nLab spin structure

Idea

A spin structure on a manifold $X$ with an orientation is a lift $\hat g$ of the classifying map $g : X \to B S O(n)$ of the tangent bundle through the second step $B Spin(n) \to B S O(n)$ in the Whitehead tower of $O(n)$.

$$\array{ && B Spin(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& B S O(n) }$$

Spin structures derive their name from the fact that their existence on a space $X$ make the quantum anomaly for spinning particles propagating on $X$ vanish. See there.

Definition

Let $n \in \mathbb{N}$, write

$$\mathbb{Z}_2 \to Spin(n) \to SO(n)$$

for the spin group group extension of the special orthogonal group in dimension $n$. (All of the following also applies verbatim for Lorentzian signature).

Topological

Definition

Write

$$\widetilde{SO(n)} \coloneqq [\mathbb{Z}_2 \to Spin(n)]$$

for the crossed module of smooth groups induced by the spin group extension. Write

$$\mathbf{B}\mathbb{Z}_2 \coloneqq [\mathbb{Z}_2 \to 1]$$

for the crossed module given by shifting up the group of order two in degree. Finally write

$$\array{ \widetilde{SO(n)} \\ \downarrow \\ SO(n) }$$

for the canonical morphism of crossed modules which is the terminal morphism in degree-1 and the defining projection in degree 0, and write

$$\tilde{\mathbf{w}}_2 \;\colon\; \widetilde{SO(n)} = [\mathbb{Z}_2 \to Spin(n)] \longrightarrow [\mathbb{Z}_2 \to 1] = \mathbf{B}\mathbb{Z}_2$$

for the canonical morphism of crossed modules which is the identity in degree 1 and the terminal map in degree 0.

Proposition

In the span/zig-zag

$$\array{ \widetilde{SO(n)} &\stackrel{\tilde{\mathbf{w}_2}}{\longrightarrow}& \mathbf{B}\mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ SO(n) }$$

the left leg is a weak equivalence of smooth groupoids (under the identification of crossed modules with strict 2-groups). Under delooping it presents a morphism of smooth 2-groupoids of the form

$$\mathbf{w}_2 \;\colon\; \mathbf{B}SO(n) \longrightarrow \mathbf{B}^2 \mathbb{Z}_2$$

from the universal moduli stack of smooth $SO(n)$-principal bundles to that of $\mathbf{B}\mathbb{Z}_2$-principal 2-bundles ($\mathbb{Z}_2$-bundle gerbes). The homotopy fiber of this map in Smooth∞Grpd is $\mathbf{B}Spin(n)$, for the spin group regarded as a smooth group. Under geometric realization of cohesive ∞-groupoids ${\vert - \vert} \colon Smooth\infty Grpd \longrightarrow \infty Grpd$ this maps ti the universal second Stiefel-Whitney class

$${\vert \mathbf{w}_2\vert} \simeq w_2 \;\colon\; B SO(n) \longrightarrow K(\mathbb{Z}_2,2) \,.$$

This is discussed in (dcct, section 5.1).

Proof

One checks that the homotopy fiber of $\mathbf{w}_2$ in Smooth∞Grpd is $\mathbf{B}Spin(n)$, for $Spin(n)$ the spin group regarded as a smooth group, for instance by using the techniques for computing homotopy pullbacks as discussed there. Moreover, by the general discussion at smooth ∞-groupoid – structures this homotopy fiber is preseved under geometric realization of cohesive ∞-groupoids so that the homotopy fiber of ${\vert \mathbf{w}_2 \vert}$ is the classifying space $B Spin(n)$. Since $\mathbb{K}(\mathbb{Z}_2,2)$ is connected, this characterizes ${\vert \mathbf{w}_2 \vert}$ as $w_2$.

Remark

Given a smooth manifold $X$ with an orientation, its oriented tangent bundle is modulated by a map

$$\tau_X \colon X \longrightarrow \mathbf{B}SO(n) \,.$$

Postcomposition with $\mathbf{w}_2$ from prop. gives a map in Smooth∞Grpd of the form

$$\mathbf{w}_2(\tau_X) \;\colon\; X \stackrel{\tau_X}{\longrightarrow} \mathbf{B}SO(n) \stackrel{\mathbf{w}_2}{\longrightarrow} \mathbf{B}^2 \mathbb{Z}_2 \,.$$

This modulates a $\mathbf{B}\mathbb{Z}_2$-principal 2-bundle on $X$, also called a $\mathbb{Z}_2$-bundle gerbe. By construction (the universal property of the homotopy fiber) this is the obstruction to the existence of a lift $\hat g$ in

$$\array{ && \mathbf{B} Spin(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{\tau_x}{\to}& \mathbf{B} S O(n) &\stackrel{\mathbf{w}_2}{\longrightarrow}& \mathbf{B}^2 \mathbb{Z}_2 } \,.$$

Such a lift is a choice of spin structure on $X$. Therefore as bundle gerbes, this $\mathbf{w}_2(\tau_X)$ is also called a lifting bundle gerbe.

From the perspective of lifting bundle gerbes, spin structures are discussed in (Murray-Singer 03).

Definition

For $X$ a manifold, the groupoid/homotopy 1-type $Spin(X)$ of spin structures over $X$ is the homotopy fiber in ∞Grpd $\simeq$ $L_{whe}$Top of the second Stiefel-Whitney class

$$\array{ Spin(X) &\to& * \\ {}^{\mathllap{\eta}}\downarrow && \downarrow \\ Top(X,B SO) &\stackrel{(w_2)_*}{\to}& Top(X, B^2 \mathbb{Z}_2) } \,.$$

Here an object $s \in Spin(X)$ over an $SO$-principal bundle $\eta(s)$ on $X$ is called a spin structure on $\eta(s)$ ($SO$ is the special orthogonal group).

For $\eta(s)$ the $SO$-principal bundle for which the tangent bundle $T X$ is the canonically associated bundle, one says that a spin-structure on $\eta(s)$ is a spin structure on the manifold $X$.

Remark

From the smooth geometric perspective on spin structures of remark one may also start with an affine connection on the tangent bundle given, principally, by an $SO(n)$-principal connection which in turn is modulated by a map $\nabla$ in

$$\array{ && \mathbf{B} SO(n)_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} SO(n) } \,.$$

But since $\mathbb{Z}_2$ is a discrete group, there is no non-flat $\mathbf{B}\mathbb{Z}_2$-principal 2-connection and hence no non-trivial “differential refinement” of $\mathbf{w}_2$.

Beware that sometimes in the physics literature an $SO(n)$-principal connection is already called a “spin connection” (due to the fact that often in physics only local data is connsidered, and locally there is no difference between $Spin(n)$-principal connections and $SO(n)$-principal connections, up to equivalence.)

Algebraic

In analogy to how an orientation of a real vector space $V$ equipped with an inner product is equivalently an isometry between the top exterior power of $V$ and the real numbers, so a spin structure on a real vector space $V$ equipped with an inner product is an isomorphism in the 2-category of algebras, bimodules, and intertwiners (see here) from the Clifford algebra of $V$ to the Clifford algebra of the real vector space $\mathbf{R}^n$ of the same dimension $n=\dim V$ with the canonical inner product.

Spin structures naturally form a category, with morphisms being (isometric) isomorphisms of bimodules as described above.

Properties

Over a Riemann surface

Over a Riemann surface spin structures correspond to square roots of the canonical bundle. See at Theta characteristic.

Over a Hermitian manifold / Kähler manifold

More generally:

Proposition

A spin structure on a compact Hermitian manifold (Kähler manifold) $X$ of complex dimension $n$ exists precisely if, equivalently

• there is a choice of square root $\sqrt{\Omega^{n,0}}$ of the canonical line bundle $\Omega^{n,0}$ (a “Theta characteristic”);

• there is a trivialization of the first Chern class $c_1(T X)$ of the tangent bundle.

In this case one has:

Proposition

There is a natural isomorphism

$$S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}_X}$$

of the sheaf of sections of the spinor bundle $S_X$ on $X$ with the tensor product of the Dolbeault complex with the corresponding Theta characteristic;

Moreover, the corresponding Dirac operator is the Dolbeault-Dirac operator $\overline{\partial} + \overline{\partial}^\ast$.

This is due to (Hitchin 74). A textbook account is for instance in (Friedrich 74, around p. 79 and p. 82).

As quantum anomaly cancellation condition

In the context of quantum field theory the existence of a spin structure on a Riemannian manifold $X$ arises notably as the condition for quantum anomaly cancellation of the sigma-model for the spinning particle – the superparticle – propagating on $X$.

It is the generalization of this anomaly computation from the worldlines of superparticles to superstrings that leads to string structure, and then further the generalizaton to the worldvolume anomaly of fivebranes that leads to fivebrane structure.

Examples

On the 2-sphere

The 2-sphere $S^2$ is famously a complex manifold: the Riemann sphere. One standard way to exhibit the complex structure is to cover $S^2$ with two copies of the complex plane with coordinate transition functions on the overlap $\mathbb{C} - \{0\}$ given by

$$z_2 = z_1^{-1} \,.$$

The 2-sphere is moreover a Kähler manifold and of course compact. Therefore by prop. a spin structure on $S^2$ is equivalently a square root $\sqrt{\Omega^{1,0}} \in \mathbf{Line}_{\mathbb{C}}(S^2)$ of the canonical line bundle $\Omega^{1,0}$, which here is simply the holomorphic 1-form bundle.

Now hermitian complex line bundles on the 2-sphere are classified by $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$. By the clutching construction a line bundle given by two trivializing sections $\sigma_1, \sigma_2$ of the trivial line bundle on the coordinate patch $\mathbb{C}$ has class the winding number of the transition function

$$S^1 \hookrightarrow \mathbb{C} - \{0\} \stackrel{s_2 s_1^{-1}}{\to} \mathbb{C}^\times \,.$$

Now the canonical section of the holomorphic 1-form bundle on $\mathbb{C}$ is simply the canonical 1-form $d z$ itself. By the above coordinate charts we have on $\mathbb{C} - \{0\}$

$$d z_2 = d (z_1^{-1}) = - z_1^{-2} d z_1$$

and so the transition function of the canonical bundle in this local trivialization is

$$- z^{-2} \colon \mathbb{C}-\{0\} \to \mathbb{C}^\times \,.$$

This has winding number $\pm 2$. Therefore the first Chern class of the holomorphic 1-form bundle $\Omega^{1,0}$ is $c_1(\Omega^{1,0}) = \pm 2$ (the sign being an arbitrary convention, determined by the identification $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$).

And so it follows that there is a unique spin structure, namely given by choosing $\sqrt{\Omega^{1,0}}$ to be the line bundle on $S^2$ with first Chern class $\pm 1$.

To construct $\sqrt{\Omega^{1,0}}$ by local sections analogous to how we got $\Omega^{1,0}$ from the two sections $d z_1$ and $d z_2$, slice open $\mathbb{C}$ to $\mathbb{C} - [0,\infty)$ and consider one of the two $z_1^{-1/2} d z_1$ as a local section of the trivial complex line bundle. Do the same on the other patch. Then

$$\begin{aligned} z_2^{-1/2} d z_2 &= z_1^{1/2} (-z_1^{-2} d z_1) \\ &= -z_1^{-1}\left(z_1^{-1/2} dz_1\right) \end{aligned} \,.$$

This is well defined also over the cut and so we can patch the cut with any small neighbourhood with any section chosen over it and conclude that these sections are the local sections locally trivializing a bundle of class $\pm 1$ and hence that of $\sqrt{\Omega^{1,0}}$.

Notice also that the canonical vector field on the first patch given by $z_1 \partial_{z_1}$ transforms on the overlap to

$$\begin{aligned} z_1 \partial_{z_1} & = z_2^{-1} \frac{\partial z_2}{\partial z_1 } \partial_{z_2} \\ & = z_2^{-1} (- z_1^{-2}) \partial_{z_2} \\ & = - z_2 \partial_{z_2} \end{aligned}$$

and hence continues canonically to a well-defined vector field on all of $S^2$. If $L_k$ is the rank $k$ line bundle on $S^2$ given by the clutching construction by the transition function $z^k$, then holomorphic sections of this bundle are expressed in terms of canonical bases $z_1^{a_1}$, $z_2^{a_2}$ with $a_i \geq 0$ satisfying

$$z_2^{a_2} = z_2^{k} z_2^{-a_1}$$

and hence for

$$a_1 + a_2 = k \,.$$

This gives a $(k+1)$-dimensional space of holomorphic sections.

For more along these lines see also at geometric quantization of the 2-sphere.

On the $n$-sphere

Generally:

Example

The n-sphere, for each $n \in \mathbb{N}$, carries a canonical spin structure, induced from its coset space-realization $S^n \simeq Spin(n+1)/Spin(n)$ (here), as a special case of the canonical $H$-structure on $G/H$ (this example).

Other ways to see this:

• Nikolai Nowaczyk, Theorem A.6.6 in: Dirac Eigenvalues of higher Multiplicity (arXiv:1501.04045)

• S. Gutt, Killing spinors on spheres and projective spaces, p. 238-248 in: A. Trautman, G. Furlan (eds.) Spinors in Geometry and Physics – Trieste 11-13 September 1986, World Scientific 1988 (doi:10.1142/9789814541510, GBooks, p. 243)

Higher spin structures

Spin structures are one step in a tower of conditions that are related to the quantum anomaly cancellation of higher dimensional spinning/super branes.

This is controled by the Whitehead tower of the classifying space/delooping of the orthogonal group $O(n)$, which starts out as

$$\array{ & Whitehead tower \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ second frac Pontr. class & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ first frac Pontr. class & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ second SW class & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & B^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ first SW class & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & Postnikov tower }$$

where the stages are the deloopings of

… $\to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group,

where lifts through the stages correspond to

• orthogonal structure

• orientation

• spin structure

• string structure

• fivebrane structure

and where the obstruction classes are the universal characteristic classes

• first Stiefel-Whitney class$w_1$

• second Stiefel-Whitney class$w_2$

• first fractional Pontryagin class$\tfrac{1}{2}p_1$

• second fractional Pontryagin class$\tfrac{1}{6}p_2$

and where every possible square in the above is a homotopy pullback square (using the pasting law).

Notice that for instance $w_2$ is identified as such by using that $[S^2,-]$ preserves homotopy pullbacks and sends $B O \to \tau_{\leq 2} B O$ to a equivalence, so that $B SO \to B^2 \mathbb{Z}$ is an isomorphism on the second homotopy group and hence by the Hurewicz theorem is also an isomorphism on the cohomology group $H^2(-,\mathbb{Z}_2)$. Analogously for the other characteristic maps.

In summary, more concisely, the tower is

$$\array{ \vdots \\ \downarrow \\ B Fivebrane \\ \downarrow \\ B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z} \\ \downarrow \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z} \\ \downarrow \\ B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2 \\ \downarrow \\ B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ B GL } \,,$$

where each “hook” is a fiber sequence.

Related concepts

• metaplectic structure, metalinear structure

• (twisted differential) c-structure

  • orientation

  • spin structure, twisted spin structure, differential spin structure

    spin^c structure, twisted spin^c structure

    spin^h structure

    Spin Chern-Simons theory

  • 2-framing

  • string structure, differential string structure

  • fivebrane structure, differential fivebrane structure

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)

References

General

Standard texbooks include

• H. Blaine Lawson, Marie-Louise Michelsohn, chapter II of Spin geometry, Princeton University Press (1989)

• Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)

A discussion of the full groupoid of spin structures is in

• Johannes Ebert, Characteristic classes of spin surface bundles:

  Applications of the Madsen-Weiss theory_ Phd thesis (2006) (pdf)

Discussion of lifting bundle gerbes for spin structures is in

• Michael Murray, Michael Singer, Gerbes, Clifford modules and the index theorem (arXiv:math/0302096)

Discussion of spin structures in terms of smooth moduli stacks as above is in

• Urs Schreiber, section 5.1 of differential cohomology in a cohesive topos.

Discussion of spin structures on surfaces is in

• Dennis Johnson, Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22 (1980), no. 2, 365–373.

See also this MO comment.

Discussion of spin structure on Kähler manifolds is in

• Nigel Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1 − 55.

• Christian Bär, On harmonic spinors (pdf)

A textbook account is in (Friedrich 97, section 3.4)

A survey is also at

• Theta characteristics and framings

• Spin manifolds (pdf)

Spin structures on orbifolds:

• Daniel J. Acosta, A Furuta-like inequality for spin orbifolds and the minimal genus problem, Topology and its Applications Volume 114, Issue 1, 16 July 2001, Pages 91-106 (doi:10.1016/S0166-8641(00)00030-4)

• Chongying Dong, Kefeng Liu, Xiaonan Ma, On orbifold elliptic genus (pdf), p. 87-105 In: Alejandro Adem, Jack Morava, Yongbin Ruan, Orbifolds in Mathematics and Physics, Contemporary Mathematics 310 AMS (2002)(ams:conm-310)

• Florin Belgun, Nicolas Ginoux, Hans-Bert Rademacher, A Singularity Theorem for Twistor Spinors, Annales de l’Institut Fourier, Volume 57 (2007) no. 4, p. 1135-1159 (arXiv:math/0409136, numdam:AIF_2007__57_4_1135_0)

In quantum anomaly cancellation

Discussions of spin structures as anomaly cancellation for the spinning particle (see also the references hereuantum mechanics#ReferencesRelationToMorseTheory) at supersymmetric quantum mechanics):

• Edward Witten, p. 65-68 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913]

• Luis Alvarez-Gaumé, p. 165 of: Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90 2 (1983) 161-173 [doi:10.1007/BF01205500, euclid]

• Daniel Friedan, P. Windey, Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly, Nucl. Phys. B 235 (1984) 395 [doi:10.1016/0550-3213(84)90506-6]