In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.

Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry.

Overview

In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M have a spin structure.^[1][2][3] This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M vanishes. Furthermore, if w₂(M) = 0, then the set of the isomorphism classes of spin structures on M is acted upon freely and transitively by H¹(M, Z₂) . As the manifold M is assumed to be oriented, the first Stiefel–Whitney class w₁(M) ∈ H¹(M, Z₂) of M vanishes too. (The Stiefel–Whitney classes w_i(M) ∈ Hⁱ(M, Z₂) of a manifold M are defined to be the Stiefel–Whitney classes of its tangent bundle TM.)

The bundle of spinors π_S: S → M over M is then the complex vector bundle associated with the corresponding principal bundle π_P: P → M of spin frames over M and the spin representation of its structure group Spin(n) on the space of spinors Δ_n. The bundle S is called the spinor bundle for a given spin structure on M.

A precise definition of spin structure on manifold was possible only after the notion of fiber bundle had been introduced; André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case.^[4][5]

Spin structures on Riemannian manifolds

Definition

A spin structure on an orientable Riemannian manifold ${\displaystyle (M,g)}$ with an oriented vector bundle ${\displaystyle E}$ is an equivariant lift of the orthonormal frame bundle ${\displaystyle P_{\operatorname {SO} }(E)\rightarrow M}$ with respect to the double covering ${\displaystyle \rho :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)}$. In other words, a pair ${\displaystyle (P_{\operatorname {Spin} },\phi )}$ is a spin structure on the SO(n)-principal bundle ${\displaystyle \pi :P_{\operatorname {SO} }(E)\rightarrow M}$ when

a) ${\displaystyle \pi _{P}:P_{\operatorname {Spin} }\rightarrow M}$ is a principal Spin(n)-bundle over ${\displaystyle M}$, and

b) ${\displaystyle \phi :P_{\operatorname {Spin} }\rightarrow P_{\operatorname {SO} }(E)}$ is an equivariant 2-fold covering map such that

${\displaystyle \pi \circ \phi =\pi _{P}\quad }$and${\displaystyle \quad \phi (pq)=\phi (p)\rho (q)\quad }$for all ${\displaystyle p\in P_{\operatorname {Spin} }}$ and ${\displaystyle q\in \operatorname {Spin} (n)}$.

Two spin structures ${\displaystyle (P_{1},\phi _{1})}$ and ${\displaystyle (P_{2},\phi _{2})}$ on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map ${\displaystyle f:P_{1}\rightarrow P_{2}}$ such that

${\displaystyle \phi _{2}\circ f=\phi _{1}\quad }$ and ${\displaystyle \quad f(pq)=f(p)q\quad }$ for all ${\displaystyle p\in P_{1}}$ and ${\displaystyle q\in \operatorname {Spin} (n)}$.

In this case ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$ are two equivalent double coverings.

The definition of spin structure on ${\displaystyle (M,g)}$ as a spin structure on the principal bundle ${\displaystyle P_{\operatorname {SO} }(E)\rightarrow M}$ is due to André Haefliger (1956).

Obstruction

Haefliger^[1] found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold (M,g). The obstruction to having a spin structure is a certain element of H²(M, Z₂) . For a spin structure the class is the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M. Hence, a spin structure exists if and only if the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M vanishes.

Spin structures on vector bundles

Let M be a paracompact topological manifold and E an oriented vector bundle on M of dimension n equipped with a fibre metric. This means that at each point of M, the fibre of E is an inner product space. A spinor bundle of E is a prescription for consistently associating a spin representation to every point of M. There are topological obstructions to being able to do it, and consequently, a given bundle E may not admit any spinor bundle. In case it does, one says that the bundle E is spin.

This may be made rigorous through the language of principal bundles. The collection of oriented orthonormal frames of a vector bundle form a frame bundle P_SO(E), which is a principal bundle under the action of the special orthogonal group SO(n). A spin structure for P_SO(E) is a lift of P_SO(E) to a principal bundle P_Spin(E) under the action of the spin group Spin(n), by which we mean that there exists a bundle map ${\displaystyle \phi }$ : P_Spin(E) → P_SO(E) such that

${\displaystyle \phi (pg)=\phi (p)\rho (g)}$, for all p ∈ P_Spin(E) and g ∈ Spin(n),

where ρ : Spin(n) → SO(n) is the mapping of groups presenting the spin group as a double-cover of SO(n).

In the special case in which E is the tangent bundle TM over the base manifold M, if a spin structure exists then one says that M is a spin manifold. Equivalently M is spin if the SO(n) principal bundle of orthonormal bases of the tangent fibers of M is a Z₂ quotient of a principal spin bundle.

If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle.

Obstruction and classification

For an orientable vector bundle ${\displaystyle \pi _{E}:E\to M}$ a spin structure exists on ${\displaystyle E}$ if and only if the second Stiefel–Whitney class ${\displaystyle w_{2}(E)}$ vanishes. This is a result of Armand Borel and Friedrich Hirzebruch.^[6] Furthermore, in the case ${\displaystyle E\to M}$ is spin, the number of spin structures are in bijection with ${\displaystyle H^{1}(M,\mathbb {Z} /2)}$. These results can be easily proven^{[7]pg 110-111} using a spectral sequence argument for the associated principal ${\displaystyle \operatorname {SO} (n)}$-bundle ${\displaystyle P_E\to <!--\; \to \; -->M}$Zdroj:https://en.wikipedia.org?pojem=Spin_structure
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