Killing Superalgebras in 2 Dimensions

TL;DR

This paper explores Killing superalgebras on 2D pseudo-Riemannian manifolds, computing cohomology and deformations, and linking these to geometric Killing spinors in different signatures.

Contribution

It provides explicit examples, computes Spencer cohomology, and analyzes deformations of superalgebras in 2D geometries, extending the theoretical framework.

Findings

Killing superalgebras are constructed for various 2D geometries.

Spencer cohomology groups are computed for these superalgebras.

Deformations of the superalgebras relate to geometric Killing spinors.

Abstract

We provide some examples of Killing superalgebras on 2-dimensional pseudo-Riemannian manifolds within the theoretical framework established in [SIGMA 21 (2025), 081, 61 pages, arXiv:2409.11306]. We compute the Spencer cohomology group $\mathsf{H}^{2,2}(\mathfrak{s}_-;\mathfrak{s})$ and filtered deformations of the non-chiral flat model (Euclidean and Poincar\'e) superalgebra $\mathfrak{s}$ for various Dirac currents and show these arise as Killing superalgebras for (imaginary) geometric Killing and skew-Killing spinors in both Riemannian and Lorentzian signature.