# David Skinner: Supersymmetry

This is the site for the Part III course on Supersymmetry, offered to MASt students in Maths at Cambridge. The notes here are in progress and will be updated as the course proceeds. It's likely that they'll contain somewhat more than we'll have time to cover in the lectures.

# Contents

**Introduction: PDF File**

Preliminaries. Motiviation. What is supersymmetry? Why study it?.**Supersymmetry in Zero Dimensions: PDF File**

Fermions and super vector spaces. Differentiation and Berezin integration. QFT in zero dimensions. Method of steepest descent for bosons. Gaussian integrals for fermions. A simple supersymmetric theory. Localization. Landau-Ginzburg theories and the chiral ring. The Duistermaat-Heckmann localization formula. A glimpse of two-dimensional Yang-Mills theory.**Supersymmetric Quantum Mechanics: PDF File**

SQM with a Potential. Supersymmetric Ground States and the Witten Index. The Path Integral Approach to Quantum Mechanics. Nonlinear Sigma Models. The Witten Index of a NLSM. The Atiyah-Singer Index Theorem.**Supersymmetric Quantum Field Theory: PDF File**

Dirac spinors in two dimensions. Superspace and Superfields. Chiral Superfields. Supersymmetric Actions in d=2. The Wess-Zumino Model. U(1) Axial and Vector Symmetries. The Vacuum Moduli Space. Seiberg Non-Renormalization Theorems.**Nonlinear Sigma Models: PDF File**

Kahler Manifolds as Complex Manifolds with a compatible Symplectic Structure. Supersymmetric NLSM on a Kahler Manifold. Anomalies in R-Symmetries. The Beta Function of a NLSM.

# Problem Sheets

**Problem Sheet 1: PDF File****Problem Sheet 2: PDF File**

**Problem Sheet 3: PDF File**

**Specimen Exam Paper: PDF File**

# Recommended Books

The main book for the course is *Mirror Symmetry* by K. Hori. C. Vafa *et al.* You can download a pdf copy of the book from the Clay Maths Institute site. We'll mostly be looking at chapters 8-16.