Introduction to Supersymmetry Christian SÄamann Notes by Anton Ilderton August 24, 2009 1 Contents 1 Introduction 5 1.1 A SUSY toy model ............................... 5 1.2 What is it good for? .............................. 10 2 Spinors 12 2.1 The Poincar¶egroup .............................. 12 2.2 Spin and pin groups .............................. 18 2.3 Summary .................................... 22 3 The SUSY algebra 24 3.1 SUSY algebra on R1;3 ............................. 24 3.2 Representations ................................. 25 3.3 The Wess{Zumino model ........................... 28 4 Superspace and super¯elds 32 4.1 Reminder: Gra¼mann numbers. ........................ 32 4.2 Flat superspace ................................. 33 4.3 Super¯elds ................................... 35 5 SUSY{invariant actions from super¯elds 38 5.1 Actions from chiral s¯elds ........................... 38 5.2 Actions with vector super¯elds ........................ 42 6 SUSY quantum ¯eld theories 49 6.1 Abstract considerations ............................ 49 6.2 S¯eld quantisation of chiral s¯elds ...................... 51 6.3 Quantisation of Super Yang{Mills theory . 58 7 Maximally SUSY Yang{Mills theories 63 7.1 Spinors in arbitrary dimensions ........................ 63 7.2 Actions and constraint equations ....................... 64 7.3 d = 4, N = 4 super Yang{Mills theory .................... 67 8 Seiberg-Witten Theory 72 8.1 The moduli space of pure N = 2 SYM theory . 72 8.2 Duality ..................................... 74 8.3 The exact e®ective action ........................... 75 A Conventions and identities 78 B Solutions to exercises 81 2 Preface Supersymmetry, or SUSY for short, is an extension of the classical symmetries of ¯eld theories. SUSY was discovered in the early 1970's and has attracted growing attention ever since, even though there is still no experimental evidence for its existence up to this day. There are essentially two reasons why high-energy physicists keep interested in supersymmetry: From a phenomenological point of view, the supersymmetric extension of the Standard Model provides very reasonable solutions to some of the remaining puzzles in particle physics. On the other hand, supersymmetric ¯eld theories have many intriguing features which often can be accessed analytically, making them ideal toy models for theorists. These lecture notes are based on a series of lectures given by myself from April to June 2009 in the School of Mathematics at Trinity College, Dublin. The material covered in these notes was presented during eleven lectures, each lasting 90 mins, which was a little ambitious, retrospectively. The aim of the lectures was to give a reasonable overview of this topic while being thorough enough to provide a graduate student with the necessary tools to do research involving supersymmetric ¯eld theories herself. I put some emphasis on motivating spinors, as I still remember struggling with the reasons for their existence when I ¯rst encountered them. Besides the standard material presented in almost any course on supersymmetry, I chose super¯eld quantisation, maximally supersymmetric Yang-Mills theories and Seiberg-Witten theory as additional topics for these lectures. I had just used supergraphs in a research project myself and their usefulness and simplicity was still fresh in my mind. The maximally supersymmetric Yang-Mills theories with their amazing properties were included because of the important role they play in string theory today. Finally, the section on Seiberg-Witten theory demonstrates why supersymmetric ¯eld theories are indeed beautiful toy models. If you should ¯nd any typos or mistakes in the text, please let us know by sending an email to [email protected] or [email protected]. The most recent version of these lecture notes can be found on the lecture series homepage, http://www.christiansaemann.de/supersymmetry.html . On this webpage, there is also a (still preliminary) version of a Mathematica notebook which performs many of the tedious computations necessary for understanding these lectures automatically. I am very grateful to Anton Ilderton for doing such an amazing job with writing up these notes, correcting prefactors and providing typed-up solutions to all the steps left as exercises during the lectures. I would also like to thank all the people attending the lectures at Trinity for making them more interesting and lively by asking questions. Christian SÄamann 3 General remarks Our metric is (¡; +; +; +), which di®ers by a sign from the metric of most quantum ¯eld theory textbooks, as e.g. Peskin{Schroeder. The remainder of our conventions are introduced in the text, as they are needed. A summary is included in the appendices. Numerous exercises are included, the intent of which is mainly to make the user familiar with the kind of methods, tricks and identities needed to perform SUSY calculations. Full solutions can be found in the appendices. Recommended textbooks and lecture notes (with metric conventions) Recall that the two di®erent choices for Minkowski metric, the mostly plus (¡; +; +; +) and the mostly minus (+; ¡; ¡; ¡), are often referred to as the East Coast Metric (ECM) and the West Coast Metric (WCM), respectively. [1] J. Wess and J. Bagger, \Supersymmetry and supergravity", Princeton, USA: Univ. Pr. (1992) 259 p. ECM, the essentials. [2] S. Weinberg, \The quantum theory of ¯elds. Vol. 3: Supersymmetry", Cambridge, UK: Univ. Pr. (2000) 419 p. ECM, uses Dirac spinors, more physics. [3] I. L. Buchbinder and S. M. Kuzenko, \Ideas and methods of supersymmetry and supergravity: Or a walk through superspace", Bristol, UK: IOP (1998) 656 p. ECM, supermathematics, NR theorems, super Feynman rules. [4] S. J. Gates, Marcus T. Grisaru, M. Rocek and W. Siegel, \Superspace, or one thousand and one lessons in supersymmetry," Front. Phys. 58 (1983) 1 [hep- th/0108200]. ECM, many useful things, in particular super Feynman rules. [5] A. Van Proeyen, \Tools for supersymmetry", arXiv:hep-th/9910030. ECM, more algebraic. [6] S. P. Martin, \A Supersymmetry Primer", arXiv:hep-ph/9709356. ECM, particle physics, MSSM. [7] A. Bilal, \Introduction to supersymmetry", arXiv:hep-th/0101055. WCM, for particle physics conventions. [8] J. D. Lykken, \Introduction to supersymmetry", arXiv:hep-th/9612114. WCM, useful as a reference. Cover: an otter, and it's supersymmetric partner, the sotter. 4 1 Introduction 1.1 A SUSY toy model At the beginning of the 1970's, people started looking at SUSY toy models. In this sec- tion we will discuss a simple model which will illustrate many of the important physical properties of SUSY theories which are commonly discussed in a ¯eld theory context. It will also serve to illustrate the fundamentals of many calculations we will later perform in ¯eld theory, in a simple and accessible setting. Much of the following discussion can be found in [9, 10] in more detail. x1 De¯nition. Our model is quantum mechanical. We have a hermitian Hamiltonian H and non{hermitian operators Q, Qy related through the anti{commutator 1 1 H = fQ; Qyg ´ (QQy + QyQ) ; (1) 2 2 where the operators obey the following \0 + 1 dimensional SUSY algebra", fQ; Qg = fQy;Qyg = 0 (2) [Q; H] = [Qy;H] = 0 : The Q and Qy are called supercharges and generate supersymmetry transformations. We always count real supercharges, so here we have two of them. It is a direct consequence 2 y2 of (2) that Q = Q = 0. ¡ ¢ x2 Properties. Consider a Hilbert space H; h j i carrying a representation of H, Q and Qy. From the algebra (2), it follows that H is positive de¯nite, for, given any state j ª i 2 H, 2h ª jHj ª i = h ª jfQ; Qygj ª i = h ª jQQyj ª i + h ª jQyQj ª i (3) = kQj ª ik2 + kQyj ª ik2 ¸ 0 : We see that states in a SUSY theory have non{negative energy; note that the minimum may or may not be obtained. To further examine the state space, we diagonalise H and consider the eigenstates j n i such that Hj n i = Enj n i: (4) We must treat the two cases En = 0 and En > 0 separately. Wep begin with thep case y y En = E > 0. We may here introduce the scaled operators a = Q= 2E and a = Q = 2E which, within the space of states of energy E, obey the algebra fa; ayg = 1 ; fa; ag = fay; ayg = 0 : (5) This is a simple example of a Cli®ord algebra, which we will study later on. We now construct all the states with energy E, in analogy to the construction of harmonic 5 oscillator states via creation operators acting on the vacuum. Again, we clearly have a2 = ay2 = 0, from which it follows that the only eigenvalue of a is 0. Call the state with this eigenvalue j ¡ i. We can create no more states by acting on this with a, and we can create only one more by acting with ay (since ay2 = 0), which we call j + i ´ ayj ¡ i. Hence we have a subsystem of two states obeying ayj ¡ i = j + i ; aj + i = j ¡ i ; aj ¡ i = ayj + i = 0 : A simple 2d representation of the algebra (5) is given by the following matrices and vectors, Ã ! Ã ! Ã ! Ã ! 0 0 0 1 1 0 a = ; ay = ; j + i = ; j ¡ i = : (6) 1 0 0 0 0 1 We see here a basic example of the existence of two types of states in SUSY theories; `+' states and `¡' states which will later be bosons and fermions. They are transformed into each other by the action of the SUSY generators, and more generally we will see that Qj boson i = j fermion i ;Qj fermion i = j boson i ; and similarly for the action of Qy. We also have an example of the SUSY property that states of non{zero energy are degenerate and appear in pairs; a possible non{zero spectrum of a SUSY theory is sketched in Fig.