Spin Structure
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a spin structure on an
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
allows one to define associated
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
s, giving rise to the notion of a
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
in differential geometry. Spin structures have wide applications to
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, in particular to
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
where they are an essential ingredient in the definition of any theory with uncharged
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s. They are also of purely mathematical interest in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, and
K theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
. They form the foundation for
spin geometry In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in math ...
.


Overview

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (''M'',''g'') admits
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s. One method for dealing with this problem is to require that ''M'' has 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 In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
''w''2(''M'') ∈ H2(''M'', Z2) of ''M'' vanishes. Furthermore, if ''w''2(''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 ''w''1(''M'') ∈ H1(''M'', Z2) of ''M'' vanishes too. (The Stiefel–Whitney classes ''wi''(''M'') ∈ H''i''(''M'', Z2) of a manifold ''M'' are defined to be the Stiefel–Whitney classes of its
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''TM''.) The bundle of spinors π''S'': ''S'' → ''M'' over ''M'' is then the
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
associated with the corresponding
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
πP: P → ''M'' of spin frames over ''M'' and the spin representation of its structure group Spin(''n'') on the space of spinors Δ''n''. 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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
had been introduced;
André Haefliger André Haefliger (born 22 May 1929 in Nyon, Switzerland) is a Swiss mathematician who works primarily on topology. Education and career Haefliger went to school in Nyon and then attended his final years at Collège Calvin in Geneva. He studied ...
(1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and
Max Karoubi __NOTOC__ Max Karoubi () is a French mathematician, topologist, who works on K-theory, cyclic homology and noncommutative geometry and who founded the first European Congress of Mathematics. In 1967, he received his Ph.D. in mathematics (Doct ...
(1968) extended this result to the non-orientable pseudo-Riemannian case.


Spin structures on Riemannian manifolds


Definition

A spin structure on an
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
(M,g) with an oriented vector bundle E is an
equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
''lift'' of the orthonormal frame bundle P_(E) \rightarrow M with respect to the double covering \rho : \operatorname(n) \rightarrow \operatorname(n). In other words, a pair (P_, \phi) is a spin structure on the SO(''n'')-principal bundle \pi: P_(E) \rightarrow M when :a) \pi_ : P_ \rightarrow M is a principal Spin(''n'')-bundle over M, and :b) \phi: P_ \rightarrow P_(E) is an
equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
2-fold
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
such that
\pi\circ \phi=\pi_P \quad and\quad \phi(pq) = \phi(p)\rho(q) \quadfor all p \in P_ and q \in \operatorname(n) .
Two spin structures (P_1, \phi_1) and (P_2, \phi_2) on the same oriented
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
are called "equivalent" if there exists a Spin(''n'')-equivariant map f: P_1 \rightarrow P_2 such that :\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(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_(E) \rightarrow M is due to
André Haefliger André Haefliger (born 22 May 1929 in Nyon, Switzerland) is a Swiss mathematician who works primarily on topology. Education and career Haefliger went to school in Nyon and then attended his final years at Collège Calvin in Geneva. He studied ...
(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 In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
''w''2(''M'') ∈ H2(''M'', Z2) of ''M''. Hence, a spin structure exists if and only if the second Stiefel–Whitney class ''w''2(''M'') ∈ H2(''M'', Z2) of ''M'' vanishes.


Spin structures on vector bundles

Let ''M'' be a
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
and ''E'' an oriented vector bundle on ''M'' of dimension ''n'' equipped with a
fibre metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. Definition If ''M'' is a topological manifol ...
. This means that at each point of ''M'', the fibre of ''E'' is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
. A spinor bundle of ''E'' is a prescription for consistently associating a
spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
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 bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
s. The collection of oriented
orthonormal frame In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If ''M'' is a manifold equipped with a metric ''g'', then an orthonormal frame at a point ...
s of a vector bundle form a
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
''P''SO(''E''), which is a principal bundle under the action of the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
SO(''n''). A spin structure for ''P''SO(''E'') is a ''lift'' of ''P''SO(''E'') to a principal bundle ''P''Spin(''E'') under the action of the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...
Spin(''n''), by which we mean that there exists a bundle map ''\phi'' : ''P''Spin(''E'') → ''P''SO(''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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''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 In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of the tangent fibers of ''M'' is a Z2 quotient of a principal spin bundle. If the manifold has a
cell decomposition Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
or a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
over the 1-
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
\pi_E:E \to M a spin structure exists on E if and only if the second
Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
w_2(E) vanishes. This is a result of
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
and
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
. Furthermore, in the case E \to M is spin, the number of spin structures are in bijection with H^1(M,\mathbb/2). These results can be easily provenpg 110-111 using a spectral sequence argument for the associated principal \operatorname(n)-bundle P_E \to M. Notice this gives a
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
\operatorname(n) \to P_E \to M
hence the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
can be applied. From general theory of spectral sequences, there is an exact sequence
0 \to E_3^ \to E_2^ \xrightarrow E_2^ \to E_3^ \to 0
where
\begin E_2^ &= H^0(M, H^1(\operatorname(n),\mathbb/2)) = H^1(\operatorname(n),\mathbb/2) \\ E_2^ &= H^2(M, H^0(\operatorname(n),\mathbb/2)) = H^2(M,\mathbb/2) \end
In addition, E_\infty^ = E_3^ and E_\infty^ = H^1(P_E,\mathbb/2)/F^1(H^1(P_E,\mathbb/2)) for some filtration on H^1(P_E,\mathbb/2), hence we get a map
H^1(P_E,\mathbb/2) \to E_3^
giving an exact sequence
H^1(P_E,\mathbb/2) \to H^1(\operatorname(n),\mathbb/2) \to H^2(M,\mathbb/2)
Now, a spin structure is exactly a double covering of P_E fitting into a commutative diagram
\begin \operatorname(n) & \to & \tilde_E & \to & M \\ \downarrow & & \downarrow & & \downarrow \\ \operatorname(n) & \to & P_E & \to & M \end
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(\pi_1(E), \mathbb/2). But, from
Hurewicz theorem In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
and change of coefficients, this is exactly the cohomology group H^1(P_E,\mathbb/2). Applying the same argument to \operatorname(n), the non-trivial covering \operatorname(n) \to \operatorname(n) corresponds to 1 \in H^1(\operatorname(n),\mathbb/2) = \mathbb/2, and the map to H^2(M,\mathbb/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/2) \to H^1(\operatorname(n),\mathbb/2)
is the set of double coverings giving spin structures. Now, this subset of H^1(P_E,\mathbb/2) can be identified with H^1(M,\mathbb/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(n)) \to \pi_1(P_E) \to \pi_1(M) \to 1
and applying \text(-,\mathbb/2), giving the sequence of cohomology groups
0 \to H^1(M,\mathbb/2) \to H^1(P_E,\mathbb/2) \to H^1(\operatorname(n),\mathbb/2)
Because H^1(M,\mathbb/2) is the kernel, and the inverse image of 1 \in H^1(\operatorname(n),\mathbb/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 In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...
is isomorphic to H1(''M'',Z2). More precisely, the space of the isomorphism classes of spin structures is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
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 ''w''2 vanishes then these choices may be extended over the two-
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
, then (by
obstruction theory In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the exis ...
) they may automatically be extended over all of ''M''. In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
this corresponds to a choice of periodic or antiperiodic
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s for
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
going around each loop. Note that on a complex manifold X the second Stiefel-Whitney class can be computed as the first
chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
\text 2.


Examples

# A
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
''g''
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
admits 22''g'' inequivalent spin structures; see
theta characteristic In mathematics, a theta characteristic of a non-singular algebraic curve ''C'' is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles ''L'' on a connected compact Riemann surface, it is therefore ''L'' such ...
. # If ''H''2(''M'',Z2) vanishes, ''M'' is ''spin''. For example, ''S''''n'' is ''spin'' for all n\neq 2 . (Note that ''S''2 is also ''spin'', but for different reasons; see below.) # The
complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
CP2 is not ''spin''. # More generally, all even-dimensional
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
s CP2''n'' are not ''spin''. # All odd-dimensional
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
s CP2n+1 are ''spin''. # All compact,
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
s of dimension 3 or less are ''spin''. # All
Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
s are ''spin''.


Properties

* The
 genus In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the ...
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 In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the ...
is a rational invariant, defined for any manifold, but it is not in general an integer. *:This was originally proven by
Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
and
Borel Borel may refer to: People * Borel (author), 18th-century French playwright * Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding ...
, and can be proven by the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
, by realizing the
 genus In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the ...
as the index of a
Dirac operator In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formall ...
– 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, but uses the SpinC group, which is defined instead by the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
:1 \to\mathbb Z_2\to \operatorname^(n) \to \operatorname(n)\times\operatorname(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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
:\kappa\times i\colon (n)\times (1)\to (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 :^(n) = (n)\times_ (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. 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 Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
or a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
, a spinC structure can be equivalently thought of as a homotopy class of complex structure over the 2-
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
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 .


Obstruction

A spinC structure exists when the bundle is orientable and the second
Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
of the bundle ''E'' is in the image of the map (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 In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
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 . Thus, spinC-structures correspond to elements of although not in a natural way.


Geometric picture

This has the following geometric interpretation, which is due to
Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
. 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 In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
. Instead it is sometimes −1. This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed
spin bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\col ...
. Therefore, the triple products of transition functions of the full ''spin''''c'' bundle, which are the products of the triple product of the ''spin'' and U(1) component bundles, are either or 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
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
, where the second
arrow An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers c ...
is
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
by 2 and the third is reduction modulo 2. This induces a
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
on cohomology, which contains ::\dots \longrightarrow \textrm H^2(M;\mathbf Z) \stackrel \textrm H^2(M;\mathbf Z) \longrightarrow \textrm H^2(M;\mathbf Z_2) \stackrel \longrightarrow \textrm H^3(M;\mathbf Z) \longrightarrow \dots , where the second
arrow An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers c ...
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 ''w''2 of . 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 In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topolo ...
picture of ''w''2. To cancel this obstruction, one tensors this ''spin'' bundle with a U(1) bundle with the same obstruction ''w''2. 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 In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
, 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 , 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 ''w''2. In particular, if ''w''2 is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to ''w''2 and so the obstruction cannot be cancelled. By exactness, ''w''2 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 :::W_3=\beta w_2=0 where we have used the fact that the third integral Stiefel–Whitney class ''W''3 is the Bockstein of the second Stiefel–Whitney class ''w''2 (this can be taken as a definition of ''W''3).


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 oriented
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s of dimension 4 or less are spinC. # All
almost complex manifold In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...
s are spinC. # All ''spin'' manifolds are spinC.


Application to particle physics

In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
implies that the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
of an uncharged
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
is a section of the
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
, but for the fermions on the worldvolumes of
D-branes In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
it is a
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...
. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
charged spinors are sections of associated ''spin''''c'' bundles, and in particular no charged spinors can exist on a space that is not ''spin''''c''. An exception arises in some
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
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

* Metaplectic structure *
Orthonormal frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
*
Spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...


References


Further reading

* * * * * {{Cite book , last1=Scorpan , first1=Alexandru , title=The wild world of 4-manifolds , publisher= American Mathematical Society , year=2005 , pages=174–189 , chapter=4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of , chapter-url=https://books.google.com/books?id=VgG9AwAAQBAJ&pg=PA173 , isbn=9780821837498


External links


Something on Spin Structures
by Sven-S. Porst is a short introduction to
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
and spin structures for mathematics students. Structures on Riemannian manifolds Structures on manifolds Algebraic topology K-theory Mathematical physics