hep-th/9611003   03 Nov 1996

Supersymmetry, Supercurrent,

and Scale Invariance

Olivier Piguet1;2 Instituto da F'isica, Universidade Cat'olica de Petr'opolis

25610-130 Petr'opolis, RJ, Brazil

and Centro Brasileiro de Pesquisas F'isicas (CBPF) Rua Xavier Sigaud 150, 22290-180 Urca, RJ, Brazil

CBPF-NF-072/96 UGVA-DPT 1996/08-938

hep-th/9611003

Lectures given at the Catholic University of Petr'opolis (RJ) and at the CBPF (Rio

de Janeiro), September 1995 and July-August 1996(Notes written in collaboration with Oswaldo Monteiro Del Cima)

Contents 1 Introduction 4 2 Generalities 6

2.1 Extended Supersymmetry Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 N =1 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Invariant Actions and Ward Identity Operators . . . . . . . . . . . . . . . . . . . . 7

3 The Baby Model 9

3.1 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 On leave of absence from: D'epartement de Physique Th'eorique - Universit'e de Gen`eve, 24 quai E. Ansermet - CH-1211 Gen`eve 4, Switzerland.

2 Supported in part by the Swiss National Science Foundation and the Brazilian National Research Council

(CNPq)

3.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Free Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Super Yang-Mills Theory 11

4.1 Pure Super Yang-Mills Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 The Wess-Zumino Gauge: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Gauge Fixing and BRS Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.4 Matter Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.5 R Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.6 Slavnov-Taylor identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.7 Rigid Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.8 Ward Identities and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.9 General Classical Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.10 Soft Breakings of Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17