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