Spin structure

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. 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 wi(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 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(n) 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 (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. ==Spin structures on Riemannian manifolds==

Definition A spin structure on an orientable Riemannian manifold (M,g) with an oriented vector bundle E is an equivariant lift of the orthonormal frame bundle P_{\operatorname{SO}}(E) \rightarrow M with respect to the double covering \rho : \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n). In other words, a pair (P_{\operatorname{Spin}}, \phi) is a spin structure on the SO(n)-principal bundle \pi: P_{\operatorname{SO}}(E) \rightarrow M when {{ordered list|type=lower-alpha|\pi_{P} : P_{\operatorname{Spin}} \rightarrow M is a principal Spin(n)-bundle over M, and {{block indent|\pi\circ \phi=\pi_P \quad and\quad \phi(pq) = \phi(p)\rho(q) \quadfor all p \in P_{\operatorname{Spin}} and q \in \operatorname{Spin}(n) .}}}} Two spin structures (P_1, \phi_1) and (P_2, \phi_2) on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map f: P_1 \rightarrow P_2 such that {{block indent|\phi_2\circ f=\phi_1 \quad and \quad f(p q) = f(p)q \quad for all p\in P_1 and q \in \operatorname{Spin}(n) .}} In this case \phi_1 and \phi_2 are two equivalent double coverings. The definition of spin structure on (M,g) as a spin structure on the principal bundle P_{\operatorname{SO}}(E) \rightarrow M is due to André Haefliger (1956). Obstruction Haefliger 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 [k] of H2(M, Z2) . For a spin structure the class [k] 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 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 \phi : PSpin(E) → PSO(E) such that :\phi(pg) = \phi(p)\rho(g), for all and , where 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 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 \pi_E:E \to M a spin structure exists on E if and only if the second Stiefel–Whitney class w_2(E) vanishes. This is a result of Armand Borel and Friedrich Hirzebruch. Furthermore, in the case E \to M is spin, the number of spin structures are in bijection with H^1(M,\mathbb{Z}/2). These results can be easily provenpg 110-111 using a spectral sequence argument for the associated principal \operatorname{SO}(n)-bundle P_E \to M. Notice this gives a fibration \operatorname{SO}(n) \to P_E \to M, hence the Serre spectral sequence can be applied. From general theory of spectral sequences, there is an exact sequence 0 \to E_3^{0,1} \to E_2^{0,1} \xrightarrow{d_2} E_2^{2,0} \to E_3^{2,0} \to 0 where \begin{align} E_2^{0,1} &= H^0(M, H^1(\operatorname{SO}(n),\mathbb{Z}/2)) = H^1(\operatorname{SO}(n),\mathbb{Z}/2) \\ E_2^{2,0} &= H^2(M, H^0(\operatorname{SO}(n),\mathbb{Z}/2)) = H^2(M,\mathbb{Z}/2). \end{align} In addition, E_\infty^{0,1} = E_3^{0,1} and E_\infty^{0,1} = H^1(P_E,\mathbb{Z}/2)/F^1(H^1(P_E,\mathbb{Z}/2)) for some filtration on H^1(P_E,\mathbb{Z}/2), hence we get a map H^1(P_E,\mathbb{Z}/2) \to E_3^{0,1} giving an exact sequence H^1(P_E,\mathbb{Z}/2) \to H^1(\operatorname{SO}(n),\mathbb{Z}/2) \to H^2(M,\mathbb{Z}/2). Now, a spin structure is exactly a double covering of P_E fitting into a commutative diagram \begin{matrix} \operatorname{Spin}(n) & \to & \tilde{P}_E & \to & M \\ \downarrow & & \downarrow & & \downarrow \\ \operatorname{SO}(n) & \to & P_E & \to & M \end{matrix} where the two left vertical maps are the double covering maps. Now, double coverings of P_E are in bijection with index 2 subgroups of \pi_1(P_E), which is in bijection with the set of group morphisms \text{Hom}(\pi_1(E), \mathbb{Z}/2). But, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group H^1(P_E,\mathbb{Z}/2). Applying the same argument to \operatorname{SO}(n), the non-trivial covering \operatorname{Spin}(n) \to \operatorname{SO}(n) corresponds to 1 \in H^1(\operatorname{SO}(n),\mathbb{Z}/2) = \mathbb{Z}/2, and the map to H^2(M,\mathbb{Z}/2) is precisely the w_2 of the second Stiefel–Whitney class, hence w_2(1) = w_2(E). If it vanishes, then the inverse image of 1 under the map H^1(P_E,\mathbb{Z}/2) \to H^1(\operatorname{SO}(n),\mathbb{Z}/2) is the set of double coverings giving spin structures. Now, this subset of H^1(P_E,\mathbb{Z}/2) can be identified with H^1(M,\mathbb{Z}/2), showing this latter cohomology group classifies the various spin structures on the vector bundle E \to M. This can be done by looking at the long exact sequence of homotopy groups of the fibration \pi_1(\operatorname{SO}(n)) \to \pi_1(P_E) \to \pi_1(M) \to 1 and applying \text{Hom}(-,\mathbb{Z}/2), giving the sequence of cohomology groups 0 \to H^1(M,\mathbb{Z}/2) \to H^1(P_E,\mathbb{Z}/2) \to H^1(\operatorname{SO}(n),\mathbb{Z}/2). Because H^1(M,\mathbb{Z}/2) is the kernel, and the inverse image of 1 \in H^1(\operatorname{SO}(n),\mathbb{Z}/2) 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 (not canonical) with the elements of H1(M,Z2), which by the universal coefficient theorem is isomorphic to H1(M,Z2). More precisely, the space of the isomorphism classes of spin structures is an affine space over 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 (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. Note that on a complex manifold X the second Stiefel-Whitney class can be computed as the first chern class \text{mod } 2. Examples • A genus g Riemann surface admits 22g inequivalent spin structures; see theta characteristic. • If H2(M,Z2) vanishes, M is spin. For example, Sn is spin for all n\neq 2 . (Note that S2 is also spin, but for different reasons; see below.) • The complex projective plane CP2 is not spin. • More generally, all even-dimensional complex projective spaces CP2n are not spin. • All odd-dimensional complex projective spaces CP2n+1 are spin. • All compact, orientable manifolds of dimension 3 or less are spin. • 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 :1 \to\mathbb Z_2\to \operatorname{Spin}^{\mathbf{C}}(n) \to \operatorname{SO}(n)\times\operatorname{U}(1) \to 1. To motivate this, suppose that is a complex spinor representation. The center of U(N) consists of the diagonal elements coming from the inclusion , i.e., the scalar multiples of the identity. Thus there is a homomorphism :\kappa\times i\colon {\mathrm {Spin}}(n)\times {\mathrm U}(1)\to {\mathrm U}(N). This will always have the element (−1,−1) in the kernel. Taking the quotient modulo this element gives the group SpinC(n). This is the twisted product :{\mathrm {Spin}}^{\mathbb C}(n) = {\mathrm {Spin}}(n)\times_{\mathbb Z_2} {\mathrm U}(1)\, , where U(1) = SO(2) = S1. In other words, the group SpinC(n) is a central extension of SO(n) by S1. Viewed another way, SpinC(n) 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(n), and a bundle over SO(n) with fibre a circle. • All almost complex manifolds are spinC. • All spin manifolds are spinC. ==Application to particle physics==

In particle physics the spin–statistics theorem implies that the wavefunction of an uncharged fermion can be described as 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. 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. The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references. It turns out that the standard notion of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial structure for the mathematical formulation of these theories is a "Lipschitz structure". ==See also==