Introduction to Rigid Supersymmetric Theories

Abstract

In these lectures we discuss the supersymmetry algebra and its irreducible representations. We construct the theories of rigid supersymmetry and gave their superspace formulations. The perturbative quantum properties of the extended supersymmetric theories are derived, including the superconformal invariance of a large class of these theories as well as the chiral effective action for N = 2 Yang-Mills theory. The superconformal transformations in four dimensional superspace are derived and encoded into one superconformal Killing superfield. It is also shown that the anomalous dimensions of chiral operators in a superconformal quantum field are related to their R weight.Some of this material follows the book of reference [0] by the author. Certain chapters of this book are reproduced, however, in other sections the reader is referred to the relevant parts of reference [0]. In this review, chapter 6 on superconformal theories and three sections of chapter 5 on flat directions, non-holomorphicity and the chiral effective action for N = 2 Yang-Mills theory are new material.The aim of the lectures is to provide the reader with the material required to understand more recent developments in the non-perturbative properties of quantum extended supersymmetric theories. ‡ This material is based on lectures presented at the Nato conference on "Confinement, Duality, and Nonperturbative Aspects of QCD", the Issac Newton Institute, Cambridge, UK and at the TASI 1998 Summer School, Boulder, Colorado, USA.

On compactified harmonic/projective superspace, 5D superconformal theories, and all that

“…However, since we are aiming at future applications to brane-world physics, a more pragmatic course is chosen here, which is based on the introduction of the relevant superconformal Killing vectors and elaborating associated building blocks. The concept of superconformal Killing vectors [28,29,30,31,32,21,33], has proved to be extremely useful for various studies of superconformal theories in four and six dimensions, see e.g. [34,35,36].…”

The energy-momentum multiplet of supersymmetric defect field theories

“…More precisely, this equation is equivalent to the superconformal Killing equations [30]. For example, letting v µ | = µ + ib µ and − 1 2 D α λ α | = Λ (1) + iΛ (2) , one can verify that (7.6) implies…”

“…It follows from the first relation that U R = 1 2 R a K a Φ a is a real multiplet. The gauge transformation of a chiral superfield is now [26,27,30] δΦ a = L + Φ a + 1 2 σR a Φ a . (7.14) Here there is no sum over a.…”

Anomalous superconformal Ward identities

“…The main difference between the two choices is the fact that L ij still contains a conserved vector current (associated with unbroken central charge transformations), whereas the vector current in L ijkl is not conserved. The first choice yields the N=2 superconformal anomaly relation in the standard form [10] i…”