Killing sections and sigma models with Lie algebroid targets

TL;DR

This paper generalizes the concept of Killing vector fields to Killing sections in Riemannian Lie algebroids and explores their role in the symmetries of sigma models with such targets, interpreting solutions as generalized harmonic maps.

Contribution

It introduces the notion of Killing sections for Riemannian Lie algebroids and applies this to analyze symmetries and critical points in related sigma models.

Findings

Killing sections generalize classical Killing vector fields.

Symmetries of sigma models with Lie algebroid targets are characterized.

Critical points correspond to generalized harmonic maps.

Abstract

We define and examine the notion of a Killing section of a Riemannian Lie algebroid as a natural generalisation of a Killing vector field. We show that the various expression for a vector field to be Killing naturally generalise to the setting of Lie algebroids. As an application we examine the internal symmetries of a class of sigma models for which the target space is a Riemannian Lie algebroid. Critical points of these sigma models are interpreted as generalised harmonic maps.