I'm pretty confused about the motivations behind defining a spin structure on a manifold. Let me explain.
In quantum mechanics, particles are represented by irreducible unitary projective representations of the Poincaré group (i.e. the states that characterize a particle are vectors belonging to a representation of $ SO^+(1,3)$). Knowing that there is a correspondence between the projective representations of a group and the "true" representations of its universal covering group, we are more interested in the "true" representations of the universal covering group (since they are easier to find) of $ SO^+(1,3) $ i.e. $ SL(2, \mathbb{C}) $.
Now let's turn to QFT and consider that our spacetime is characterized by 4-dimensional Lorentzian manifold $ (M,g) $. Particles are therefore characterized by fields $\phi$ defined over $M$ and we naturally expect that for any $x \in M$
$$ \phi(x) \in V $$
where $V$ is a representation space of $ SL(2, \mathbb{C})$. So, if we want to speak in the language of fiber bundle, we expect the fields representing the particles to be sections of a vector bundle $E$ associated to an $SL(2,\mathbb{C})$ principal bundle.
So here's my question:
Why do we define a spin structure as:
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a $SL(2, \mathbb{C})$-principal bundle $P$
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AND a double covering $\Lambda : P \longrightarrow FM $ where $FM$ is the frame bundle of $M$ i.e. a $SO^+(1,3)$-principal bundle such
$$ \Lambda(pq) = \Lambda(p) \lambda(q)$$
for $q \in SL(2,\mathbb{C})$ and $\lambda: SL(2,\mathbb{C}) \longrightarrow SO^+(1,3) $ the universal covering map.
In other words, why is the second point important? Why can't we just consider any $SL(2,\mathbb{C})$-principal bundle and describe our particles with associated vector bundle?
Best Answer
You need the second condition because this $\mathrm{SL}(2,\mathbb{C})$ isn't just a random internal symmetry group (like the gauge groups of Yang-Mills theories, where you indeed just have a $G$-principal bundle without further conditions), it's supposed to be the spacetime symmetry group - the universal cover of the local $\mathrm{SO}(1,3)$-symmetry induced by the pseudo-Riemannian structure of spacetime.
As you correctly stated, we get to wanting representations of $\mathrm{SL}(2,\mathbb{C})$ by considering projective representations of $\mathrm{SO}(1,3)$, and that $\mathrm{SO}(1,3)$ is precisely the $\mathrm{SO}(1,3)$ from the frame bundle. So the manifold already comes with a $\mathrm{SO}(1,3)$-bundle, and of course our $\mathrm{SL}(2,\mathbb{C})$-bundle needs to be compatible with this and the precise meaning of compatibility is exactly that we have the covering map $P\to FM$.
The way an ordinary tensor field transforms under the spacetime symmetry that the $\mathrm{SL}(2,\mathbb{C})$ represents is already fixed by it being the tensor field on this manifold, and one way to state that transformation behavior is by considering both $TM$ and $T^\ast M$ (and their tensor powers) as associated bundles to the frame bundle $FM$. Through the projection $P\to FM$, all those bundles also become associated bundles to $P$ - the action $P$ on them is just projecting down to $FM$ and then using the action of $FM$.
This kind of "attaching" a principal bundle to the tangent/frame bundle of a manifold is called soldering and distinguishes spacetime symmetries from internal symmetries.