Spin structure
In
Spin structures have wide applications to
Overview
In
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.[4][5]
Spin structures on Riemannian manifolds
Definition
A spin structure on an
- a) is a principal Spin(n)-bundle over , and
- b) is an covering mapsuch that
andfor all and .
Two spin structures and on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map such that
- and for all and .
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 (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 [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
This may be made rigorous through the language of
- , for all p ∈ PSpin(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
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 (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
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 . (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 3or less are spin.
- All Calabi–Yau manifolds are spin.
Properties
- The  genusof a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.
- In general the  genusis a rational invariant, defined for any manifold, but it is not in general an integer.
- This was originally proven by  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.
- In general the
SpinC structures
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. Thus there is a homomorphism
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
where U(1) = SO(2) = S1. In other words, the group SpinC(n) is a
Viewed another way, SpinC(n) is the quotient group obtained from Spin(n) × Spin(2) with respect to the normal Z2 which is generated by the pair of covering transformations for the bundles Spin(n) → SO(n) and Spin(2) → SO(2) 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.[10][11]
The fundamental group π1(SpinC(n)) 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 TN ⊕ L.
Obstruction
A spinC structure exists when the bundle is orientable and the second Stiefel–Whitney class of the bundle E is in the image of the map H2(M, Z) → H2(M, Z/2Z) (in other words, the third integral Stiefel–Whitney class vanishes). In this case one says that E is spinC. 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.
Classification
When a manifold carries a spinC structure at all, the set of spinC structures forms an affine space. Moreover, the set of spinC structures has a free transitive action of H2(M, Z). Thus, spinC-structures correspond to elements of H2(M, Z) although not in a natural way.
Geometric picture
This has the following geometric interpretation, which is due to
This failure occurs at 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 (−1)2 = 1 and so the spinC bundle satisfies the triple overlap condition and is therefore a legitimate bundle.
The details
The above intuitive geometric picture may be made concrete as follows. Consider the
where the second arrow is induced by 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 Čech 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
where we have used the fact that the third integral Stiefel–Whitney class W3 is the Bockstein of the second Stiefel–Whitney class w2 (this can be taken as a definition of W3).
Integral lifts of Stiefel–Whitney classes
This argument also demonstrates that second Stiefel–Whitney class defines elements not only of Z2 cohomology but also of integral cohomology in one higher degree. In fact this is the case for all even Stiefel–Whitney classes. It is traditional to use an uppercase W for the resulting classes in odd degree, which are called the integral Stiefel–Whitney classes, and are labeled by their degree (which is always odd).
Examples
- All smooth manifolds of dimension 4 or less are spinC.[12]
- All almost complex manifolds are spinC.
- All spin manifolds are spinC.
Application to particle physics
In
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.[13][14] 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".[13][15]
See also
- Metaplectic structure
- Orthonormal frame bundle
- Spinor
References
- ^ a b Haefliger, A. (1956). "Sur l'extension du groupe structural d'un espace fibré". C. R. Acad. Sci. Paris. 243: 558–560.
- ^ J. Milnor (1963). "Spin structures on manifolds". L'Enseignement Mathématique. 9: 198–203.
- .
- .
- ^ Alagia, H. R.; Sánchez, C. U. (1985), "Spin structures on pseudo-Riemannian manifolds" (PDF), Revista de la Unión Matemática Argentina, 32: 64–78
- JSTOR 2372795.
- ^ Pati, Vishwambhar. "Elliptic complexes and index theory" (PDF). Archived (PDF) from the original on 20 Aug 2018.
- ^ "Spin manifold and the second Stiefel-Whitney class". Math.Stachexchange.
- ISBN 978-0-691-08542-5.
- S2CID 6906852.
- ISBN 978-0-8218-2055-1.
- ISBN 0-8218-0994-6.
- ^ S2CID 119598006..
- S2CID 104292702.
- S2CID 118698159.
Further reading
- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. ISBN 978-0-691-08542-5.
- Friedrich, Thomas (2000). Dirac Operators in Riemannian Geometry. ISBN 978-0-8218-2055-1.
- Karoubi, Max (2008). K-Theory. Springer. pp. 212–214. ISBN 978-3-540-79889-7.
- Greub, Werner; Petry, Herbert-Rainer (2006) [1978]. "On the lifting of structure groups". Differential Geometrical Methods in Mathematical Physics II. Lecture Notes in Mathematics. Vol. 676. Springer-Verlag. pp. 217–246. ISBN 9783540357216.
- Scorpan, Alexandru (2005). "4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of". The wild world of 4-manifolds. American Mathematical Society. pp. 174–189. ISBN 9780821837498.
External links
- Something on Spin Structures by Sven-S. Porst is a short introduction to orientation and spin structures for mathematics students.