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Spin structure

In differential geometry, a spin structure on an orientable Riemannian manifold M is a geometric structure that allows the consistent definition of spinor fields over M, enabling the construction of spinor bundles and associated Dirac operators.^[1] It arises as a lift of the structure group of the tangent bundle from the orthogonal group SO(n) to its double cover, the spin group Spin(n), addressing the challenge of defining spinors globally on manifolds where the transition functions may not preserve spinor representations.^[2] The existence of a spin structure is obstructed by topological invariants, such as the second Stiefel–Whitney class, and spin structures play crucial roles in index theory, topology, and the study of Dirac operators, with applications in quantum field theory and string theory.^[1]

Overview

Basic concepts and motivation

In differential geometry, the special orthogonal group SO(n) serves as the structure group for the oriented orthonormal frame bundle of an n-dimensional Riemannian manifold, describing rotations of tangent spaces. For n \geq 3, SO(n) is not simply connected, possessing a fundamental group isomorphic to \mathbb{Z}_2, which admits a unique non-trivial double cover known as the spin group \operatorname{Spin}(n). This double cover is a Lie group homomorphism \rho: \operatorname{Spin}(n) \to SO(n) with kernel \{\pm 1\}, ensuring that \operatorname{Spin}(n) captures "square root" rotations not visible in SO(n).^[3] The primary motivation for spin structures arises in the study of spinors, which are mathematical objects essential for describing fermionic particles in quantum field theory and relativistic physics, originating from Dirac's equation for electrons. On a manifold, defining spinor fields requires lifting the frame bundle from SO(n) to \operatorname{Spin}(n), as spinors transform under representations of \operatorname{Spin}(n) rather than SO(n). Without such a lift, parallel transport of spinors around closed loops introduces a sign ambiguity due to the \pm 1 kernel, potentially leading to inconsistent global definitions; a spin structure resolves this by providing a consistent choice of lift, enabling well-defined Clifford multiplication and Dirac operators.^[4]^[3] Geometrically, a spin structure can be intuited as a choice of "square root" for the orientation of the tangent bundle, extending the local orientation data to a global structure compatible with spinor transport. Spin structures exist only on orientable manifolds (where the first Stiefel-Whitney class w_1(TM) = 0 in H^1(M; \mathbb{Z}_2)) with vanishing second Stiefel-Whitney class w_2(TM) = 0 in H^2(M; \mathbb{Z}_2). The condition w_1 = 0 obstructs non-orientability, a prerequisite for spin structures, while w_2 = 0 is the further obstruction to their existence.^[5]^[6] For non-spin manifolds, spin^c structures generalize this concept by incorporating a line bundle to bypass the obstruction.^[3]

Historical development

The historical development of spin structures originated in the realm of quantum physics with Paul Dirac's seminal 1928 paper, where he formulated the relativistic wave equation for the electron. This equation incorporated the electron's intrinsic angular momentum, or spin, necessitating mathematical representations with half-integer values to reconcile quantum mechanics with special relativity.^[7] In the 1930s, French mathematician Élie Cartan built upon these physical insights by integrating spinors into differential geometry. Cartan developed the theory of spinors on Riemannian manifolds, introducing spin frames as a means to locally adapt orthonormal frames for handling spinorial objects in curved spaces. His work, detailed in his 1938 monograph, laid the geometric foundation for describing spin in non-flat geometries.^[8] Post-World War II advancements in the 1950s were led by André Lichnerowicz, who formalized spinor bundles over general manifolds and extended the Dirac operator to curved spacetimes. This formulation enabled the analysis of spinor fields within the framework of general relativity, providing tools for studying quantum fields on arbitrary backgrounds. Lichnerowicz's contributions, emerging around the mid-1950s, marked a shift toward global geometric structures for spin.^[9] The 1960s and 1970s saw topological refinements by mathematicians including Michael Atiyah, Raoul Bott, and John Milnor, who linked spin structures to characteristic classes such as Stiefel-Whitney classes. Milnor's 1963 article explicitly introduced the term "spin structure" and explored its topological properties on manifolds. Atiyah and Bott further advanced this in the early 1970s through their work on Riemann surfaces, emphasizing cohomological classifications. These developments culminated in the formalization of spin structures via applications of the Atiyah-Singer index theorem, which connected analytic indices of Dirac operators to topological invariants around 1963 onward.^[10]^[11]

Spin structures on manifolds

Definition on oriented Riemannian manifolds

A spin structure on an oriented Riemannian manifold (M, g) of dimension n is defined in terms of its tangent bundle TM, which admits a principal SO(n)-bundle structure known as the oriented frame bundle P_{SO(n)}(TM) \to M.^[12] This bundle consists of all oriented orthonormal frames in TM, with the structure group SO(n) acting by right multiplication.^[12] Formally, a spin structure consists of a principal Spin(n)-bundle P \to M equipped with a bundle homomorphism \pi: P \to P_{SO(n)}(TM) that covers the identity map on M and is equivariant with respect to the canonical double covering homomorphism \xi: Spin(n) \to SO(n).^[13] Equivalently, a spin structure is a reduction of the structure group of the oriented frame bundle P_{SO(n)}(TM) from SO(n) to its universal cover Spin(n), lifting the SO(n)-action to a Spin(n)-action.^[12] This construction ensures that P double covers P_{SO(n)}(TM) fiberwise, as \xi is a 2:1 homomorphism.^[13] Associated to such a spin structure is the spinor bundle S = P \times_{Spin(n)} \mathbb{C}^{2^{n/2}} when n is even, obtained via the spinor representation of Spin(n) on \mathbb{C}^{2^{n/2}}.^[12] This complex vector bundle of rank $2^{n/2} carries a Clifford multiplication map TM \otimes S \to S, induced by the representation of the Clifford algebra Cl(TM) on the spinor space, which satisfies the anticommutation relations \{c(X), c(Y)\} = 2g(X,Y) \mathrm{Id}_S for vector fields X, Y \in \Gamma(TM).^[14] When a spin structure exists on (M, g), it is unique up to isomorphism if M is simply connected, but non-simply connected manifolds may admit multiple non-isomorphic spin structures, forming an affine space over H^1(M; \mathbb{Z}/2\mathbb{Z}).^[13]

Obstruction to existence

The existence of a spin structure on an oriented Riemannian manifold M^n requires that the tangent bundle TM admits a reduction of its structure group from \mathrm{SO}(n) to \mathrm{Spin}(n). A necessary condition for this is that M is orientable, which is equivalent to the vanishing of the first Stiefel-Whitney class w_1(TM) = 0 \in H^1(M; \mathbb{Z}/2\mathbb{Z}); if w_1(TM) \neq 0, no orientation exists, and thus no spin structure can be defined.^[15] Beyond orientability, the primary topological obstruction is the vanishing of the second Stiefel-Whitney class w_2(TM) = 0 \in H^2(M; \mathbb{Z}/2\mathbb{Z}).^[13] For simply connected manifolds, the condition simplifies further: since simple connectivity implies \pi_1(M) = 0, it follows that w_1(TM) = 0, making M automatically orientable; thus, such a manifold admits a spin structure if and only if w_2(TM) = 0.^[13] In this case, the spin structure, if it exists, is unique up to isomorphism.^[13] In general, the obstructions arise from the double covering \{\pm 1\} \to \mathrm{Spin}(n) \to \mathrm{SO}(n) for n \geq 3. The homotopy groups satisfy \pi_1(\mathrm{Spin}(n)) = 0 and \pi_1(\mathrm{SO}(n)) = \mathbb{Z}/2\mathbb{Z}, with \pi_k(\mathrm{Spin}(n)) \cong \pi_k(\mathrm{SO}(n)) for k \geq 2. The primary obstruction to lifting the structure group thus lies in H^2(M; \mathbb{Z}/2\mathbb{Z}) and is precisely w_2(TM); higher-dimensional obstructions vanish due to the isomorphism of higher homotopy groups.^[16] Consequently, an oriented manifold is spin if and only if w_2(TM) = 0.^[15] Manifolds admitting spin structures are necessarily orientable with w_2(TM) = 0, but the converse does not hold: there exist oriented manifolds with w_2(TM) \neq 0 that fail to be spin yet admit \mathrm{Spin}^c structures. A canonical example is the complex projective plane \mathbb{CP}^2, which is orientable but has w_2(T\mathbb{CP}^2) \neq 0, obstructing a spin structure while allowing a \mathrm{Spin}^c structure.^[17] Explicit computations confirm the existence in standard cases: all spheres S^n are spin, as w_2(TS^n) = 0 for every n \geq 1; similarly, all tori T^n are spin, since their tangent bundles are trivial and thus have vanishing Stiefel-Whitney classes.^[13]

Spin structures on vector bundles

Definition and lifting

In the context of an oriented real vector bundle E \to B of rank r \geq 3 with structure group \mathrm{SO}(r), a spin structure is defined via the associated principal \mathrm{SO}(r)-bundle P_{\mathrm{SO}}(E) \to B.^[12] Specifically, a spin structure on E consists of a principal \mathrm{Spin}(r)-bundle P_{\mathrm{Spin}}(E) \to B equipped with a bundle homomorphism \Lambda: P_{\mathrm{Spin}}(E) \to P_{\mathrm{SO}}(E) that lifts the canonical double covering map \mathrm{Spin}(r) \to \mathrm{SO}(r) and is equivariant with respect to the respective group actions.^[18] This lifting ensures that the structure group reduces from \mathrm{SO}(r) to its universal cover \mathrm{Spin}(r), allowing the bundle to carry spinorial data.^[12] Equivalently, the existence of a spin structure on E is characterized by the existence of a spinor bundle, which is the vector bundle associated to P_{\mathrm{Spin}}(E) via a spinor representation \mu: \mathrm{Spin}(r) \to \mathrm{GL}(V) for some complex vector space V of dimension $2^{\lfloor r/2 \rfloor}.^[12] In local trivializations \{U_i\} of E, this perspective manifests through lifts of the transition functions g_{ij}: U_i \cap U_j \to \mathrm{SO}(r) to elements \tilde{g}_{ij}: U_i \cap U_j \to \mathrm{Spin}(r) satisfying the cocycle condition

\tilde{g}_{ij} \tilde{g}_{jk} = \epsilon_{ijk} \tilde{g}_{ik}, \quad \epsilon_{ijk} \in \{\pm 1\},

where consistent global choices of signs \epsilon_{ijk} ensure the lifts define a well-formed principal bundle.^[18] Such lifts are possible only if the first and second Stiefel-Whitney classes vanish, i.e., w_1(E) = 0 and w_2(E) = 0 \in H^2(B; \mathbb{Z}/2\mathbb{Z}), with w_1(E) = 0 already implied by the orientability of E.^[12] This general framework applies in particular to the tangent bundle TM \to M of an oriented Riemannian manifold M, where a spin structure on TM yields spinors on M.^[12]

Classification via cohomology

Spin structures on a vector bundle E \to B with structure group SO(n) exist provided the second Stiefel-Whitney class w_2(E) = 0 \in H^2(B; \mathbb{Z}/2\mathbb{Z}).^[19] Assuming this obstruction vanishes, the isomorphism classes of spin structures on E are in bijective correspondence with elements of the first Čech cohomology group H^1(B; \mathbb{Z}/2\mathbb{Z}).^[12]^[19] This classification arises from the short exact sequence of Lie groups

$1 \to \mathbb{Z}/2\mathbb{Z} \to \operatorname{Spin}(n) \to \operatorname{SO}(n) \to 1,

which induces a long exact sequence in cohomology. The connecting homomorphism \delta: H^1(B; \operatorname{SO}(n)) \to H^2(B; \mathbb{Z}/2\mathbb{Z}) recovers w_2(E) from the class of the SO(n)-bundle, and when this image is zero, the spin structures—equivalently, lifts of the structure group to Spin(n)—are parametrized by the kernel of the subsequent map, yielding the torsor structure over H^1(B; \mathbb{Z}/2\mathbb{Z}).^[19] To construct this explicitly, consider an open cover \{U_i\}_{i \in I} of the base B over which the SO(n)-bundle is defined by transition functions g_{ij}: U_i \cap U_j \to \operatorname{SO}(n) satisfying the cocycle condition g_{ij} g_{jk} = g_{ik} on triple intersections. A spin structure corresponds to a choice of lifts \tilde{g}_{ij}: U_i \cap U_j \to \operatorname{Spin}(n) such that \pi(\tilde{g}_{ij}) = g_{ij}, where \pi: \operatorname{Spin}(n) \to \operatorname{SO}(n) is the double covering map. These lifts satisfy the twisted cocycle relation

\tilde{g}_{ij} \tilde{g}_{jk} = \epsilon_{ijk} \tilde{g}_{ik}

on U_i \cap U_j \cap U_k, where \epsilon_{ijk} \in \{ \pm 1 \} \cong \mathbb{Z}/2\mathbb{Z} lies in the center of Spin(n). The collection \{\epsilon_{ijk}\} forms a \mathbb{Z}/2\mathbb{Z}-valued 2-cocycle whose class in H^2(\{U_i\}; \mathbb{Z}/2\mathbb{Z}) is w_2(E); since this vanishes, \epsilon = \delta \lambda for some \mathbb{Z}/2\mathbb{Z}-valued 1-cochain \lambda = \{\lambda_{ij}\} on the cover. Different spin structures arise from modifying the lifts by such cochains: if \tilde{g}'_{ij} = \tilde{g}_{ij} \cdot \sigma(\lambda_{ij}) for a section \sigma: \mathbb{Z}/2\mathbb{Z} \to \operatorname{Spin}(n) of the center, then \{\tilde{g}'\} defines an equivalent spin structure modulo isomorphism if \lambda is a coboundary, yielding the classification modulo coboundaries in Z^1(\{U_i\}; \mathbb{Z}/2\mathbb{Z}).^[19]^[12] The group H^1(B; \mathbb{Z}/2\mathbb{Z}) acts freely and transitively on the set of spin structures, so if at least one exists, the total number is |H^1(B; \mathbb{Z}/2\mathbb{Z})|.^[12]^[19] The difference between any two spin structures is represented by a unique class in H^1(B; \mathbb{Z}/2\mathbb{Z}), which corresponds to the cohomology class of the \mathbb{Z}/2\mathbb{Z}-gerbe measuring the relative twist in their double covers.^[19] For a simply connected base B with \pi_1(B) = 0, it follows that H^1(B; \mathbb{Z}/2\mathbb{Z}) = 0, so there is at most one spin structure on E whenever w_2(E) = 0.^[12]^[19]

Key examples

The trivial vector bundle of rank n over any base space admits a unique spin structure, obtained via the canonical homomorphism \mathrm{SO}(n) \to \mathrm{Spin}(n) applied to the trivial principal \mathrm{SO}(n)-bundle.^[20] The tangent bundle of the 2-sphere S^2 provides a concrete example of a non-trivial vector bundle admitting a spin structure, as its second Stiefel-Whitney class w_2 = 0 and the first cohomology group H^1(S^2; \mathbb{Z}/2\mathbb{Z}) = 0, yielding a unique such structure; in contrast, the tangent bundle of the 3-sphere S^3 also admits a unique spin structure, consistent with the fact that all oriented 3-manifolds are spin.^[20]^[20] The realification of the tautological (Hopf) complex line bundle over \mathbb{CP}^1 \cong S^2 is an oriented rank-2 vector bundle with w_2 \neq 0 (specifically, the nonzero generator of H^2(\mathbb{CP}^1; \mathbb{Z}/2\mathbb{Z})), and thus admits no spin structure, demonstrating the obstruction posed by a nontrivial second Stiefel-Whitney class. On the real projective 3-space \mathbb{RP}^3, the tangent bundle (or more generally, the Clifford bundle associated to the metric) admits exactly two spin structures; one of these is bounding, meaning it extends over the 4-ball bounding \mathbb{RP}^3, while the other does not.^[21] A broader classification shows that real projective spaces \mathbb{RP}^n admit spin structures if and only if n \equiv 3 \pmod{4}.^[22] The tangent bundle of the 2-torus T^2 admits exactly four spin structures, computed via the cohomological classification as |H^1(T^2; \mathbb{Z}/2\mathbb{Z})| = 4; this finite number corrects the occasional misconception of infinitely many such structures on tori.^[11]

Spin^c structures

Definition and construction

A Spin^c(n)-structure on an oriented real vector bundle E \to X of rank n is defined as a principal \operatorname{Spin}^c(n)-bundle P \to X together with an isomorphism of the associated bundle P / U(1) \cong P_{\mathrm{SO}(n)}(E), where P_{\mathrm{SO}(n)}(E) is the oriented frame bundle of E.^[23] The group \operatorname{Spin}^c(n) is constructed as the quotient \operatorname{Spin}(n) \times_{\mathbb{Z}/2\mathbb{Z}} U(1) = (\operatorname{Spin}(n) \times U(1)) / \{(1,1), (-1,-1)\}, where \mathbb{Z}/2\mathbb{Z} acts diagonally by multiplication by -1 on both factors.^[24] This group fits into the short exact sequence $1 \to U(1) \to \operatorname{Spin}^c(n) \to \mathrm{SO}(n) \to 1, making \operatorname{Spin}^c(n)a central extension of\mathrm{SO}(n)by the circle group, and it provides a double cover of the quotient group\mathrm{SO}(n) \times_{\mathbb{Z}/2\mathbb{Z}} U(1), where \mathbb{Z}/2\mathbb{Z}$ acts by sign reversal on both components.^[23] Equivalently, a \operatorname{Spin}^c(n)-structure on E can be constructed as a pair consisting of a complex line bundle L \to X and a spin structure on the underlying real vector bundle E \oplus L_{\mathbb{R}} \to X, where L_{\mathbb{R}} denotes the realification of L as a rank-2 oriented bundle.^[23] This equivalence arises because the \mathbb{Z}/2\mathbb{Z}-action aligns the structure groups such that lifting the frame bundle of E \oplus L_{\mathbb{R}} to \operatorname{Spin}(n+2) projects back to a \operatorname{Spin}^c(n)-lift for E twisted by the U(1)-structure on L.^[24] When L is the trivial line bundle, this reduces to a genuine spin structure on E.^[23] The associated spinor bundle for a \operatorname{Spin}^c(n)-structure is a complex vector bundle S \to X obtained by taking the associated bundle P \times_{\rho} V, where \rho: \operatorname{Spin}^c(n) \to U(m) is the spinor representation (with m = 2^{n/2} for n even) and V = \mathbb{C}^m is the defining representation space.^[25] This bundle decomposes as S = S^+ \oplus S^-, the chiral components twisted by the square root of the determinant line bundle \det(L)^{1/2} \to X, yielding "charged" spinors that transform under the U(1)-action induced by L.^[24] Unlike spin structures, which may not exist on all oriented bundles, \operatorname{Spin}^c(n)-structures always exist on the tangent bundles of any oriented Riemannian manifold.^[26] This follows from the fact that the second Stiefel-Whitney class w_2(TM) always lifts to an integral cohomology class in H^2(M; \mathbb{Z}), allowing the required \mathbb{Z}/2\mathbb{Z}-cocycle to be trivialized via the U(1)-extension.^[26]

Obstruction and classification

The existence of a Spin^c structure on an oriented vector bundle E \to B requires only that E is orientable, meaning the first Stiefel-Whitney class w_1(E) = 0 in H^1(B; \mathbb{Z}/2\mathbb{Z}); there is no additional condition involving w_2(E), as the Spin^c group extension allows lifting over all oriented bundles.^[2]^[27] The isomorphism classes of Spin^c structures on E are classified by the second cohomology group H^2(B; \mathbb{Z}), parametrized by the first Chern class c_1(L) \in H^2(B; \mathbb{Z}) of the auxiliary complex line bundle L \to B used in the construction, satisfying the consistency relation

c_1(L) \equiv w_2(E) \pmod{2}.

^[2]^[27] More precisely, the set of Spin^c structures forms a torsor over H^2(B; \mathbb{Z}), where the difference between two such structures corresponds to an integral lift of w_2(E) to H^2(B; \mathbb{Z}).^[2]^[27] For a closed oriented Riemannian manifold M, the number of inequivalent Spin^c structures is thus infinite unless H^2(M; \mathbb{Z}) = 0; a canonical choice exists when w_2(TM) = 0, corresponding to the trivial line bundle L.^[2]^[27]

Relation to complex structures

On an almost complex manifold, the complex structure induces a canonical Spin^c structure. The complexified tangent bundle decomposes as TM \otimes \mathbb{C} = T^{1,0}M \oplus T^{0,1}M, where T^{1,0}M and T^{0,1}M are the eigenspaces of the almost complex structure J extended complex linearly, with eigenvalues i and -i, respectively. The associated spinor bundle is then \Sigma_M = \Lambda^{0,\bullet}M = \bigoplus_r \Lambda^r(T^{0,1}M)^*, the bundle of antiholomorphic forms, and the determinant line bundle is L = \det(T^{0,1}M)^* = K_M^{-1}, where K_M is the canonical bundle. Calabi-Yau manifolds, being Ricci-flat Kähler manifolds with trivial canonical bundle, admit a canonical Spin^c structure that is compatible with holomorphic spinors; in particular, the parallel holomorphic (n,0)-form serves as a covariantly constant spinor in this structure, reflecting the manifold's SU(n) holonomy.^[28] The Dirac operator associated to a Spin^c structure incorporates the auxiliary U(1) connection from the determinant line bundle L. Specifically, it takes the form i\not{\nabla} + \omega, where \not{\nabla} is the spin Dirac operator from the Levi-Civita connection lifted to the spinor bundle, and \omega is the connection 1-form on L.^[29] Geometrically, a Spin^c structure on a complex manifold can be viewed as providing a square root of the anticanonical bundle, K^{-1/2}, with the spinor bundle S satisfying \det S = K^{-1} for the canonical choice where L = K^{-1}. This perspective arises in contexts like Riemann surfaces, where an odd Spin^c structure \delta is a square root \bar{K}^{1/2}_\delta of the anticanonical bundle \bar{K}, enabling the construction of determinants for twisted \bar{\partial}-operators. On Kähler surfaces, Spin^c structures play a central role in Seiberg-Witten theory through their determinant line bundle L, which for the canonical structure satisfies L^2 = K^{-1}, the anticanonical bundle. The Seiberg-Witten monopole equations, involving a connection A on L and spinors in W^+ \otimes L \cong \Theta \oplus K^{-1} (where \Theta is the cotangent bundle), reduce in the Kähler case to conditions on holomorphic sections, linking solutions to the geometry of the surface via the index of the twisted Dirac operator D^+_A.^[30]

Applications

In differential geometry and topology

In differential geometry and topology, spin structures play a crucial role in index theory, particularly through the Atiyah-Singer index theorem, which relates the analytical index of the Dirac operator on a compact spin manifold to a topological invariant known as the Â-genus. For a closed, oriented Riemannian spin manifold M of dimension n, the theorem states that the index of the Dirac operator D satisfies \operatorname{ind}(D) = \hat{A}(M), where \hat{A}(M) is the Â-genus, a characteristic number derived from the Pontryagin classes of M. This equality demonstrates that the existence of a spin structure allows the Â-genus to be realized as an integer, providing a bridge between elliptic partial differential equations and topological invariants. The theorem's proof involves heat kernel methods and equivariant extensions, highlighting how spin structures enable the construction of twisted Dirac operators on vector bundles over M.^[31] Spin structures also underpin the study of spin bordism groups, which classify manifolds up to cobordism and connect to real K-theory via the connective spectrum ko. The spin bordism groups \Omega_*^{\text{Spin}}(pt) are isomorphic to the homotopy groups of the Thom spectrum MSpin, and in low dimensions, they align with the stable homotopy groups of real K-theory, \pi_*(ko) \cong KO_*(pt), up to dimension 7 due to the 7-connected map MSpin \to ko. For instance, in dimension 4, spin bordism classes are determined by the signature, leading to Rokhlin's theorem, which asserts that the signature \sigma(M) of any closed, smooth, oriented spin 4-manifold M is divisible by 16. This divisibility arises from the index of the Dirac operator and imposes a strong topological constraint on the intersection form of M, with the minimal non-zero example being \sigma = \pm 16 for the K3 surface.^[32] In equivariant settings, the G-index theorem extends these ideas to twisted spin complexes under group actions, computing the equivariant index of Dirac operators via fixed-point formulas involving representations. For a compact Lie group G acting on a spin manifold M, the G-index of a twisted Dirac operator D_E associated to an equivariant vector bundle E is given by a localization formula over the fixed-point set, incorporating equivariant characteristic classes. This theorem facilitates computations in cobordism and index theory for manifolds with symmetry, such as those arising in representation theory. Finally, spin structures are essential in the positive mass theorem for asymptotically flat manifolds, where they enable the use of spinor fields to prove non-negativity of the ADM mass. In the proof by Witten, a spin structure on the asymptotically flat spin manifold (M, g) with non-negative scalar curvature allows the construction of a harmonic spinor vanishing at infinity, leading to the inequality m \geq 0, with equality only for the Euclidean metric. This result, building on Schoen and Yau's geometric approach, relies on the completeness of the spinor bundle at infinity and provides a key tool for understanding gravitational energy in general relativity through differential geometry.

In quantum field theory and particle physics

In quantum field theory on curved spacetimes, spin structures are essential for consistently defining Dirac fields, which describe spin-1/2 fermions. The Dirac equation takes the form i \gamma^\mu \nabla_\mu \psi = m \psi, where \psi is the spinor field, m is the fermion mass, the \gamma^\mu matrices satisfy the Clifford algebra \{ \gamma^\mu, \gamma^\nu \} = 2 g^{\mu\nu} with metric g^{\mu\nu}, and the covariant derivative \nabla_\mu includes the spin connection \omega_\mu^{ab} derived from the chosen spin structure on the frame bundle.^[33] This connection ensures local Lorentz invariance for the spinors, as the spin structure lifts the orthogonal frame bundle to a Spin bundle, allowing the Clifford module to be well-defined globally.^[34] Without a compatible spin structure, the fermionic theory would suffer inconsistencies, such as ill-defined parallel transport for spinors along non-contractible loops.^[33] Spin structure choices also play a key role in resolving global anomalies in chiral gauge theories. In four dimensions, these anomalies arise from large gauge transformations and can be detected via the eta invariant of the Dirac operator, where \eta(0) \mod 2 must vanish for consistency; the appropriate spin structure selection ensures this mod-2 condition holds, canceling the anomaly.^[35] For instance, in SU(2) chiral theories, the Witten anomaly, a Z_2 global anomaly, is tied to the topology of the gauge group and spacetime, with spin structures providing the necessary framing to make the path integral well-defined. In superstring theory, spin structures on the worldsheet torus determine the boundary conditions for worldsheet fermions, distinguishing the Ramond sector (with periodic fermions yielding spacetime supersymmetry) from the Neveu-Schwarz sector (with antiperiodic fermions giving bosonic states).^[36] Summing over the 2^4 = 16 possible spin structures at one loop ensures modular invariance and anomaly cancellation in the type II superstring spectrum.^[37] Similarly, in four-dimensional quantum field theories, Spin^c structures are crucial for the electroweak model, where the SU(2)_L × U(1)_Y gauge group acts on the Spin^c frame bundle, and the Higgs field serves as a section of the associated complex line bundle, enabling symmetry breaking via the Higgs mechanism.^[38] An important application arises in quantum chromodynamics (QCD), where instanton configurations require spin structures to compute fermionic zero modes via the Atiyah-Singer index theorem. For a single BPST instanton in the SU(3) gauge theory, the index of the Dirac operator yields N_f left-handed zero modes (with N_f the number of flavors), reflecting the chiral asymmetry induced by the self-dual topology and the chosen spin structure on Euclidean R^4.^[39] These zero modes contribute to non-perturbative effects like the U(1)_A anomaly and eta' mass generation. The geometric index theorem further applies this framework to curved backgrounds, linking instanton contributions to gravitational anomalies in QCD.

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### Summary of Spin Structures from the Document
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[PDF] Spin Structures and the Second Stiefel-Whitney Class - UTK Math
Vector bundles and w2. Def. A spin structure on an oriented rank n vector bundle E → X is a principal Spinn bundle PSpin(E) over X, together with a 2-fold ...
[20]
https://doi.org/10.5169/seals-38784
[21]
[PDF] Spinor structures on spheres and projective spaces
An explicit construction of spinor structures on real, complex, and quaternionic projective spaces is given for all cases when they exist. The construction is ...
[22]
[PDF] arXiv:2103.00617v1 [math.DG] 28 Feb 2021
Feb 28, 2021 · Here we consider a class of non-spin manifolds with “almost spin” structure, namely those with spinc or. Pin±-structures. It turns out that in ...
[23]
[PDF] Spin and Spinc structures
A Spinc structure on E → X is a complex line bundle L → X together with a spin structure on E ⊕ LR. Proof. From (f,g) : X → BSO(n) × BU(1) we can push ...
[24]
[PDF] 1. Introduction 2. Geometric formulation of Spinc
In this paper I will review the def- inition of spinc-structures on manifolds from both a geometric and algebraic point of view, and prove their existence in ...
[25]
https://arxiv.org/pdf/math/9910052.pdf
[26]
[PDF] ALL 4-MANIFOLDS HAVE SPINc STRUCTURES
In this note we prove that every orientable 4-manifold allows spinc-structures. This was shown in the closed case by Hirzebruch and Hopf in [3]. They use.
[27]
https://webhomes.maths.ed.ac.uk/~v1ranick/papers/mellor.pdf
[28]
[PDF] 1. Introduction 2. Geometric formulation of Spinc
Then the double cover of O(n) is just the group of signed products, which is called P inn (a play on SO(n) and Spinn which stuck). We will define the Clifford ...<|control11|><|separator|>
[29]
[PDF] Notes on Spinors and Non-Kähler Threefolds
The standard definition in the literature of a Calabi-Yau manifold requires a complex manifold with trivial canonical bundle and Kähler metric g satisfying ...<|separator|>
[30]
[PDF] Lectures on Dirac Operators and Index Theory - UCSB Math
Jan 7, 2015 · Similar procedure for Clifford algebra leads us to the groups P in and Spin. For simplicity, we restrict ourselves to Cln = Cln,0. We define Cl×.
[31]
[PDF] Lecture Notes on Seiberg-Witten Invariants (Revised Second Edition)
manifold with a spinc structure which has determinant line bundle L2. Given a connection d2A on L2, there is a unique Spin(4)c-connection on W ⊗. L which ...
[32]
[PDF] The Index of Elliptic Operators: I - MF Atiyah, IM Singer
Feb 25, 2002 · This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main ...
[33]
[PDF] The Atiyah-Hirzebruch spectral sequence
Aug 12, 2020 · É The Atiyah-Hirzebruch spectral sequence collapses at E2 without extension problems. É Spin bordism: There is a 7-connected map MSpin → ko.
[34]
Theory of Spinors in Curved Space-Time - MDPI
This paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection.
[35]
[PDF] Fermions in Curved Spacetimes
In this thesis we study a formulation of Dirac fermions in curved spacetime that respects general coordinate invariance as well as invariance under local spin ...
[36]
[PDF] Discrete gauge anomalies revisited - arXiv
Aug 8, 2018 · Abstract: We revisit discrete gauge anomalies in chiral fermion theories in 3+1 dimen- sions. We focus on the case that the full symmetry ...
[37]
[PDF] TASI Lectures on Perturbative String Theory and Ramond ... - arXiv
The Green-Schwarz formulation of the superstring dispenses with the need to sum over different spin stuctures. (related to the NS and R sectors) in the one-loop ...<|separator|>
[38]
[PDF] Superstrings
The theory is supersymmetric: the massless fermionic sector contains the same number of degrees of freedom, which are two spin 3/2 gravitinos of opposite ...
[39]
[hep-th/0207007] Chiral Fermions and Spinc structures on Matrix ...
Jun 30, 2002 · Motivated by the chiral nature of the standard model spectrum we investigate manifolds that do not admit spinors but do admit Spin^c structures.