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Killing Superalgebras in 2 Dimensions

Andrew D. K. Beckett

TL;DR

This work advances the 2D theory of Killing superalgebras by constructing admissible spinor connections with Dirac currents on $(0,2)$ and $(1,1)$ manifolds and framing their Killing structures as filtered subdeformations of flat model superalgebras. Using Spencer cohomology, it identifies the inequivalent infinitesimal deformations in degree $(2,2)$, finding a one-parameter class when the spinor module is non-chiral, and showing trivial cohomology in certain chiral Lorentzian cases. The maximal supersymmetry analysis yields explicit deformations of the 2D isometry algebras, producing homogeneous backgrounds such as $H^2$, $AdS_2$, and $dS_2$ with curvature $R$ determined by a Killing function $b$. The results give concrete Killing spinor equations $\nabla_X\epsilon= bX\cdot\epsilon$ or $b(*X)\cdot\epsilon$ and clarify the role of Dirac currents and admissible bilinears in shaping the associated geometry and symmetry algebras. Overall, the paper provides a clear, computable 2D realization of Killing superalgebras, their deformations, and the corresponding maximally symmetric geometries within the general framework developed in prior work.

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 Poincaré) 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.

Killing Superalgebras in 2 Dimensions

TL;DR

This work advances the 2D theory of Killing superalgebras by constructing admissible spinor connections with Dirac currents on and manifolds and framing their Killing structures as filtered subdeformations of flat model superalgebras. Using Spencer cohomology, it identifies the inequivalent infinitesimal deformations in degree , finding a one-parameter class when the spinor module is non-chiral, and showing trivial cohomology in certain chiral Lorentzian cases. The maximal supersymmetry analysis yields explicit deformations of the 2D isometry algebras, producing homogeneous backgrounds such as , , and with curvature determined by a Killing function . The results give concrete Killing spinor equations or and clarify the role of Dirac currents and admissible bilinears in shaping the associated geometry and symmetry algebras. Overall, the paper provides a clear, computable 2D realization of Killing superalgebras, their deformations, and the corresponding maximally symmetric geometries within the general framework developed in prior work.

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 and filtered deformations of the non-chiral flat model (Euclidean and Poincaré) superalgebra 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.
Paper Structure (15 sections, 6 theorems, 38 equations, 3 tables)

This paper contains 15 sections, 6 theorems, 38 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Constructions, conventions and formulae for spinor modules
  3. Signature-agnostic Clifford algebra conventions
  4. Pinor and spinor modules, admissible bilinears and Dirac currents
  5. Signature $\boldsymbol{(1,1)}$
  6. Signature $\boldsymbol{(0,2)}$
  7. Adjoints and Fierz identity
  8. Spencer cohomology and filtered subdeformations
  9. Spencer cohomology
  10. Maximally supersymmetric filtered subdeformations
  11. Admissible connections and Killing superalgebras
  12. Admissible connections
  13. Further insights from integrability
  14. Maximally supersymmetric case
  15. Summary of results

Key Result

Proposition 2.2

Let $\mathsf{S}$ be the real irreducible pinor module in signature $(1,1)$ or $(0,2)$ and $B$ either of the admissible bilinears from the appropriate table above. Then for $\epsilon,\epsilon'\in \mathsf{S}$, we have

Theorems & Definitions (9)

  • Definition 2.1: admissible bilinear
  • Definition 2.1: admissible bilinear
  • Proposition 2.2: Fierz identity
  • Corollary 2.3
  • Proposition 3.1
  • Definition 4.1
  • Lemma 4.2
  • Proposition 4.3
  • Lemma 4.4