spin geometry

string geometry

cohomology

# Contents

## 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).

###### 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. 1 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 1 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.)

## 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

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. 2 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.

## 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

and where the obstruction classes are the universal characteristic classes

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.

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$\vdots$
$\downarrow$
ninebrane 10-group$\mathbf{B}Ninebrane$ninebrane structurethird fractional Pontryagin class
$\downarrow$
fivebrane 6-group$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$fivebrane structuresecond fractional Pontryagin class
$\downarrow$
string 2-group$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$string structurefirst fractional Pontryagin class
$\downarrow$
spin group$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$spin structuresecond Stiefel-Whitney class
$\downarrow$
special orthogonal group$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$orientation structurefirst 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

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

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

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.

Discussion of spin structure on Kähler manifolds is in

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

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

A survey is also at

### In quantum anomaly cancellation

Discussions of spin structures in the context of quantum anomaly cancellation for the spinning particle date back to

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

• Luis Alvarez-Gaumé, Communications in Mathematical Physics 90 (1983) 161

• D. Friedan, P. Windey, Nucl. Phys. B235 (1984) 395

Revised on June 2, 2014 02:29:59 by Urs Schreiber (89.204.135.76)