$2d$ Sigma Models and Geometry

TL;DR

This review explores how two-dimensional supersymmetric sigma models relate to various complex and Riemannian geometries, highlighting the rich geometric structures emerging from different supersymmetry configurations.

Contribution

It provides a comprehensive overview of the connections between 2D supersymmetry and diverse geometric frameworks, including torsionful and generalized geometries.

Findings

Supersymmetry imposes specific geometric structures on target spaces.

Torsionful geometries arise naturally in 2D supersymmetric models.

The study links supersymmetry to advanced geometric concepts like generalized geometry.

Abstract

Supersymmetric nonlinear sigma models have target spaces that carry interesting geometry. The geometry is richer the more supersymmetries the model has. The study of models with two dimensional world sheets is particularly rewarding since they allow for torsionful geometries. In this review I describe and exemplify the relation of $2d$ supersymmetry to Riemannian, complex, bihermitian, $(p,q)$ Hermitean, K\"ahler, hyperk\"ahler, generalised geometry and more