Irreducible holonomy algebras of Riemannian supermanifolds

TL;DR

This paper classifies the possible irreducible holonomy algebras of Riemannian supermanifolds, extending classical Berger's theorem to the supergeometry setting, under specific algebraic assumptions.

Contribution

It generalizes Berger's classical classification to Riemannian supermanifolds with certain algebraic structures, identifying all possible irreducible holonomy algebras.

Findings

Classification of irreducible holonomy algebras in supergeometry

Extension of Berger's theorem to supermanifolds

Identification of algebraic structures involved

Abstract

Possible irreducible holonomy algebras $\g\subset\osp(p,q|2m)$ of Riemannian supermanifolds under the assumption that $\g$ is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.