In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought of as a Lie groupoid (with object manifold $M$) with source and target maps that coincide. The group at a point $m\in M$ is the group underlying the vector space $\bigwedge^2T_mM$. They describe this as the 'space of infinitesimal parallelograms at $m$'.

Now this paper is from almost ten years ago, and I know Urs has moved on to higher things, but this definition, or at least the idea of it, seems to me to look something like derived geometry. Namely, in theoretical physics (and by now, geometry in various guises) people are interested in the free loop space $LM$ of a manifold $M$ together with the circle action given by rotations of loops. Various interesting quantities (actions, for instance) should make sense as living on the space of infinitesimal loops.

The fixed-point space of this action, ordinarily speaking, is just the manifold $M$, included into $LM$ by the constant loops map. However, in derived geometry, when we take limits, such as fixed-point spaces, we get something else, and usually the 'right' thing. However, I don't really know anything about derived geometry, so I don't know if what I'm looking at the the Baez-Schreiber paper could be a derived space, or something related to one. Also, looking at the nLab page derived loop space, perhaps I don't want the fixed points, but just the derived free loop space itself — but this is a long shot.

My question is this:

Is the Lie groupoid described in the first paragraph above a model for a derived space? Or does it give rise to one in a canonical way? And if so, is it the derived fixed-point locus for the circle action on $LM$?

  • $\begingroup$ Hi David. It's not quite clear what you're asking exactly. Can you be more precise? $\endgroup$ – André Henriques May 21 '14 at 7:01
  • $\begingroup$ @AndréHenriques- well, it's in the title, but I guess I can make it more clear.<br> -Done! $\endgroup$ – David Roberts May 21 '14 at 7:02
  • $\begingroup$ I thought that the derived loop space $map(S^1,M)=map((\text{cohomology of $S^1$}),M)=map((\text{odd line}),M)=\Pi TM$ is the odd tangent bundle of $M$ (the space such that functions on it are differential forms on $M$). Is that right? That doesn't look like the gerbe you describe in your first paragraph. $\endgroup$ – André Henriques May 21 '14 at 7:59
  • $\begingroup$ I'm not convinced that your groupoid is very interesting... Maybe a better question is: what is the derived fixed-point locus of the free loop space? -- is there a down-to-earth (i.e. finite dimensional) description of that derived space? $\endgroup$ – André Henriques May 21 '14 at 8:15
  • $\begingroup$ @AndréHenriques - it's not my groupoid. A physicist told me that higher bundles over this thing should be interesting, and my guess as outlined in the post (that it is somehow the derived fixed-point locus) is the only reason I could think why. $\endgroup$ – David Roberts May 21 '14 at 11:52

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