TITLE: Topology of the Universal Spinor Field Bundle
QUESTION [9 upvotes]: While reading article [1] below I came across the notion of a universal spinor bundle. This is defined at the beginning of section 6 (p.14) in [1] as follows: Let $M$ be a spin manifold and $\mathcal{G}$ be a set of smooth-semi Riemannian metrics that is open in the $\mathcal{C}^\infty$-topology (for simplicity let $\mathcal{G} = \mathcal{R}(M)$, the set of all Riemannian metrics). Define $E_{(g,x)} :=\Sigma^g_x M$ to be the spinor space at $x \in M$ with respect to $g \in \mathcal{R}(M)$. The author claims that
i) The map $\pi:E \to \mathcal{R}(M) \times M$, $\psi \in \Sigma^g_xM \mapsto (g,x)$ is a fibre bundle.
ii) The space $\mathcal{S}_g$ of sections (probably smooth sections?) of $\pi^{-1}(\{g\} \times M)$ is a Frechet manifold.
iii) These spaces assemble to a Frechet fibre bundle $\mathcal{S}:=\bigcup_{g \in \mathcal{R}(M)} \to \mathcal{R}(M)$.
Prior to these claims the author refers the reader to [2,p.153ff] for more details. But unfortunately, I can't find a proof of these claims in there. Probably [2] is supposed to be a general introduction to Lagrangian field theory, which is an important subject in the rest of the section. I am however interested in the Universal Spinor Bundle in its own right. Therefore this raises the following 
Question: How are (i)-(iii) proven? More explicitely I am asking 
1) How exactly are the spaces $E$, $\mathcal{S}_g$ and $\mathcal{S}$ topologized? Equivalently, how do local trivializations look like and why are their transition functions continuous resp. smooth?
2) Are $\mathcal{S}_g$ really the smooth sections and can this construction be generalized to $L^2$-sections?
3) Can someone give a reference for more details on the Universal Spinor bundle?
4) Why does one not consider $E$ as a bundle over $\mathcal{R}(M)$? Here the later I would give the $\mathcal{C}^1$-topology (this is commmon in spin geometry).
Possible Solution: I thought about it for a while and came across the idea that the identification of the spinor bundles with different metrics as discussed in [1, Section 5] or [2, Section 2] could be helpful to construct local trivializations. But I am not sure what formal argument to use in order to show that they depend continuously on the metrics or in what sense one should define continuity here. I am also unsure, if this way of thinking is not way too complicated.

[1] Bär, Gauduchon, Moroianu - Generlized Cylinders in Semi-Riemannian and Spin Geometry, http://arxiv.org/abs/math/0303095
[2] Deligne. Quantum fields and strings
[3] Maier: Generic Metrics and Connection on Spin- and Spin-c-manifolds

REPLY [2 votes]: If you are happy with Frechet bundles here is an alternative approach.  Let $\pi \colon {\mathcal G} \times M  \to M$ be the projection and consider $\pi^{-1}(TM) \to {\mathcal G} \times M$ a  real vector bundle of rank $n$. Following the notation in the paper let $P_{GL^+} \to M$ be the 
$GL^+(n, {\mathbb R})$ bundle of oriented frames of $TM$.  The bundle of oriented frames of $\pi^{-1}(TM)$
is $\pi^{-1}(P_{GL^+})$.  As in the paper pick a lift $P_{\widetilde{GL}^+} \to M$ of $P_{GL^+}$ to $\widetilde{GL}^+(n, {\mathbb R}) \to M$ of bundles over $M$. This exists because we assume $M$ is spin. Then  $\pi^{-1}(P_{\widetilde{GL}^+})$ is a lift of $\pi^{-1}(P_{GL^+})$ to $\widetilde{GL}^+(n, {\mathbb R})$.  
If $g \in {\mathcal G}$ and $m \in M$ then $\pi^{-1}(TM)_{(g, m)} = T_m M$ so has on it an inner product defined by $g(m)$. Denote this "universal" inner product on $\pi^{-1}(TM)$ by $g$.  It will be smooth for the usual reason with Frechet manifolds which is  because if $M$ and $N$ are finite-dimensional bundles then the evaluation map
$$
M \times C^\infty(M, N) \to N
$$
is a smooth map of Frechet manifolds [1]. Again following the approach in the paper we let 
$P_{SO} \subset \pi^{-1}(P_{GL^+}) $ be the subbundle of oriented orthonormal frames for the metric 
 $g$.  Taking the pre-image of this in $\pi^{-1}(P_{\widetilde{GL}^+})$ gives us a $Spin(r, s)$
bundle over $\mathcal{G} \times M$. The associated  vector bundle to this using the spin representation  gives us  $E$ as a smooth, finite rank, Frechet vector bundle. 
Finally you want a theorem that says that when you "push-down" $E$ with $\pi$ the result is a smooth Frechet vector bundle on ${\mathcal G}$.  This seems reasonably but I'm not sure where to find it.  I can't see it in [1].
Sorry this is a bit sketchy but that reflects the sketchiness of my knowledge of Frechet manifolds.  
[1] Richard Hamilton -- The Inverse Function Theorem of Nash Moser. http://dx.doi.org/10.1090%2FS0273-0979-1982-15004-2