Exploring the Supersymmetric Sigma Model

TL;DR

This paper introduces the non-linear sigma model, discusses its supersymmetric extension, and explores specific cases like the $SU(2) imes U(1)$ model, highlighting its topological and conformal properties.

Contribution

It provides a clear presentation of the supersymmetric sigma model, including its equations of motion and specific cases, with insights into its topological and anomaly cancellation aspects.

Findings

The model's conformal anomaly is canceled when obeying Einstein equations.

Supersymmetric extension involves chiral and twisted chiral fields.

Explicit analysis of the $SU(2) imes U(1)$ case.

Abstract

The purpose of this work is to present some basic concepts about the non-linear sigma model in a simple and direct way. We start with showing the bosonic model and the Wess-Zumino-Witten term, making some comments about its topological nature, and its association with the torsion. It is also shown that to cancel the quantum conformal anomaly the model should obey the Einstein equations. We provide a quick introduction about supersymmetry in chapter 2 to help the understanding the supersymmetric extension of the model. In the last chapter we present the supersymmetric model and its equations of motion. Finally we work-out the two-supersymmetry case, introducing the chiral as well as the twisted chiral fields, expliciting the very specific $SU(2)\otimes U(1)$ case.