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Spinor bundle

A spinor bundle is a complex vector bundle associated to the frame bundle of a smooth Riemannian manifold equipped with a spin structure, constructed via the spinor representation of the double cover Spin(n) of the special orthogonal group SO(n).^[1]^[2] This structure lifts the oriented orthonormal frame bundle from SO(n) to Spin(n), enabling the definition of spinors as sections of the bundle, which transform under double-valued representations and are essential for describing fermionic fields in quantum field theory on curved spaces.^[1]^[2] The existence of a spin structure—and thus a spinor bundle—on an n-dimensional oriented manifold requires the vanishing of the second Stiefel-Whitney class w_2 in H^2(M; \mathbb{Z}_2), a topological obstruction that is satisfied, for example, by all parallelizable manifolds and n-spheres.^[2] For even dimensions, the spinor representation decomposes into chiral components S_n = S_n^+ \oplus S_n^-, leading to positive and negative spinor bundles whose sections correspond to left- and right-handed spinors in particle physics.^[2] In odd dimensions, the representation is irreducible, yielding a single spinor bundle.^[1] Spinor bundles underpin the Dirac operator, defined as \delta = C \circ \nabla, where C denotes Clifford multiplication by tangent vectors and \nabla is the Levi-Civita connection lifted to the bundle; this operator governs the dynamics of spinor fields and appears in the index theorem for manifolds, relating analytic and topological invariants.^[2] Extensions include Spin^c bundles, which incorporate a U(1) factor for charged spinors and exist on all oriented 4-manifolds via a determinant line bundle with first Chern class congruent to w_2 modulo 2.^[2]^[1] These structures are fundamental in general relativity and quantum field theory, modeling electrons and other half-integer spin particles on spacetimes like Minkowski space with signature (3,1).^[1]

Background Concepts

Spinors and Clifford Algebras

The Clifford algebra \mathrm{Cl}(V, g) associated to a finite-dimensional real vector space V equipped with a non-degenerate symmetric bilinear form g: V \times V \to \mathbb{R} (or equivalently, the induced quadratic form Q(v) = g(v,v)) is the associative unital algebra generated by V subject to the relations v \cdot v = Q(v) \cdot 1 for all v \in V, where \cdot denotes the algebra multiplication and $1 is the unit element.^[3] More precisely, it is constructed as the quotient of the tensor algebra T(V) by the two-sided ideal generated by elements of the form v \otimes v - Q(v) \cdot 1.^[3] This algebra satisfies a universal property: for any associative unital algebra A and any linear map \iota: V \to A such that \iota(v)^2 = Q(v) \cdot 1_A for all v \in V, there exists a unique algebra homomorphism \Phi: \mathrm{Cl}(V, g) \to A extending \iota.^[4] Clifford algebras were introduced by William Kingdon Clifford in his 1878 paper, where he generalized the algebras of complex numbers, quaternions, and Grassmann's exterior algebra to higher dimensions as "geometric algebras" for synthesizing vector operations.^[5] In the specific case of the Euclidean space \mathbb{R}^n with the standard positive definite inner product (denoted \mathrm{Cl}(n) = \mathrm{Cl}(\mathbb{R}^n, \langle \cdot, \cdot \rangle)), the algebra \mathrm{Cl}(n) is a matrix algebra over \mathbb{R}, \mathbb{C}, or the quaternions \mathbb{H}.^[3]^[6] Spinors arise as elements of the spinor space S, which provides the spinor representation of the spin group \mathrm{Spin}(n) \leq \mathrm{Cl}(n)^\times, the connected double cover of the orthogonal group \mathrm{SO}(n).^[7] For even dimensions n = 2m, S decomposes as S = S^+ \oplus S^-, where S^+ and S^- are the chiral (half-spinor) components, each of dimension $2^{m-1} over \mathbb{C}; the full space S has dimension $2^m = 2^{n/2}.^[7] For odd dimensions n = 2m + 1, there is a single irreducible spinor space S of dimension $2^m = 2^{\lfloor n/2 \rfloor}.^[7] This representation is realized over \mathrm{Cl}(n), where \mathrm{Spin}(n) acts by left multiplication on minimal left ideals of the algebra. The classification of these spin representations was established by Élie Cartan in 1913, who identified them as the fundamental non-tensorial representations of the orthogonal Lie algebras \mathfrak{so}(n), with spinors transforming under projective representations of \mathrm{SO}(n).^[8] In low dimensions, real spinors admit explicit matrix realizations via the generators of \mathrm{Cl}(n). For n=2, \mathrm{Cl}(2) \cong M_2(\mathbb{R}) and the spinor space S is \mathbb{R}^2, with generators represented as

\gamma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \gamma_2 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

satisfying \gamma_i^2 = I and \{ \gamma_i, \gamma_j \} = 0 for i \neq j.^[9] The group \mathrm{Spin}(2) \cong \mathrm{SO}(2) acts via rotations on this 2-dimensional real space. For n=3, \mathrm{Cl}(3) \cong M_2(\mathbb{C}) and the spinor space S is \mathbb{C}^2 \cong \mathbb{R}^4, with generators

\gamma_1 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \gamma_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \gamma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

where \gamma_i^2 = I and anticommutation holds off-diagonal; \mathrm{Spin}(3) \cong \mathrm{SU}(2) acts faithfully on this space, corresponding to the quaternion structure underlying 3D rotations.^[9]

Riemannian Manifolds and Spin Structures

A Riemannian manifold is a pair (M, g), where M is a smooth n-dimensional manifold and g is a Riemannian metric on M, providing a smoothly varying positive definite inner product on each tangent space T_p M.^[10] For the manifold to be oriented, the tangent bundle TM is equipped with a consistent choice of orientation on each fiber, making TM an oriented vector bundle.^[11] The associated frame bundle P_{\mathrm{SO}(n)}(M) is the principal \mathrm{SO}(n)-bundle over M whose fibers consist of all ordered oriented orthonormal bases (frames) of the tangent spaces T_p M with respect to g.^[11] A spin structure on the oriented Riemannian manifold (M, g) is a principal \mathrm{Spin}(n)-bundle P_{\mathrm{Spin}(n)}(M) over M together with a bundle homomorphism \Phi: P_{\mathrm{Spin}(n)}(M) \to P_{\mathrm{SO}(n)}(M) covering the identity map on M, such that \Phi is a fiber-preserving local diffeomorphism with each fiber map being the canonical double covering \mathrm{Spin}(n) \to \mathrm{Spin}(n)/\{\pm 1\} \cong \mathrm{SO}(n).^[11] This lift exists if and only if the second Stiefel-Whitney class w_2(TM) \in H^2(M; \mathbb{Z}/2\mathbb{Z}) vanishes.^[12] When w_2(TM) = 0, the set of isomorphism classes of spin structures forms a torsor over H^1(M; \mathbb{Z}/2\mathbb{Z}), meaning spin structures differ by elements of this cohomology group, which classifies their equivalence.^[11] For example, the 3-sphere S^3 admits a unique spin structure, as its cohomology groups H^1(S^3; \mathbb{Z}/2\mathbb{Z}) = 0 and H^2(S^3; \mathbb{Z}/2\mathbb{Z}) = 0 imply w_2(TS^3) = 0 with no nontrivial choices.^[13] In contrast, the complex projective plane \mathbb{CP}^2 does not admit a spin structure because w_2(T\mathbb{CP}^2) \neq 0 in H^2(\mathbb{CP}^2; \mathbb{Z}/2\mathbb{Z}), specifically the nonzero class corresponding to the generator.^[12]

Construction of Spinor Bundles

Definition via Spin Structures

Let (M, g) be an n-dimensional Riemannian manifold equipped with a spin structure, which provides a principal \mathrm{Spin}(n)-bundle P_{\mathrm{Spin}(n)}(M) that double covers the orthonormal frame bundle P_{\mathrm{SO}(n)}(M). The spinor bundle S(M) is defined as the associated complex vector bundle S(M) = P_{\mathrm{Spin}(n)}(M) \times_{\mathrm{Spin}(n)} S, where S denotes the complex spinor representation space of \mathrm{Spin}(n), a faithful representation on a complex vector space of dimension $2^{\lfloor n/2 \rfloor}.^[14]^[15] This construction yields a complex vector bundle over M of rank $2^{\lfloor n/2 \rfloor}, with fibers isomorphic to S.^[14] For even dimension n = 2k, the spinor representation space S decomposes into chiral components S = S^+ \oplus S^-, where S^+ and S^- are the eigenspaces of the chirality operator \omega = i^k e_1 \cdots e_n in the Clifford algebra (acting via left multiplication) with eigenvalues +1 and -1, respectively; each has complex dimension $2^{k-1}.^[14] The corresponding decomposition of the spinor bundle is S(M) = S^+(M) \oplus S^-(M), where S^\pm(M) = P_{\mathrm{Spin}(n)}(M) \times_{\mathrm{Spin}(n)} S^\pm, and the transition functions between local trivializations are induced by the canonical projection \mathrm{Spin}(n) \to \mathrm{SO}(n), reflecting the double cover structure.^[14]^[15] In local coordinates, over an open set U \subset M where the orthonormal frame bundle is trivialized by a frame \{e_i\}, the spinor bundle restricts to a trivial bundle U \times S, and sections (spinor fields) transform under changes of frame via the spinor representation matrices \rho(g) \in \mathrm{GL}(S, \mathbb{C}) for g \in \mathrm{Spin}(n).^[2] For a fixed spin structure, the spinor bundle is unique up to isomorphism as a complex vector bundle associated to that structure; however, distinct spin structures on M (classified by H^1(M; \mathbb{Z}/2\mathbb{Z})) generally yield non-isomorphic spinor bundles.^[14]^[2]

Associated Bundle Approach

The associated bundle construction offers a general method to form vector bundles from principal bundles equipped with group representations. For a principal G-bundle P \to M over a smooth manifold M and a representation \rho: G \to \mathrm{GL}(V) on a finite-dimensional vector space V, the associated vector bundle E = P \times_G V is the quotient space (P \times V)/G under the right action (p, v) \cdot g = (p g, \rho(g^{-1}) v) for g \in G. Fibers of E over points in M are isomorphic to V, and a connection on P induces a compatible connection on E.^[16] Applying this to spinor bundles, the group G = \mathrm{Spin}(n) acts via the principal \mathrm{Spin}(n)-bundle P_{\mathrm{Spin}(n)}(M) arising from a spin structure on the Riemannian manifold (M, g) of dimension n, with the representation \rho being the spinor representation of \mathrm{Spin}(n) on the complex spinor space \Delta_n \cong \mathbb{C}^{2^{\lfloor n/2 \rfloor}}. The spinor bundle S(M) is then the associated bundle P_{\mathrm{Spin}(n)}(M) \times_{\mathrm{Spin}(n)} \Delta_n, whose fibers consist of spinors transforming under the double cover of rotations.^[17] In even dimensions n=2m, the spinor representation decomposes into chiral half-spinor representations \Delta_n = \Delta_n^+ \oplus \Delta_n^- on spaces of dimension $2^{m-1} each, yielding half-spinor bundles S^+(M) = P_{\mathrm{Spin}(n)}(M) \times_{\mathrm{Spin}(n)} \Delta_n^+ and S^-(M) = P_{\mathrm{Spin}(n)}(M) \times_{\mathrm{Spin}(n)} \Delta_n^- as associated bundles. Tensor powers of these, such as S(M) \otimes S(M) or S^+(M) \otimes S^-(M), serve as associated bundles for higher-spin fields via the corresponding symmetric or exterior representations of \mathrm{Spin}(n).^[17] The spinor bundle S(M) inherits a Clifford module structure over the tangent bundle TM, where sections of TM act on sections of S(M) via fiberwise Clifford multiplication X \cdot \psi = \rho(\gamma(X)) \psi for X \in TM_x and \psi \in S_x(M), with \gamma the Clifford map from the tangent space to the Clifford algebra \mathrm{Cl}(n). This action is skew-Hermitian with respect to the Hermitian metric on S(M) induced from \Delta_n and compatible with the Levi-Civita connection on TM.^[18]

Properties and Operations

Sections as Spinor Fields

A section of the spinor bundle S(M) over a Riemannian spin manifold M is referred to as a spinor field, denoted \psi: M \to S(M), which assigns to each point x \in M a spinor in the fiber S_x(M).^[17] For smoothness, such sections are required to be smooth maps that transform under local frames according to the spinor representation of the Spin group, ensuring compatibility with the bundle's structure group.^[17] Locally, spinor fields can be expressed in trivializations of the bundle using the spinor representation \Delta_n: \mathrm{Spin}(n) \to \mathrm{GL}(2^{\lfloor n/2 \rfloor}, \mathbb{C}). The space of all smooth global sections, denoted \Gamma(S(M)), forms a complex vector space under pointwise addition and scalar multiplication, providing the arena for defining spinor fields across the entire manifold.^[17] To endow this space with a Hilbert space structure, one considers the L^2 completion of \Gamma(S(M)) with respect to the inner product induced by the Riemannian metric g on M, given by \langle \psi, \phi \rangle_{L^2} = \int_M \langle \psi(x), \phi(x) \rangle_{S_x} \, d\mathrm{vol}_g(x), where \langle \cdot, \cdot \rangle_{S_x} is the Hermitian inner product on the fiber.^[17] This L^2(S(M)) Hilbert space is essential for functional analytic studies on spin manifolds. On Euclidean space \mathbb{R}^n equipped with the flat metric, the spinor bundle is trivial, allowing constant spinor fields that are invariant under parallel transport and serve as basic examples of global sections.^[17] More generally, on manifolds with reduced holonomy, such as Calabi-Yau manifolds with holonomy in \mathrm{SU}(n), there exist covariantly constant spinor fields, which are parallel sections preserved by the spin connection due to the special holonomy representation.^[17] These spinors play a key role in understanding the geometry of such spaces. (Joyce, Riemannian Holonomy Groups and Calabi-Yau Manifolds) In analytic contexts, the smooth sections \Gamma(S(M)) are dense in the Sobolev spaces H^k(S(M)) for any k \geq 0, a property that facilitates approximation techniques and regularity results in the study of spinor bundles over compact manifolds.^[17] This density ensures that smooth spinor fields can approximate more general L^2 or Sobolev spinors, supporting applications in partial differential equations on spin manifolds.

Clifford Multiplication and Dirac Operators

Clifford multiplication provides an algebraic action of the tangent space on the spinor bundle, realized through the associated Clifford bundle. Specifically, it defines a bundle map \mathrm{Cl}(TM) \otimes S(M) \to S(M), where \mathrm{Cl}(TM) is the complexified Clifford bundle of the tangent bundle TM equipped with the metric g, and S(M) is the spinor bundle. Locally, with respect to an orthonormal frame \{e_i\} on TM, Clifford multiplication by a vector v = \sum v^i e_i acts on a spinor section \psi \in \Gamma(S(M)) as v \cdot \psi = \sum v^i (e_i \cdot \psi), satisfying the defining anticommutation relations \{e_i \cdot, e_j \cdot\} \psi = 2 g_{ij} \psi for all \psi.^[16]^[19] The Dirac operator D is the primary differential operator on sections of the spinor bundle, acting as a first-order elliptic differential operator D: \Gamma(S(M)) \to \Gamma(S(M)). It is defined using the spin connection \nabla, the Levi-Civita connection lifted to S(M), via the local expression D\psi = \sum_i e_i \cdot \nabla_{e_i} \psi, where the sum is over an orthonormal frame \{e_i\} and employs the Einstein summation convention. This construction ensures D is formally self-adjoint with respect to the L^2 inner product on \Gamma(S(M)) induced by the Riemannian volume form, making D a self-adjoint Fredholm operator on the Hilbert space L^2(M, S(M)) when M is compact.^[16]^[19]^[16] The principal symbol of the Dirac operator \sigma_D(x, \xi): S_x(M) \to S_x(M) is given by Clifford multiplication by the cotangent vector \xi, specifically \sigma_D(x, \xi) = \xi^\sharp \cdot, where \xi^\sharp is the metric dual vector in T_x M. This symbol renders D a Dirac-type operator, with ellipticity arising from the non-degeneracy of the Clifford action. In even dimensions, the index of the chiral Dirac operator D^+: \Gamma(S^+(M)) \to \Gamma(S^-(M)) on a compact spin manifold M is computed by the Atiyah-Singer index theorem as \mathrm{index}(D^+) = \int_M \hat{A}(M), connecting the analytical index to the topological invariants of M.^[19]^[16]^[20]

Applications

In Mathematical Physics

In theoretical physics, spinor bundles provide the geometric framework for describing fermionic fields on curved spacetimes, enabling the formulation of relativistic quantum mechanics and quantum field theory in the presence of gravity. A key application is the Dirac equation for spin-1/2 particles on a pseudo-Riemannian manifold M, where the wave function \psi is a section of the spinor bundle S(M). The equation takes the form

i \hbar \gamma^\mu \left( \partial_\mu + \frac{1}{4} \omega_{\mu ab} \gamma^{ab} \right) \psi = m \psi,

with \gamma^\mu obtained via Clifford multiplication by the vielbein fields, incorporating the spin connection \omega_{\mu ab} to ensure covariance under local Lorentz transformations. This formulation captures the coupling of spinors to gravity, essential for understanding particle dynamics in strong gravitational fields.^[21] In general relativity, spinor bundles facilitate the local approximation of curved spacetimes as flat via Fermi coordinates, where the metric reduces to the Minkowski form along a geodesic, allowing spinors to behave as in special relativity within small neighborhoods.^[22] This is crucial for analyzing quantum effects near black holes, such as Hawking radiation for Dirac fields, computed using Bogoliubov transformations that mix positive- and negative-frequency modes of spinor solutions across the event horizon, leading to thermal particle emission observed by distant observers.^[23] In quantum field theory, fermionic fields are realized as sections of spinor bundles over Minkowski or curved backgrounds, enabling the quantization of Dirac fields while respecting the spin-statistics theorem.^[24] Anomalies in these theories, arising from the non-invariance of the path integral measure under chiral transformations, are quantified and canceled using the Atiyah-Singer index theorem applied to the Dirac operator on the spinor bundle, ensuring consistency in gauge theories like the Standard Model. During the 1980s, spinor bundles gained prominence in supersymmetry and supergravity, where the gravitino—a spin-3/2 field—is a section of the tensor product of the spinor bundle with the cotangent bundle, requiring the manifold to admit a spin structure for supersymmetric vacua.^[25] This framework linked spinor geometry to higher-dimensional supergravity theories, such as 11-dimensional supergravity, facilitating the unification of bosonic and fermionic degrees of freedom on spin manifolds.

In Differential Geometry

In differential geometry, spinor bundles play a crucial role in studying the geometric properties of Riemannian manifolds, particularly through the analysis of special types of spinor sections and associated operators. Killing spinors are sections \psi of the spinor bundle satisfying the equation

\nabla_X \psi = \lambda X \cdot \psi

for all vector fields X, where \nabla denotes the Levi-Civita connection lifted to the spinor bundle, \cdot is Clifford multiplication, and \lambda is a real constant.^[26] The existence of such non-trivial Killing spinors imposes strong restrictions on the manifold's geometry, implying reduced holonomy groups. For \lambda = 0, the equation reduces to parallel spinors, which characterize manifolds with special holonomy, such as G_2 in dimension 7, where the holonomy representation preserves a parallel spinor.^[26] In general, the integrability conditions derived from the Killing equation lead to constraints on the curvature tensor, linking the spinor bundle's structure to the manifold's holonomy.^[27] The presence of a non-zero parallel spinor \psi (i.e., \nabla \psi = 0) further implies that the manifold is Ricci-flat. This follows from the Weitzenböck formula applied to spinors, which relates the connection Laplacian to the scalar curvature:

|\nabla \psi|^2 = \frac{1}{4} \mathrm{Scal} \, |\psi|^2.

For a parallel spinor, the left side vanishes, forcing \mathrm{Scal} = 0. Combined with the action of the curvature endomorphism on the spinor—via the integrability condition \nabla_X \nabla_Y \psi - \nabla_Y \nabla_X \psi = R(X,Y) \psi = 0—this yields vanishing Ricci curvature, as the spinor generates an irreducible representation under which the Ricci tensor must act trivially.^[28] Such manifolds admit spin structures with reduced holonomy in the Ricci-flat holonomy groups like \mathrm{SU}(n), \mathrm{Sp}(n), G_2, or \mathrm{Spin}(7).^[29] A fundamental relation in this context is the Lichnerowicz formula for the Dirac operator D acting on sections of the spinor bundle:

D^2 = \nabla^* \nabla + \frac{\mathrm{Scal}}{4},

where \nabla^* \nabla is the rough Laplacian on spinors.^[16] This formula connects the spectrum of D to the scalar curvature \mathrm{Scal}, showing that positive scalar curvature implies no zero eigenvalues for D on compact manifolds, with the lowest eigenvalue bounded below by \inf \mathrm{Scal}/4.^[16] It provides a bridge between spinorial analysis and curvature obstructions, enabling vanishing theorems for harmonic spinors (kernels of D) under curvature assumptions. Spinor bundles also feature prominently in the positive energy theorem for asymptotically flat Riemannian manifolds, which asserts that the ADM mass is non-negative and zero only for flat metrics. Witten's proof constructs harmonic spinors \psi (satisfying D\psi = 0) on the manifold, extending from asymptotic behavior at infinity, and integrates a positive expression involving the spinor to bound the mass. Specifically, the dominant energy condition ensures the integral of |\nabla \psi|^2 plus terms involving the energy density and the square of the mean curvature is non-negative, yielding the mass via boundary terms.^[30] An alternative geometric proof by Schoen and Yau uses minimal hypersurface techniques but aligns with spinorial methods in restricting negative mass configurations.^[31] These results highlight spinor bundles' role in rigidity theorems for asymptotically flat spaces with non-negative scalar curvature.

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