Spin structures derive their name from the fact that their existence on a space make the quantum anomaly for spinning particles propagating on vanish. See there.
Let , write
from the universal moduli stack of smooth -principal bundles to that of -principal 2-bundles (-bundle gerbes). The homotopy fiber of this map in Smooth∞Grpd is , for the spin group regarded as a smooth group. Under geometric realization of cohesive ∞-groupoids this maps ti the universal second Stiefel-Whitney class
This is discussed in (dcct, section 5.1).
One checks that the homotopy fiber of in Smooth∞Grpd is , for 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 is the classifying space . Since is connected, this characterizes as .
From the perspective of lifting bundle gerbes, spin structures are discussed in (Murray-Singer 03).
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 -principal connection which in turn is modulated by a map in
Beware that sometimes in the physics literature an -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 -principal connections and -principal connections, up to equivalence.)
In this case one has:
There is a natural isomorphism
In the context of quantum field theory the existence of a spin structure on a Riemannian manifold arises notably as the condition for quantum anomaly cancellation of the sigma-model for the spinning particle – the superparticle – propagating on .
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 is famously a complex manifold: the Riemann sphere. One standard way to exhibit the complex structure is to cover with two copies of the complex plane with coordinate transition functions on the overlap given by
The 2-sphere is moreover a Kähler manifold and of course compact. Therefore by prop. 2 a spin structure on is equivalently a square root of the canonical line bundle , which here is simply the holomorphic 1-form bundle.
Now hermitian complex line bundles on the 2-sphere are classified by . By the clutching construction a line bundle given by two trivializing sections of the trivial line bundle on the coordinate patch has class the winding number of the transition function
Now the canonical section of the holomorphic 1-form bundle on is simply the canonical 1-form itself. By the above coordinate charts we have on
and so the transition function of the canonical bundle in this local trivialization is
This has winding number . Therefore the first Chern class of the holomorphic 1-form bundle is (the sign being an arbitrary convention, determined by the identification ).
And so it follows that there is a unique spin structure, namely given by choosing to be the line bundle on with first Chern class .
To construct by local sections analogous to how we got from the two sections and , slice open to and consider one of the two 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 and hence that of .
Notice also that the canonical vector field on the first patch given by transforms on the overlap to
and hence continues canonically to a well-defined vector field on all of . If is the rank line bundle on given by the clutching construction by the transition function , then holomorphic sections of this bundle are expressed in terms of canonical bases , with satisfying
and hence for
This gives a -dimensional space of holomorphic sections.
For more along these lines see also at geometric quantization of the 2-sphere.
where the stages are the deloopings of
where lifts through the stages correspond to
Notice that for instance is identified as such by using that preserves homotopy pullbacks and sends to a equivalence, so that is an isomorphism on the second homotopy group and hence by the Hurewicz theorem is also an isomorphism on the cohomology group . Analogously for the other characteristic maps.
In summary, more concisely, the tower is
where each “hook” is a fiber sequence.
| | | | | | | | | | | fivebrane 6-group | | fivebrane structure | second fractional Pontryagin class | | | | | | | string 2-group | | string structure | first fractional Pontryagin class | | | | | | | spin group | | spin structure | second Stiefel-Whitney class | | | | | | | special orthogonal group | | orientation structure | first Stiefel-Whitney class | | | | | | | orthogonal group | | orthogonal structure/vielbein/Riemannian metric | | | | | | | | general linear group | | smooth manifold | |
(all hooks are homotopy fiber sequences)
Standard texbooks include
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
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