Spin structure


In differential geometry, a spin structure on an orientable Riemannian manifold 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 admits spinors. One method for dealing with this problem is to require that M have a spin structure. 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 ∈ H2 of M vanishes. Furthermore, if w2 = 0, then the set of the isomorphism classes of spin structures on M is acted upon freely and transitively by H1. As the manifold M is assumed to be oriented, the first Stiefel–Whitney class w1 ∈ H1 of M vanishes too. ∈ Hi
The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP: P → M of spin frames over M and the spin representation of its structure group Spin on the space of spinors Δn. 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 found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi extended this result to the non-orientable pseudo-Riemannian case.

Spin structures on Riemannian manifolds

Definition

A spin structure on an orientable Riemannian manifold with an oriented vector bundle is an equivariant lift of the orthonormal frame bundle with respect to the double covering. In other words, a pair is a spin structure on the SO-principal bundle when
Two spin structures and on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin-equivariant map such that
In this case and are two equivalent double coverings.
The definition of spin structure on as a spin structure on the principal bundle is due to André Haefliger.

Obstruction

Haefliger found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold. The obstruction to having a spin structure is a certain element of H2. For a spin structure the class is the second Stiefel–Whitney class w2 ∈ H2 of M. Hence, a spin structure exists if and only if the second Stiefel–Whitney class w2 ∈ H2 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 PSO, which is a principal bundle under the action of the special orthogonal group SO. A spin structure for PSO is a lift of PSO to a principal bundle PSpin under the action of the spin group Spin, by which we mean that there exists a bundle map ' : PSpin → PSO such that
where is the mapping of groups presenting the spin group as a double-cover of SO.
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 principal bundle of orthonormal bases of the tangent fibers of M is a Z'''2 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 a 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 a spin structure exists on if and only if the second Stiefel–Whitney class vanishes. This is a result of Armand Borel and Friedrich Hirzebruch. Furthermore, in the case is spin, the number of spin structures are in bijection with. These results can be easily provenpg 110-111 using a spectral sequence argument for the associated principal -bundle. Notice this gives a fibration
hence the Serre spectral sequence can be applied. From general theory of spectral sequences, there is an exact sequence
where
In addition, and for some filtration on, hence we get a map
giving an exact sequence
Now, a spin structure is exactly a double covering of fitting into a commutative diagram
where the two left vertical maps are the double covering maps. Now, double coverings of are in bijection with index subgroups of, which is in bijection with the set of group morphisms. But, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group. Applying the same argument to, the non-trivial covering corresponds to, and the map to is precisely the of the second Stiefel–Whitney class, hence. If it vanishes, then the inverse image of under the map
is the set of double coverings giving spin structures. Now, this subset of can be identified with, showing this latter cohomology group classifies the various spin structures on the vector bundle. This can be done by looking at the long exact sequence of homotopy groups of the fibration
and applying, giving the sequence of cohomology groups
Because is the kernel, and the inverse image of is in bijection with the kernel, we have the desired result.

Remarks on classification

When spin structures exist, the inequivalent spin structures on a manifold have a one-to-one correspondence with the elements of H1, which by the universal coefficient theorem is isomorphic to H1. More precisely, the space of the isomorphism classes of spin structures is an affine space over H1.
Intuitively, for each nontrivial cycle on M a spin structure corresponds to a binary choice of whether a section of the SO bundle switches sheets when one encircles the loop. If w2 vanishes then these choices may be extended over the two-skeleton, then they may automatically be extended over all of M. In particle physics this corresponds to a choice of periodic or antiperiodic boundary conditions for fermions going around each loop. Note that on a complex manifold the second Stiefel-Whitney class can be computed as the first chern class.

Examples

  1. A genus g Riemann surface admits 22g inequivalent spin structures; see theta characteristic.
  2. If H2 vanishes, M is spin. For example, Sn is spin for all.
  3. The complex projective plane CP2 is not spin.
  4. More generally, all even-dimensional complex projective spaces CP2n are not spin.
  5. All odd-dimensional complex projective spaces CP2n+1 are spin.
  6. All compact, orientable manifolds of dimension 3 or less are spin.
  7. All Calabi–Yau manifolds are spin.

    Properties

  • The  genus of a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.
  • :In general the  genus is a rational invariant, defined for any manifold, but it is not in general an integer.
  • :This was originally proven by Hirzebruch and Borel, and can be proven by the Atiyah–Singer index theorem, by realizing the  genus as the index of a Dirac operator – a Dirac operator is a square root of a second order operator, and exists due to the spin structure being a "square root". This was a motivating example for the index theorem.

    SpinC structures

A spinC structure is analogous to a spin structure on an oriented Riemannian manifold, but uses the SpinC group, which is defined instead by the exact sequence
To motivate this, suppose that is a complex spinor representation. The center of U consists of the diagonal elements coming from the inclusion, i.e., the scalar multiples of the identity. Thus there is a homomorphism
This will always have the element in the kernel. Taking the quotient modulo this element gives the group SpinC. This is the twisted product
where U = SO = S1. In other words, the group SpinC is a central extension of SO by S1.
Viewed another way, SpinC is the quotient group obtained from with respect to the normal Z2 which is generated by the pair of covering transformations for the bundles and respectively. This makes the SpinC group both a bundle over the circle with fibre Spin, and a bundle over SO with fibre a circle.
The fundamental group π1 is isomorphic to Z if n ≠ 2, and to Z ⊕ Z if n = 2.
If the manifold has a cell decomposition or a triangulation, a spinC structure can be equivalently thought of as a homotopy class of complex structure over the 2-skeleton that extends over the 3-skeleton. Similarly to the case of spin structures, one takes a Whitney sum with a trivial line bundle if the manifold is odd-dimensional.
Yet another definition is that a spinC structure on a manifold N is a complex line bundle L over N together with a spin structure on.