Spin structure
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.They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors.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) .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.on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant mapHaefliger[1] found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold (M,g).Let M be a paracompact topological manifold and E an oriented vector bundle on M of dimension n equipped with a fibre metric.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.These results can be easily proven[7]pg 110-111 using a spectral sequence argument for the associated principalBut, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group, showing this latter cohomology group classifies the various spin structures on the vector bundleIntuitively, 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[8] vanishes then these choices may be extended over the two-skeleton, then (by obstruction theory) 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.A spinC structure is analogous to a spin structure on an oriented Riemannian manifold,[9] but uses the SpinC group, which is defined instead by the exact sequence To motivate this, suppose that κ : Spin(n) → U(N) is a complex spinor representation.The center of U(N) consists of the diagonal elements coming from the inclusion i : U(1) → U(N), i.e., the scalar multiples of the identity.Taking the quotient modulo this element gives the group SpinC(n).Similarly to the case of spin structures, one takes a Whitney sum with a trivial line bundle if the manifold is odd-dimensional.Intuitively, the lift gives the Chern class of the square of the U(1) part of any obtained spinC bundle.By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spinC structure.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.Consider the short exact sequence 0 → Z → Z → Z2 → 0, where the second arrow is multiplication by 2 and the third is reduction modulo 2.The lift does not exist when the product of these three signs on a triple overlap is −1, which yields the Čech cohomology picture of w2.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.To cancel the corresponding obstruction in the spin bundle, this image needs to be w2.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.An exception arises in some supergravity theories where additional interactions imply that other fields may cancel the third Stiefel–Whitney class.The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references.