check http://www.damtp.cam.ac.uk/user/wolf Acknowledgements. I am very grateful to J. Bedford, N. Bouatta, D. Correa, N. Dorey, M. Dunajski, L. Mason, R. Ricci and C. S¨amann for many helpful discussions and suggestions. Special thanks go to J. Bedford for various discussions and comments on the manuscript. I would also like to thank those who attended the lectures for asking various interesting questions. This work was supported by an STFC Postdoctoral Fellowship and by a Senior Research Fellowship at the Wolfson College, Cambridge, U.K. Cambridge, 21st January 2010 Martin Wolf 1 Literature Amongst many others (see bibliography at the end of this article), the following lecture notes and books have been used when compiling this article and are recommended as references and for additional reading (chronologically ordered). Complex geometry: (i) P. Griffiths & J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York, 1978 (ii) R. O. Wells, Differential analysis on complex manifolds, Springer Verlag, New York, 1980 (iii) M. Nakahara, Geometry, topology and physics, The Institute of Physics, Bristol–Philadel- phia, 2002 (iv) V. Bouchard, Lectures on complex geometry, Calabi–Yau manifolds and toric geometry, arXiv:hep-th/0702063 Supermanifolds and supersymmetry: (i) Yu. I. Manin, Gauge field theory and complex geometry, Springer Verlag, New York, 1988 (ii) C. Bartocci, U. Bruzzo & D. Hernand´ez-Ruip´erez, The geometry of supermanifolds, Kluwer, Dordrecht, 1991 (iii) J. Wess & J. Bagger, Supersymmetry and supergravity, Princeton University Press, Prin- ceton, 1992 (iv) C. S¨amann, Introduction to supersymmetry, Lecture Notes, Trinity College Dublin, 2009 Twistor geometry: (i) R. S. Ward & R. O. Wells, Twistor geometry and field theory, Cambridge University Press, Cambridge, 1989 (ii) S. A. Huggett & K. P. Tod, An introduction to twistor theory, Cambridge University Press, Cambridge, 1994 (iii) L. J. Mason & N. M. J. Woodhouse, Integrability, self-duality, and twistor theory, Clarendon Press, Oxford, 1996 (iv) M. Dunajski, Solitons, instantons and twistors, Oxford University Press, Oxford, 2009 Tree-level gauge theory scattering amplitudes and twistor theory: (i) F. Cachazo & P. Svrˇcek, Lectures on twistor strings and perturbative Yang–Mills theory, PoS RTN2005 (2005) 004, arXiv:hep-th/0504194 (ii) J. A. P. Bedford, On perturbative field theory and twistor string theory, arXiv:0709.3478, PhD thesis, Queen Mary, University of London (2007) (iii) C. Vergu, Twistors, strings and supersymmetric gauge theories , arXiv:0809.1807, PhD thesis, Universit´eParis IV–Pierre et Marie Curie (2008) 2 Contents Part I: Twistor re-formulation of gauge theory 1. Twistorspace..................................... ..... 6 1.1. Motivation ..................................... ... 6 1.2. Preliminaries .................................. ..... 6 1.3. Twistorspace................................... .... 8 2. MasslessfieldsandthePenrosetransform . ............ 11 2.1. Integral formulæ for massless fields . ........... 11 2.2. Cechˇ cohomology groups and Penrose’s theorem – a sketch . ......... 12 3. Self-dual Yang–Mills theory and the Penrose–Ward transform............... 17 3.1. Motivation ..................................... 17 3.2. Penrose–Wardtransform. ....... 19 3.3. Example: Belavin–Polyakov–Schwarz–Tyupkin instanton............... 23 4. Supertwistorspace ................................ ....... 24 4.1. A brief introduction to supermanifolds . ............ 24 4.2. Supertwistorspace .............................. ...... 27 4.3. Superconformalalgebra . ....... 28 5. Supersymmetric self-dual Yang–Mills theory and the Penrose–Ward transform . 32 5.1. Penrose–Wardtransform. ....... 32 5.2. HolomorphicChern–Simonstheory . ......... 36 6. = 4 supersymmetric Yang–Mills theory from supertwistor space ............ 48 N 6.1. Motivation ..................................... 48 6.2. = 4 supersymmetric Yang–Mills theory from supertwistor space ......... 49 N Part II: Tree-level gauge theory scattering amplitudes 7. Scattering amplitudes in Yang–Mills theories . ............... 55 7.1. Motivation and preliminaries . ......... 55 7.2. Colourordering................................. ..... 57 7.3. Spinor-helicity formalism re-visited . .............. 59 8. MHVamplitudesandtwistortheory . ......... 61 8.1. Tree-level MHV amplitudes . ....... 61 8.2. Tree-level MHV superamplitudes . ......... 64 8.3. Witten’s half Fourier transform . .......... 68 8.4. MHV superamplitudes on supertwistor space . ........... 72 9. MHVformalism ..................................... 74 9.1. Cachazo–Svrˇcek–Witten rules . .......... 74 9.2. Examples ....................................... 76 9.3. MHVdiagramsfromtwistorspace . ....... 78 9.4. Superamplitudes in the MHV formalism . ......... 85 9.5. Localisationproperties . ......... 89 10. Britto–Cachazo–Feng–Witten recursion relations . .................. 91 10.1. Recursion relations in pure Yang–Mills theory . ............... 91 10.2. Recursion relations in maximally supersymmetric Yang–Mills theory . 94 3 Appendices A.Vectorbundles .................................... ..... 98 B. Characteristicclasses. ..........101 C.Categories....................................... 103 D.Sheaves .......................................... 104 4 Part I Twistor re-formulation of gauge theory 1. Twistor space 1.1. Motivation Usually, the equations of motion of physically interesting theories are complicated systems of coupled non-linear partial differential equations. This thus makes it extremely hard to find explicit solutions. However, among the theories of interest are some which are completely solvable in the sense of allowing for the construction (in principle) of all solutions to the corresponding equations of motion. We shall refer to these systems as integrable systems. It should be noted at this point that there are various distinct notions of integrability in the literature and here we shall use the word ‘integrability’ in the loose sense of ‘complete solvability’ without any concrete assumptions. The prime examples of integrable theories are the self-dual Yang–Mills and gravity theories in four dimensions including
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