spin geometry, string geometry, fivebrane geometry
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $O(n)$.
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.
Let $n \in \mathbb{N}$, write
for the spin group group extension of the special orthogonal group in dimension $n$. (All of the following also applies verbatim for Lorentzian signature).
Write
for the crossed module of smooth groups induced by the spin group extension. Write
for the crossed module given by shifting up the group of order two in degree. Finally write
for the canonical morphism of crossed modules which is the terminal morphism in degree-1 and the defining projection in degree 0, and write
for the canonical morphism of crossed modules which is the identity in degree 1 and the terminal map in degree 0.
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
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
This is discussed in (dcct, section 5.1).
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 $w_2$.
Given a smooth manifold $X$ with an orientation, its oriented tangent bundle is modulated by a map
Postcomposition with $\mathbf{w}_2$ from prop. 1 gives a map in Smooth∞Grpd of the form
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
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).
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
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 $X$.
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
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.)
Over a Riemann surface spin structures correspond to square roots of the canonical bundle. See at Theta characteristic.
More generally:
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:
There is a natural isomorphism
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).
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.
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
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
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 $\mathbb{C} - \{0\}$
and so the transition function of the canonical bundle in this local trivialization is
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
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
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
and hence for
This gives a $(k+1)$-dimensional space of holomorphic sections.
For more along these lines see also at geometric quantization of the 2-sphere.
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
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
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
where each “hook” is a fiber sequence.
smooth ∞-group | Whitehead tower of smooth moduli ∞-stacks | G-structure/higher spin structure | obstruction |
---|---|---|---|
$\vdots$ | |||
$\downarrow$ | |||
fivebrane 6-group | $\mathbf{B}Fivebrane$ | 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)
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
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
See also this MO comment.
Discussion of spin structure on Kähler manifolds is in
A textbook account is in (Friedrich 97, section 3.4)
A survey is also at
Spin manifolds (pdf)
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