Why to introduce spinor fields we need this map in the definition of a spin structure?

Question

Let me start with what I currently understand. Let ${\rm SO}(1,3)$ be the proper ortochronous Lorentz group. Its universal cover is ${\rm SL}(2,\mathbb{C})$. The representations of its universal cover are labelled by pairs of integers or half-integers $(A,B)$ which label ${\frak su}_\mathbb{C}(2)$ representations. The representations with $A+B$ integer descend to true representations of ${\rm SO}(1,3)$ but the ones with $A+B$ half integer do not and these are spinor representations.

In particular we have for example $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ Weyl spinors. The spinor themselves are elements of $\mathbb{C}^2$ and given $\Lambda \in {\rm SL}(2,\mathbb{C})$ we know how it acts on them through the representations $D^{(\frac{1}{2},0)}(\Lambda)$ and $D^{(0,\frac{1}{2})}(\Lambda)$.

Now given this setup we would like to talk about spinor fields in some general spacetime $(M,g)$. Since spinors are being introduced as elements of a representation space of the universal cover of ${\rm SO}(1,3)$ it comes as no surprise that the associated fields should come as sections of an associated bundle to a principal ${\rm SL}(2,\mathbb{C})$-bundle. What happens, though, is that it is often said that one needs a spin structure, which can be defined as follows:

Definition: Let $(M,g)$ be a semi-Riemannian manifold of signature $(t,s)$ and let $F(M)$ be the associated principal ${\rm SO}(t,s)$-bundle of orthonormal frames. A spin structure on $(M,g)$ is a principal ${\rm Spin}(t,s)$-bundle $\pi_S:S(M)\to M$ together with a principal bundle map $\Phi : S(M)\to F(M)$ such that $$\Phi(s\cdot \Lambda)=\Phi(s)\cdot \rho(\Lambda),$$ where $\rho: {\rm Spin}(t,s)\to {\rm SO}(t,s)$ is the covering map.

What I don't understand is how this structure is used in practice. Why do we need this map $\Phi : S(M)\to F(M)$? Why do we need to connect the two bundles together to be able to talk about spinor fields?

Because if we just have one ${\rm Spin}(t,s)$-bundle - or in signature $(1,3)$ one ${\rm SL}(2,\mathbb{C})$-bundle - it seems we can already take the spinor representations like the Weyl representations and perform the associated bundle construction to build spinor fields. Why apart from that connecting to the frame bundle through this map $\Phi$ is necessary?

Answer 1

Why do we need to connect the two bundles together to be able to talk about spinor fields?

It's dictated by the Lorentz invariance requirement of the Dirac Lagrangian $$ L = i\bar{\psi}\not{\partial}\psi - m\bar{\psi}\psi $$

with spinor $\psi$ on $S(M)$ and the space-time derivative $\not{\partial}$ on $F(M)$. Lorentz invariance forces you to properly map $S(M) \to F(M)$.