hep-th/9612114 10 Dec 1996 FERMILAB-PUB-96/445-T INTRODUCTION TO SUPERSYMMETRY JOSEPH D. LYKKEN Fermi National Accelerator Laboratory P.O. Box 500 Batavia, IL 60510 These lectures give a self-contained introduction to supersymmetry from a modern perspective. Emphasis is placed on material essential to understanding duality. Topics include: central charges and BPS-saturated states, supersymmetric nonlinear sigma models, N=2 Yang-Mills theory, holomorphy and the N=2 Yang-Mills fi function, supersymmetry in 2, 6, 10, and 11 spacetime dimensions. 1 Introduction "Never mind, lads. Same time tomorrow. We must get a winner one day." - Peter Cook, as the doomsday prophet in "The End of the World". Supersymmetry, along with its monozygotic sibling superstring theory, has become the dominant framework for formulating physics beyond the standard model. This despite the fact that, as of this morning, there is no unambiguous experimental evidence for either idea. Theorists find supersymmetry appealing for reasons which are both phenomenological and technical. In these lectures I will focus exclusively on the technical appeal. There are many good recent reviews of the phenomenology of supersymmetry. 1 Some good technical reviews are Wess and Bagger, 2 West, 3 and Sohnius. 4 The goal of these lectures is to provide the student with the technical background requisite for the recent applications of duality ideas to supersymmetric gauge theories and superstrings. More specifically, if you absorb the material in these lectures, you will understand Section 2 of Seiberg and Witten, 5 and you will have a vague notion of why there might be such a thing as M -theory. Beyond that, you're on your own. 2 Representations of Supersymmetry 2.1 The general 4-dimensional supersymmetry algebra A symmetry of the S-matrix means that the symmetry transformations have the effect of merely reshuffling the asymptotic single and multiparticle states. The known symmetries of the S-matrix in particle physics are: 1 ffl Poincar'e invariance, the semi-direct product of translations and Lorentz rotations, with generators Pm, Mmn. ffl So-called "internal" global symmetries, related to conserved quantum numbers such as electric charge and isospin. The symmetry generators are Lorentz scalars and generate a Lie algebra, [B`; Bk] = iCj`kBj ; (1) where the Cj`k are structure constants. ffl Discrete symmetries: C, P, and T. In 1967, Coleman and Mandula 6 provided a rigorous argument which proves that, given certain assumptions, the above are the only possible symmetries of the S-matrix. The reader is encouraged to study this classic paper and think about the physical and technical assumptions which are made there. The Coleman-Mandula theorem can be evaded by weakening one or more of its assumptions. In particular, the theorem assumes that the symmetry algebra of the S-matrix involves only commutators. Weakening this assumption to allow anticommuting generators as well as commuting generators leads to the possibility of supersymmetry. Supersymmetry (or SUSY for short) is defined as the introduction of anticommuting symmetry generators which transform in the ( 12; 0) and (0; 12 ) (i.e. spinor) representations of the Lorentz group. Since these new symmetry generators are spinors, not scalars, supersymmetry is not an internal symmetry. It is rather an extension of the Poincar'e spacetime symmetries. Supersymmetry, defined as the extension of the Poincar'e symmetry algebra by anticommuting spinor generators, has an obvious extension to spacetime dimensions other than four; the Coleman-Mandula theorem, on the other hand, has no obvious extension beyond four dimensions. In 1975, Haag, Lopusza'nski, and Sohnius 7 proved that supersymmetry is the only additional symmetry of the S-matrix allowed by this weaker set of assumptions. Of course, one could imagine that a further weakening of assumptions might lead to more new symmetries, but to date no physically compelling examples have been exhibited. 8 This is the basis of the strong but not unreasonable assertion that: