Spin structure

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In geometry, a spin structure is an additional piece of data which can sometimes be specified on a vector bundle over a differentiable manifold. A spin structure on a vector bundle 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.

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[edit] Introduction

Let X be a paracompact topological manifold and E an oriented vector bundle on X of dimension n equipped with a fibre metric. At each point of X, the fibre of E is an inner product space, and so determines spinor representations. A spinor bundle of E is a prescription for consistently associating a spin representation to every point of X. Spinor bundles may not always exist. In case it does, one says that 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(E), which is a principal bundle under the action of the special orthogonal group SO(n). A spin structure for PSO(E) is a lift of PSO(E) to a principal bundle PSpin(E) under the action of the spin group Spin(n), by which we mean that there exists a bundle map φ : PSpin(E) → PSO(E) such that

φ(pg) = φ(p)ι(g), for all p ∈ PSpin(E) and g ∈ Spin(n),

where ι : 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, 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 are a Z2 quotient of a principal spin bundle.

[edit] Spin structures

[edit] Obstruction

A spin structure exists if and only if the second Stiefel-Whitney class w2 of E vanishes. This is a result of Armand Borel and Friedrich Hirzebruch[1].

[edit] Classification

When spin structures exist, the inequivalent spin structures on a manifold are in one-to-one correspondence with the elements of H1(M,Z2), which by the universal coefficient theorem is isomorphic to H1(M,Z2).

Intuitively, for each nontrivial cycle on M a spin structure corresponds to a binary choice of whether a section of the SO(N) 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.

[edit] Application to particle physics

In particle physics the spin statistics theorem implies that the wavefunction of an uncharged fermion is a section of the associated vector bundle to the spin lift of an SO(N) bundle E. Therefore the choice of spin structure is part of the data needed to define the wavefunction, and one often needs to sum over these choices in the partition function. In many physical theories E is the tangent bundle, but for the fermions on the worldvolumes of D-branes in string theory it is a normal bundle.

[edit] Examples

  1. A genus g Riemann surface admits 22g inequivalent spin structures; see theta characteristic.
  2. The complex projective space CP2 is not spin.
  3. All compact, orientable manifolds of dimension 3 or less are spin.

[edit] SpinC structures

A spinc structure is analogous, but uses the spinc group, which is the U(1)-extension of SO(n) built on top of the extension by the cyclic group of order 2 in Spin(n).

[edit] Obstruction

A spinc structure exists when the third Stiefel-Whitney class of the bundle E vanishes. In this case one says that E is spinc.

[edit] Classification

When they exist, spinc structures are classified by H2(M;Z). The vanishing of the third Stiefel-Whitney class ensures that this class has an integral lift to a class in H2(M;Z), which intuitively is the Chern class of the square of the U(1) part of the spinc bundle.

[edit] Geometric picture

This has the following geometric interpretation, which is due to Edward Witten. When the spinc structure is nonzero this square root bundle has a non-integral Chern class, which means that it fails the triple overlap condition. In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for a principal bundle. Instead it is sometimes −1.

This failure occurs at the precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed spin bundle. Therefore the triple products of transition functions of the full spinc bundle, which are the products of the triple product of the spin and U(1) component bundles, are either 12=1 or -12=1 and so the spinc bundle satisfies the triple overlap condition and is therefore a legitimate bundle.

[edit] The details

The above intuitive geometric picture may be made concrete as follows. Consider the short exact sequence Z →Z →Z2 where the first arrow is multiplication by 2 and the second is reduction modulo 2. This induces a long exact sequence on cohomology, which contains

\longrightarrow\textrm H^2(M;\mathbf Z)\longrightarrow\textrm H^2(M;\mathbf Z)\longrightarrow\textrm H^2(M;\mathbf Z_2)\longrightarrow\textrm H^3(M;\mathbf Z)\longrightarrow

where the second arrow multiplication by 2, the third is induced by restriction modulo 2 and the fourth is the associated Bockstein homomorphism β.

The obstruction to the existence of a spin bundle is an element w2 of H2(M,Z2). It reflects the fact that one may always locally lift an SO(N) bundle to a spin bundle, but one needs to choose a Z2 lift of each transition function, which is a choice of sign. The lift does not exist when the product of these three signs on a triple overlap is -1, which yields the Cech cohomology picture of w2.

To cancel this obstruction, one tensors this spin bundle with a U(1) bundle with the same obstruction w2. Notice that this is an abuse of the word bundle, as neither the spin bundle nor the U(1) bundle satisfies the triple overlap condition and so neither is actually a bundle.

A legitimate U(1) bundle is classified by its Chern class, which is an element of H2(M,Z). Identify this class with the first element in the above exact sequence. The next arrow doubles this Chern class, and so legitimate bundles will correspond to even elements in the second H2(M,Z), while odd elements will correspond to bundles that fail the triple overlap condition. The obstruction then is classified by the failure of an element in the second H2(M,Z) to be in the image of the arrow, which, by exactness, is classified by its image in H2(M,Z2) under the next arrow.

To cancel the corresponding obstruction in the spin bundle, this image needs to be w2. In particular, if w2 is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to w2 and so the obstruction cannot be cancelled. By exactness, w2 is in the image of the preceding arrow only if it is in the kernel of the next arrow, which we recall is the Bockstein homomorphism β. That is, the condition for the cancellation of the obstruction is

W3 = βw2 = 0

where we have used the fact that the third Stiefel-Whitney class W3 is the Bockstein of the second Stiefel-Whitney class w2.

[edit] Integral lifts of Stiefel-Whitney classes

This argument also demonstrates that third Stiefel-Whitney class defines elements not only of Z2 cohomology but also of integral cohomology. In fact this is the case for all odd Stiefel-Whitney classes. It is traditional to use an uppercase W for such classes.

[edit] Application to particle physics

In quantum field theory charged spinors are sections of associated spinc bundles, and in particular no charged spinors can exist on a space that is not spinc. An exception arises in some supergravity theories where additional interactions imply that other fields may cancel the third Stiefel-Whitney class.

[edit] Examples

  1. All compact, oriented, smooth manifolds of dimension 4 or less are spinc.
  2. All almost complex manifolds are spinc.
  3. All spin manifolds are spinc.

[edit] See also

[edit] References

  1. ^ Borel, A. and Hirzebruch, F. Characteristic classes and homogeneous spaces I, Amer. J. Math 80 (1958), 97-136)

[edit] Further reading

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