Locally supersymmetric D=3 non-linear sigma models

TL;DR

This paper classifies three-dimensional non-linear sigma models with various local supersymmetries, revealing their target space geometries and demonstrating two-loop finiteness for certain cases after dimensional reduction.

Contribution

It provides a comprehensive classification of N=1 to N=16 supersymmetric sigma models in three dimensions, detailing their target space geometries and finiteness properties.

Findings

Target spaces are Riemannian, Kahler, quaternionic, or symmetric spaces.

Models with N=3 and N≥5 are two-loop finite after reduction.

Unique symmetric spaces correspond to specific supermultiplet counts and exceptional groups.

Abstract

We study non-linear sigma models with N local supersymmetries in three space-time dimensions. For N=1 and 2 the target space of these models is Riemannian or Kahler, respectively. All N>2 theories are associated with Einstein spaces. For N=3 the target space is quaternionic, while for N=4 it generally decomposes into two separate quaternionic spaces, associated with inequivalent supermultiplets. For N=5,6,8 there is a unique (symmetric) space for any given number of supermultiplets. Beyond that there are only theories based on a single supermultiplet for N=9,10,12 and 16, associated with coset spaces with the exceptional isometry groups $F_{4(-20)}$, $E_{6(-14)}$, $E_{7(-5)}$ and $E_{16(+8)}$, respectively. For $N=3$ and $N\geq5$ the $D=2$ theories obtained by dimensional reduction are two-loop finite.