Apologies, I'm very new and naive to topology.
I have been learning about different covering maps between manifolds and recently was learning about the map $T^2 \rightarrow S^2$ which is a two-to-one covering map with four ramifications points of ramification index two.
my understanding is that a function on the torus when mapped to the two-sphere takes on a multi-function behavior, which fails at the ramification points. If we take said function and rotate it $2\pi$ around one of our four points it returns to the negative of it's value, only returning to its original value after two full revolutions around our ramification points.
The behavior of a function in this case appears precisely like spin $1/2$ spinor fields in quantum mechanics (physics is my background).
Can someone explain to me how these two concepts are (or aren't) related. I find myself picturing the spinors of quantum mechanics as maps of functions from some double covering space.
I know that the $Spin(n)$ groups are double covers of the $SO(n)$ groups from which arise spinor representations for connections via covariant derivatives in physics.
I'm also aware that fermions (particles represented by non-integer spin spinor fields) can be modeled as topological solitons called skyrmions which does involve some kinds of map between spaces though I don't get how/if it relates to what I'm asking here.
Here's my best crack at this problem:
I'll admit I just taught myself most of this terminology in the last couple days in an attempt to answer my question. This problem is to be applied to physics.
A spin structure is defined as:
A spin structure on an orientable Riemannian manifold $(M,g)$ is an equivariant lift of the oriented orthonormal frame bundle $F_{SO}(M)\rightarrow M$ with respect to the double covering
$\rho:Spin(n)\rightarrow SO(n)$. In other words, a pair $(P,F_{P})$ is a spin structure on the principal bundle $\pi:F_{SO}(M)\rightarrow M$ when
a) $\pi_{P}:P\rightarrow M$ is a principal $Spin(n)$ bundle over $M$,
b)$F_{P}:P\rightarrow F_{SO}(M)$ is an equivariant 2-fold covering map such that
$\pi\circ F_{\mathbf{P}}=\pi_{\mathbf{P}}$ and $F_{P}(pq)=F_{P}(p)\rho(q)$ for all $p\in P$ and $q\in Spin(n)$.
The principal bundle $\pi_{P}:P\rightarrow M$ is also called the bundle of spin frames over $M$.
Now let us consider the branched double covering map in the question: $$\rho:M\rightarrow N$$
Of which an explicit example is:
$$\rho:S^{2}\times S^{1}\rightarrow S^{3}$$
The oriented orthonormal frame bundle is trivial for both the oriented 3-dimensional cases: $$F_{SO}\left(M\right)=SO(3)\times M$$.
Now that we have our map: $$\pi:F_{SO}\left(N\right)\rightarrow N$$
We can consider pullback bundles under our (almost) double covering map $\rho$ between these spaces:
$$\pi':\rho^{*}F_{SO}\left(M\right)\rightarrow N$$
$$h:\rho^{*}F_{SO}\left(N\right)\rightarrow F_{SO}\left(N\right)$$
Which as far as I'm aware satisies the above criteria for a spin structure. We do have however a finite number of branch points (specifically ramification points) in our map where the double cover fails.
In the neighborhood of these ramified branch points by definition we are guaranteed a local trivialization such that normal coordinates take the form:
$$w=x^{k}$$
Where k is the ramification index of the point (2 in our case). For our map
The pullback bundle article also mentions that sections s of a bundle are pulled back according to:
$$\rho^{*}s=s\circ\rho$$
In pulling back a section of our frame bundle in the neighborhood of our ramification points then we obtain:
$$\rho^{*}s=\sqrt{s}$$
Which is the square root of a section of the oriented orthonormal frame bundle! One of Wikipedia's (many) definitions of a spinor is:
spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).
SUMMARY This all seems to Imply that: The pullback of a vector bundle to a branched-double covering space is a spinor bundle.
IS THIS CORRECT?!?!?!
Now I've just learned most of this fiber bundle terminology to solve this exact problem, so please correct me if I'm wrong. I would happily accept a well written explanation of why or why not I'm right or wrong. See what I did there (: