Killing SpinorEdit
Killing spinors sit at an intersection of geometry and physics, tying the shape of space to the behavior of fundamental fields. They are special spinor fields defined on differentiable manifolds that satisfy a linear differential equation linking the spinor to the manifold’s geometry. In the mathematics of spin geometry, these objects illuminate the possible shapes a space can take; in theoretical physics, they encode preserved supersymmetry in certain gravitational theories. Their existence imposes strong geometric constraints and provides a bridge between the abstract language of curvature and the concrete requirements of physical models.
Mathematical definition
A Killing spinor is defined on a spin manifold (M,g) equipped with a spinor bundle S and a spin connection ∇ derived from the Levi-Civita connection compatible with g. If ψ is a nonzero section of S, and λ is a constant (the Killing number), then ψ is a Killing spinor with Killing number λ if for every vector field X on M,
∇_X ψ = λ X·ψ,
where X· denotes the Clifford multiplication of the tangent vector X with the spinor ψ. This equation generalizes the notion of a Killing vector, hence the name. The case λ = 0 yields a parallel spinor, a particularly rigid situation that forces the underlying metric to have special holonomy.
The existence of a Killing spinor has immediate geometric consequences. In the Riemannian (positive-definite) setting, a Killing spinor with real λ implies that the metric is Einstein, with Ricci curvature proportional to the metric: Ric = c g for an explicit constant c determined by the dimension and by λ. The precise coefficient depends on conventions for the spinor normalization and curvature conventions, but the general statement is robust: nonzero Killing spinors enforce a highly symmetric curvature structure. In broader terms, Killing spinors anchor the notion of special holonomy, tying the differential equations satisfied by spinors to the holonomy group of the Levi-Civita connection. See special holonomy and holonomy for related ideas.
Two related concepts often appear in this discussion. First, the spinor bundle spinor bundle and its connection encode how spinors transform under parallel transport, with the Clifford algebra Clifford algebra governing the algebraic action X·ψ. Second, the existence of Killing spinors is tightly linked to whether the manifold admits reductions of its holonomy group to subgroups compatible with spinors of the given type Einstein manifold and Riemannian manifold structure.
Existence and geometric consequences
Spaces that admit Killing spinors form a small and highly structured class. In the Riemannian case, the presence of a nonzero Killing spinor forces the manifold to be Einstein, and often leads to a classification in terms of specific geometric constructions. The study of these manifolds is closely tied to the theory of G-structures and special holonomy, which describe how the tangent bundle can be reduced to smaller symmetry groups that preserve spinors and differential forms.
Historically, a major strand of the theory connects Killing spinors to familiar geometric models:
- Round spheres S^n with their standard metric provide classic real Killing spinors; the round metric admits spinors solving the Killing equation with a nonzero real λ (the exact value depends on normalization). See S^n.
- Sasaki-Einstein and nearly Kähler manifolds furnish rich families of examples where real Killing spinors exist, linking odd-dimensional contact geometry to spinor equations Sasaki-Einstein manifold and nearly Kähler manifold.
- Calabi-Yau manifolds and other spaces with special holonomy admit parallel spinors (λ = 0), a related but distinct situation that marks rich geometric and topological structure Calabi-Yau manifold and special holonomy.
- In Lorentzian or pseudo-Riemannian settings, variants of the Killing spinor equation appear in contexts relevant to spacetime physics, where imaginary Killing numbers or modified equations reflect different curvature signatures.
From a broader perspective, the presence of a Killing spinor constrains curvature, topology, and holonomy in ways that often enable explicit geometric construction and classification. The interplay between the analytic properties of the spinor equation and the global geometry of M is a central topic in spin geometry and relates to the spectral theory of the Dirac operator.
Examples and families
- Round sphere, S^n: The standard metric on the sphere carries real Killing spinors with λ = ±1/2 (under common normalizations). This makes S^n a benchmark example in the theory. See S^n.
- Sasaki-Einstein manifolds: These odd-dimensional manifolds furnish natural settings for real Killing spinors, connecting contact geometry with spinor equations. See Sasaki-Einstein manifold.
- Nearly Kähler manifolds: In six real dimensions, these spaces admit real Killing spinors and sit at the crossroads of complex geometry and exceptional holonomy. See nearly Kähler manifold.
- Spaces with special holonomy and parallel spinors: Calabi-Yau manifolds (SU(n) holonomy) and other restricted-holonomy spaces host parallel spinors (λ = 0). See Calabi-Yau manifold and special holonomy.
- Lorentzian backgrounds in physics: In theories with gravity and supersymmetry, certain spacetimes admit Killing spinors that preserve part of the supersymmetry; notable examples appear in AdS-like geometries and in the study of supersymmetric solutions in supergravity.
Physical significance
In theoretical physics, Killing spinors encode preserved supersymmetry in a fixed gravitational background. When a background admits a nontrivial Killing spinor, the corresponding supergravity or string theory solution preserves some fraction of the original supersymmetry, a condition that simplifies the equations of motion to a set of first-order BPS equations. This makes these backgrounds particularly tractable and physically interesting.
Classic examples include anti-de Sitter spaces and their products with spheres, such as AdS_4 × S^7 or AdS_5 × S^5 in higher-dimensional theories, where the geometry supports Killing spinors compatible with the required amount of supersymmetry. These geometries have informed the development of the AdS/CFT correspondence and related frameworks in string theory and M-theory. See supersymmetry and supergravity for broader context.
The mathematics of Killing spinors thus provides a rigorous backbone for constructing and understanding physically relevant backgrounds. It links the abstract language of curvature and holonomy to concrete, computable structures in physics, an interplay that has driven advances on both sides of the disciplinary boundary.