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| Spinor bundle | |
|---|---|
| Name | Spinor bundle |
| Field | Differential geometry |
Spinor bundle A spinor bundle is a vector bundle associated to a spin structure on a smooth manifold that carries representations of the Clifford algebra and provides the setting for spinor fields used in geometric analysis, index theory, and quantum field theory. It links classical constructions such as the tangent bundle, frame bundle, and orthonormal frame bundle with modern developments involving the Atiyah–Singer index theorem, Dirac operator, and applications in both Riemannian manifold theory and general relativity. The existence and properties of spinor bundles depend on topological invariants like the Stiefel–Whitney class and connect to structures considered by figures such as Élie Cartan, Michael Atiyah, Isadore Singer, Friedrich Hirzebruch, and Raoul Bott.
A spinor bundle arises when a manifold admits a spin structure lifting the SO(n)-principal bundle of oriented orthonormal frames to a Spin(n)-principal bundle; this lift is obstructed by the second Stiefel–Whitney class w2 in the cohomology studied by John Milnor, James Stasheff, and Henri Cartan. Given a Riemannian metric and a spin structure, one forms an associated vector bundle via an irreducible complex representation of Spin(n), producing the spinor bundle used in the work of Hermann Weyl, Paul Dirac, and Marcel Grossmann. Spinor bundles generalize classical notions connected to the Clifford bundle and interact with concepts in K-theory explored by Bott and Atiyah.
Starting from an oriented Riemannian manifold (M,g), the oriented orthonormal frame principal bundle has structure group SO(n). A spin structure is a principal Spin(n)-bundle P_spin with a 2-to-1 covering map to the SO(n) bundle; lifting obstructions are encoded by the Stiefel–Whitney class w2 described by Serre and Steenrod. Given a finite-dimensional irreducible complex representation ρ of Spin(n), the associated vector bundle S = P_spin ×_ρ V is the spinor bundle; foundational representation theory was developed by Élie Cartan and expanded by Weyl and Évariste Galois-influenced algebraists. In even dimensions one has a natural Z2-grading S = S^+ ⊕ S^- corresponding to chiral spinors linked to work by Pauli and Dirac. For manifolds with a spin^c structure—introduced in contexts studied by Michael Freedman and Edward Witten—one constructs a complex spinor bundle using a principal Spin^c(n)-bundle and an associated determinant line bundle.
On a spin manifold, the spinor bundle is locally isomorphic to a trivial bundle with fibers isomorphic to the standard spinor representation of Spin(n). Classic examples include spinor bundles on the sphere S^n, the torus T^n, and compact simply connected Lie group manifolds such as SU(2) ≅ S^3 studied in works by Hopf and Cartan. The existence of parallel spinors characterizes special holonomy groups like SU(n), Sp(n), G2, and Spin(7) examined by Berger and Bonan. On Kähler manifolds and Calabi–Yau manifolds used in string theory research by Shing-Tung Yau and Candelas, the spinor bundle structure intertwines with the canonical bundle and Dolbeault operator considered by Kodaira and Spencer. Examples in low dimensions relate to the Klein bottle, real projective space RP^n, and complex projective space CP^n where spin structures and spinor bundles have been classified by Atiyah and Singer.
A spin connection on the spinor bundle is induced functorially from the Levi-Civita connection on the tangent bundle, as developed in differential geometry by Élie Cartan and Tullio Levi-Civita. The covariant derivative defines the geometric Dirac operator, a first-order elliptic operator introduced by Paul Dirac and placed on rigorous footing by Atiyah and Singer; its square relates to the Lichnerowicz formula studied by André Lichnerowicz. Twisted Dirac operators use auxiliary bundles such as vector bundles with connections studied by Chern and Weil; index and spectral properties of these operators are central in the analysis by Bismut and Cheeger. In physics contexts, the Dirac operator couples spinor bundles to gauge fields of groups like U(1), SU(n), and SO(n), topics appearing in the work of Yang and Mills and in the Standard Model framework developed at institutions such as CERN and Caltech.
Index theory for Dirac operators on spinor bundles culminated in the Atiyah–Singer index theorem, with applications to anomalies investigated by Alvarez-Gaumé and Witten; the Â-genus and A-roof genus appear in statements by Hirzebruch and Rosenberg. K-theoretic invariants from topological K-theory studied by Atiyah and Bott classify obstructions and anomalies for families of Dirac operators as developed in the Atiyah–Patodi–Singer framework with boundary conditions introduced by Patodi. Exotic phenomena such as the Rokhlin theorem and Donaldson theory on four-manifolds link spinor bundles to gauge-theoretic moduli spaces studied by Simon Donaldson and Clifford Taubes. The role of the eta invariant and spectral flow in global analysis was advanced by Dai and Freed.
Spinor bundles are indispensable in proofs of geometric results like the positive mass theorem by Schoen and Yau and rigidity theorems explored by Lichnerowicz and Matsushima. In mathematical physics they underpin formulations of fermionic fields in quantum field theory and supersymmetry developed by Witten and Salam and of spin geometry in general relativity contexts considered by Einstein and Weyl. Compactifications in string theory use spinor bundles on Calabi–Yau and G2 manifolds studied at institutions like Institute for Advanced Study and Princeton University. In global analysis and topology, spinor bundle techniques yield invariants used in Floer homology and Seiberg–Witten theory advanced by Seiberg and Witten and further developed in research at Harvard University and MIT.