Killing Superalgebra Overview

• Killing superalgebra is a ℤ₂-graded Lie superalgebra that encodes Killing spinors, symmetry generators, and geometric flux through a bilinear squaring map.
• It employs filtered deformations and Spencer cohomology to classify supersymmetric backgrounds and reconstruct bosonic field equations in various supergravity models.
• Its adaptable structure across dimensions interlinks Killing spinors, Killing–Yano algebras, and hidden symmetries, providing a unifying framework for geometric analysis.

A Killing superalgebra is a $\mathbb{Z}_2$-graded Lie superalgebra that encodes the supersymmetries and bosonic symmetry content of (pseudo-)Riemannian or supergravity backgrounds admitting Killing spinors. These superalgebras provide a unifying algebraic framework for the paper of (rigid or local) supersymmetric field theories, supergravity, and the geometry of manifolds with special holonomy or flux. Their structure is tightly controlled by cohomological data (Spencer cohomology of filtered $\mathbb{Z}$-graded superalgebras) and is ubiquitous in the global analysis of maximally and highly supersymmetric backgrounds across all physically relevant dimensions.

1. Definitions and Construction

Fundamentally, the Killing superalgebra arises as follows. Let $(M,g)$ be a pseudo-Riemannian (spin) manifold with spinor bundle $S\to M$ and a (possibly flux-deformed) connection $D=\nabla-\beta$, where $\nabla$ is the Levi–Civita connection and $\beta$ is a Clifford-algebra-valued 1-form encoding geometric flux or supergravity couplings. The odd part of the superalgebra consists of sections $\epsilon\in \Gamma(S)$ solving the generalized Killing spinor equation

$$D_X\epsilon = \nabla_X\epsilon - \beta(X)\epsilon = 0$$

for all $X\in \Gamma(TM)$.

A bilinear "squaring" map (the Dirac current)

$\kappa: \Gamma(S)\times \Gamma(S) \to \Gamma(TM)$

is chosen, equivariant under the holonomy of $D$ and producing Killing vectors (or generalizations thereof). The even part is then given by vector fields on $M$ preserving $(g,\beta)$—i.e., infinitesimal isometries also preserving auxiliary bosonic data—possibly augmented by R-symmetry or internal gauge generators if present (Beckett, 17 Sep 2024, Beckett, 10 Nov 2025).

The defining brackets are:

• $[X,Y] = \mathcal{L}_X Y$ for $X, Y$ even (Lie algebra of symmetries);
• $[X,\epsilon]=\mathcal{L}_X\epsilon$ (the spinorial Lie derivative, specialization to $\nabla$ or a generalized version);
• $$[\epsilon_1,\epsilon_2]=\kappa(\epsilon_1,\epsilon_2)$$ (the Dirac squaring, yielding Killing vectors or further higher-degree forms in appropriate contexts).

Generalizations may include a further skew-symmetric pairing of spinors taking values in R-symmetry (or gauge) algebras, relevant for extended supergravities, or in the context of Killing superalgebras for generalized spin structures (Beckett, 10 Nov 2025).

2. Filtered Deformations and Spencer Cohomology

Killing superalgebras are not, in general, just subalgebras of the flat-space (Poincaré) superalgebra but are filtered deformations of $\mathbb{Z}$-graded subalgebras thereof (Figueroa-O'Farrill et al., 2016, Beckett, 17 Sep 2024, Beckett, 10 Nov 2025). Denoting by

$\mathfrak{s} = V \oplus S \oplus \mathfrak{so}(V)$

the flat model (with $V = \mathbb{R}^{p,q}$ and $S$ a spin module), one defines a negative grading:

• $\deg(V) = -2$, $\deg(S) = -1$, $\deg(\mathfrak{so}(V)) = 0$.

A filtered deformation is a Lie superalgebra $\mathfrak{g}$ with a filtration $$\mathfrak{g}^0 \subset \mathfrak{g}^{-1} \subset \mathfrak{g}^{-2} = \mathfrak{g}$$ such that the associated graded algebra is a subalgebra of $\mathfrak{s}$. The possible deformations are classified by the Spencer cohomology group $H^{2,2}(a_{-};a)$ of the negative part $a_{-}$ of the associated graded algebra $a$ with coefficients in $a$. The main nontrivial ingredient is the degree-(2,2) Spencer cocycle, dictating higher order corrections to the Lie brackets (Beckett, 17 Sep 2024, Figueroa-O'Farrill et al., 2016).

Concretely, the cocycle data encode the geometric content of all compatible couplings (fluxes, auxiliary fields, R-symmetry, etc) through the transformation properties of the connection $\beta$ and the Dirac current $\kappa$.

Filtered deformations naturally yield Killing superalgebras for maximally and highly supersymmetric backgrounds (i.e., those with more than half the maximal number of Killing spinors). In these cases, background fields are fixed by the Jacobi identity of the superalgebra—supersymmetry is so strong that the purely algebraic closure conditions reconstruct the field equations for the bosonic backgrounds (e.g., Einstein-Maxwell or supergravity field equations) (Figueroa-O'Farrill et al., 2016).

3. Explicit Examples and Dimensional Hierarchy

The structure and realization of Killing superalgebras depend on dimension, background type, and geometric or supergravity context:

• 2D and Chiral Theories: Killing superalgebras are classified by filtered deformations of the (Poincaré-type) flat model, with solutions parametrized by real numbers and corresponding to the isometry superalgebras of constant-curvature spaces ($\mathrm{AdS}_2,\,dS_2,\,\mathbb{R}^2$) (Beckett, 2 Oct 2024).
• 4D, 5D, 6D Lorentzian Superalgebras: For N=1 supersymmetry, Killing superalgebras are associated with minimal Poincaré blocks (Medeiros et al., 2016, Beckett et al., 2021, Medeiros et al., 2018). In 5D and 6D, extra bosonic fields (e.g., $\mathfrak{sp}(1)$-valued 1-forms, self-dual 3-forms) appear in the Spencer cohomology, providing the algebraic content for the corresponding "on-shell" or "off-shell" supergravity backgrounds.
• AdS and Warped Geometries: The Killing superalgebras of higher-dimensional AdS backgrounds are uniformly captured by constructions involving Killing spinors and the associated Dirac currents. For example, in AdS$_4$, the isometry superalgebra is always $\mathfrak{osp}(1|4)$ (minimal case), and similar uniformity is found in AdS$_5$, AdS$_6$, and AdS$_7$ (Beck et al., 2017).
• 11D Supergravity: The Killing superalgebra encodes all background flux and bosonic field equations via algebraic closure; the cohomology class determines, and is determined by, the underlying $F$-field (Figueroa-O'Farrill et al., 2016).

A distinctive feature in all maximally supersymmetric cases is that the classification of backgrounds up to local isometry reduces to the classification of filtered subdeformations (equivalently, Spencer cohomology classes) of the flat (extended) Poincaré superalgebra.

4. Relation to Killing-Yano Superalgebras and Hidden Symmetries

A related but distinct construction is the Killing–Yano (KY) superalgebra (Ertem, 2017, Açık et al., 2016, Ertem, 2017): the $\mathbb{Z}_2$-graded algebra of higher degree forms generalizing ordinary Killing vectors, defined by solutions to the KY equation

$\nabla_X \omega = \frac{1}{p+1} i_X d\omega$

for a $p$-form $\omega$. On constant-curvature manifolds, the space $\KY(M)$ with the Schouten–Nijenhuis bracket closes to a rigid Lie superalgebra, with even part given by odd-degree KY forms and odd part by even-degree forms. The rigidity (triviality of 2nd Spencer cohomology) of the KY superalgebra in these cases shows that the full algebra of hidden symmetries is a strong geometric invariant. Conformal Killing–Yano superalgebras, realized via a further modified bracket, provide another canonical class of such rigid symmetry algebras (Ertem, 2017).

In applications, the squaring map of Killing spinors yields KY forms, so the Killing superalgebra and the KY superalgebra are tightly interlinked.

5. Generalizations: Extended Symmetry Content

For backgrounds with enhanced R-symmetry, internal gauge groups, or supermanifold geometry, Killing superalgebras are defined on the full symmetry algebra—including gauge and R-symmetry generators and spinors charged under these actions (Beckett, 10 Nov 2025, Klinker, 2020). Here, the odd-odd bracket of the superalgebra produces both geometric (Killing vector) and algebraic (gauge, R-symmetry) contributions: $$[\epsilon, \zeta] = \kappa(\epsilon, \zeta) + \rho(\epsilon, \zeta)$$ where $\rho$ is an $r$-valued pairing. The admissibility conditions (closure, preservation of background fields, algebraic constraints) are strengthened, but the overall construction remains governed by filtered deformations and controlled by generalized Spencer cohomologies.

In supermanifold terms, the Killing superalgebra forms an ideal in the even-odd vector field algebra of a Batchelor supermanifold, and is central to the structure of "supersymmetric Killing structures" (Klinker, 2020).

6. Algebraic and Geometric Classification

For highly supersymmetric (i.e., maximal or near-maximal) backgrounds in Lorentzian signature, the Homogeneity Theorem ensures that the Dirac current map $\kappa$ is surjective onto the tangent bundle, and any such Killing superalgebra arises from an integrable Lie pair $(S', \beta+\gamma)$, with $\beta$ and $\gamma$ capturing the algebraic structure of the connection and the Dirac current (Beckett, 17 Sep 2024, Beckett, 10 Nov 2025).

A geometric reconstruction theorem shows that every (suitably) filtered deformation corresponds to a unique homogeneous geometry on which the Killing superalgebra is realized as symmetries preserving the spinor connection and relevant flux data.

For all the backgrounds of interest—maximally supersymmetric supergravity vacua, rigidly supersymmetric field theories on curved backgrounds, and their twistor/Killing-Yano generalizations—the Killing superalgebra thus provides a precise algebraic invariant, directly encoding the allowed geometry and underlying field-theoretic data.

7. Summary Table: Typical Structures Across Dimensions

Dimension	Odd generators	Even generators	Deformation/Extension	Typical Superalgebra
2	Pinors $S$, $S_+\oplus S_-$	Killing vectors (filtered)	Filtered deformations parametrized by $b$	Isometry+, deformations $\mathfrak{s}_b$
4	Killing spinors $S$	$\so(1,3)$-Killing vectors	Scalar/vector-valued connections ($a$, $b$, $\phi$)	Poincaré/AdS/Group-manifold types
5	Killing spinors $S$	$\so(1,4)$-Killing vectors, $\mathfrak{sp}(1)$	2-form, $\mathfrak{sp}(1)$-valued 1-form	OSP(1
6	$S$ ($\cong$ quaternionic)	$\so(1,5)$, $\mathfrak{sp}(1)$ still possible	3-form $H$, $\mathfrak{sp}(1)$-valued 1-form $\varphi$	OSP, group, extended superalgebras
$n$	Killing spinors	Killing vectors, possible gauge/R-symmetry	Encoded in Spencer $H^{2,2}$	Filtered subdeformation of Poincaré

The rigidity, algebraic classification, and reconstruction theorems for Killing superalgebras establish their centrality in geometric analysis and the structure theory of supersymmetric field theories and supergravity, serving as both geometric invariants and as organizing principles for the possible spaces of solutions and the structure of physical models (Beckett, 17 Sep 2024, Beckett, 10 Nov 2025, Figueroa-O'Farrill et al., 2016, Ertem, 2017).