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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

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2 Answers 2

I think that much of what you want to know can be summarized in the question: how do you compare spin bundles for different metrics. This question, at least in the Riemannian setting, is treated with some care in the paper:

Bourguignon, Jean-Pierre; Gauduchon, Paul, Spineurs, opérateurs de Dirac et variations de métriques. Comm. Math. Phys. 144 (1992), no. 3, 581–599.

This paper is the jumping-off point for Maier's paper ([3] in your citations). I haven't looked at Bourguignon-Gauduchon for a while, but I believe that it provides a way of identifying spin bundles for nearby metrics; this identification then tells you what the topology of the space E should be, and provide local trivializations for the bundle. The paper goes much further, and actually shows how the Dirac operators compare for different metrics, on the basis of this comparison of spinor bundles.

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Thanks for your answer. I knew this paper before and had a look at it again now. Granted, Bourguignon-Gauduchon as well as Maier provide a map $\beta_{g,h}:\Sigma^g M \to \Sigma^h M$ between the two spinor bundles with respect to two fixed Riemannian metrics $g,h$. Since $\beta_{g,h}$ is an isomorphism, it is of course continuous in its argument. They also construct a Hilbert space isomorphism $\bar \beta_{g,h}:L^2(\Sigma^gM) \to L^2(\Sigma^hM)$ between the associated spaces of $L^2$-sections. But I am asking why and in what sense these constructions are continuous in the metrics $g,h$? –  Meneldur Nov 26 '12 at 8:39

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

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Also an interesting idea. Maybe someone knows a suitable reference to Fréchet manifold theory? –  Meneldur Nov 29 '12 at 9:47
    
I've added a link for the Hamilton paper. I'd be interested in a reference for the push-down result. –  Michael Murray Nov 30 '12 at 1:58

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