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hep-th/9709062 8 Sep 97

CERN-TH/97-218

hep-th/9709062

INTRODUCTION TO SUPERSTRING THEORY

Elias Kiritsis\Lambda Theory Division, CERN, CH-1211, Geneva 23, SWITZERLAND

Abstract In these lecture notes, an introduction to superstring theory is presented. Classical strings, covariant and light-cone quantization, supersymmetric strings, anomaly cancelation, compactification, T-duality, supersymmetry breaking, and threshold corrections to low-energy couplings are discussed. A brief introduction to nonperturbative duality symmetries is also included.

Lectures presented at the Catholic University of Leuven and at the University of Padova during the academic year 1996-97.

To be published by Leuven University Press.

CERN-TH/97-218 March 1997

\Lambda e-mail: [email protected]

Contents 1 Introduction 5 2 Historical perspective 6 3 Classical string theory 9

3.1 The point particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Relativistic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Oscillator expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Quantization of the bosonic string 22

4.1 Covariant canonical quantization . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Spectrum of the bosonic string . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.5 Topologically non-trivial world-sheets . . . . . . . . . . . . . . . . . . . . . 30 4.6 BRST primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.7 BRST in string theory and the physical spectrum . . . . . . . . . . . . . . 33 4.8 Interactions and loop amplitudes . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conformal field theory 37

5.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Conformally invariant field theory . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 Radial quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.4 Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.5 The central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.6 The free fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.7 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.8 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.9 Representations of the conformal algebra . . . . . . . . . . . . . . . . . . . 54 5.10 Affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.11 Free fermions and O(N) affine symmetry . . . . . . . . . . . . . . . . . . . 59

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5.12 N=1 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 66 5.13 N=2 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 68 5.14 N=4 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 70 5.15 The CFT of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 CFT on the torus 75