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We are in the process of finalizing an article on prequantum theory in higher geometry. An early version of our writeup I have now uploaded. It needs a few more cycles of polishing, but I thought I’ll provide this here on the nForum right now as a kind of explanation for the sheer drop in the amount of noise that I have been making around the nLab as of lately ;-:
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
Regulars here will recognize various things that I/we have been talking about for a good while now. Finally they are materializing in a more pdf-kind of incarnation…
Typo:
Just had time to read the intro. A boring typo
p. 4
motivation for studying traditonal prequantum geometry.
Flicking to the end, I saw on p. 43
perfectly analogues
and
as discussed abobe
Thanks, David! Am fixing these now…
How is work going on the higher polarization front? Is there likely to be a nice higher geometric account?
David,
that’s a very good question.
Briefly, we have a bunch of hints and little examples that tell us what various aspects of the story should be. For instance for oo-CS theories induced by oo-Chern-Weil from binary L-oo-invariant polynomials it seems clear what the real higher polarizaitons are and there are low-degree examples that confirms this and give an impression of what the general story will be like, roughly.
Then there is an observation going back to Chris Rogers, which amplifies that of the many equivalent ways to think of polarized sections = quantum states in the traditional theory the one via Bohr-Sommerfeld leaves is the one that has the best chance to abstractly generalize.
But I am still not sure about some details…
A trivial typo: in proposition 2.1.2 the Cech nerve has too many arrows between $X \times_Y X \times_Y X$ and $X\times_Y X$.
Woops. Thanks for catching that! Fixed now.
As a beginning of a partial reply to David C.s question in #2 I have started to write a note on how to axiomatize in differential cohesion the notion of foliation and of foliation by isotropic submanifolds.
That note is currently at
I have tried to mark it clearly as “Under construction” and if we later find that it should better not be there, I’ll remove it again.
So it seems that isotropic foliations work well in differential cohesion: In the above note I discuss how the ordinary notion of foliation is accurately reproduced when interpreting the differential cohesive axioms in the standard model of synthetic differential infinity-groupoids.
Eventually I should also add discussion that foliations of symplectic groupoids etc. come out as they should. I think this is kind of clear, but I haven’t yet written out any details.
So given isotropic foliations in higher differential cohesion, the next step would seem to be to axiomatize what it means for them to be maximal, hence to be Lagrangian foliations and therefore real polarizations. It seems clear that one should simply consider something like the directed collection of isotropic foliations in differential cohesion and then the extremal ones would be Lagrangian foliations, hence the real polarizations.
At this rough level this seems clear, but I still need to think about how to say this formally in a nice way, and then to check that the expected notions of ordinary Lagrangians, and then of Dirac structures etc. come out correctly.
So what I described is foliations of anything. I guess you are asking if apart from foliation by isotropic submanifolds which I mentioned, this approach also handles foliations by coisotropic submanifolds?
Indeed, this is one motivation for this axiomatization in higher geometry, as follows.
A coisotropic submanifold of a Poisson manifold is equivalently a Lagrangian dg-submanifold of the correspoding Poisson Lie algebroid: the latter is an incarnation of the Poisson manifold in higher geometry as a higher structure which carries not a degree-2 cocycle but a degree 3-cocycle (a 2-plectic structure on an infinitesimal smooth groupoid). The higher analog of Lagrangian foliations of this would correspond to a coisotropic foliation of the original Poisson manifold.
The Dirac structures that I did mention above are the next higher phenomenon in this series of Lagrangian submanifolds/foliations in higher geometry.
Okay, I see what you mean.
Effectively you are asking if in homotopical symplectic reduction (or Poisson reduction) what is described at foliation – in differential cohesive homotopy type theory sees both the restriction to the “shell” as well as the quotient of that by symmetries.
That’s a good point, actually, in that this is an important class of examples to test that axiomatization of foliations against.
I think that, yes, it does come out nicely as follows:
as discussed at differential cohesion in the section differential cohesion – Derived critical locus we can (in that axiomatics) consider:
on a space $X$ an action functional $S \colon X \to \mathbb{G}$ (think $\mathbb{G} = \mathbb{R}$ the multiplicative group of real numbers for simplicity) and then form its critical locus, which is a space canonically equipped with a map to $X$, which in homotopy type theory-notation we may entirely precisely write in the naive fashion as
$\array{ \left\{ x \in X \;| \; \mathbf{d} S_x = 0 \right\} \\ \downarrow \\ X } \,.$So if $X$ here is just a 0-truncated space (a manifold), then the top guy here is just the locus where $\mathbf{d}S$ vanishes.
But if $S$ is invariant under symmetries on $X$, then we may take $X$ to be the corresponding action groupoid. So then in terms of the groupoidal-discussion at foliation the orbits of $X$ are the leaves of the foliation by symmetries.
Moreover, in this case we may think of the above map as exhibiting the corresponding induced foliation on the critical locus in just the way that I just described in foliation – in differential cohesive homotopy type theory.
In other words, by the discussion there, the foliation OF the critical leaf BY the symmetries that you are asking for would be precisely the decomposition which behind the above link I discuss is presented by the homotopy pullback denoted $Q$ in
$\array{ Q &\to& \left\{ x \in X \;| \; \mathbf{d} S_x = 0 \right\} \\ \downarrow && \downarrow \\ \flat X &\to& X } \,.$For $X$ an ordinary Lie groupoid, then $Q$ is indeed the disjoint union of the leaves of the critical locus.
Here is now the pre-arXiv version of the companion article to the above one:
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
L-∞ algebras of local observables from higher prequantum bundles (schreiber)
Abstract, links and further details behind that link.
This second article provides the $L_\infty$-algebraic technical statements that are discussed from a more general perspective in the first one Higher geometric prequantum theory (schreiber). But it does not depend on the first one and can be read independently.
It seems there may be a typo (pointed out by students in my seminar) in the paper https://arxiv.org/abs/1304.0236v2 on page 11:
The third displayed formula on page 11 should probably say $[f,g]_{H_{/Z}}$ instead of $[f,g]_{H_{/S}}$, since the latter notation is not defined anywhere in the paper.
This typo appears to be repeated several times afterward.
Also, the clause that contains this displayed formula appears to be confusingly formulated. It appears to me that the formula following “this home space” (another typo, by the way) should be taken as a definition of the internal hom in the slice category, and only then the displayed formula that follows is well-defined and becomes (a posteriori!) valid.
Thanks for alerting me and thanks to your student for careful reading!
I just checked my local file and the journal version: There these typos are fixed (or mostly: one couple of ${}_{/S}$ instead of ${}_{/Z}$ remain – grr). Apparently we didn’t update our final version on the arXiv then. I’ll contact my coauthors to check if we should do so now.
Meanwhile,I have uploaded a pdf of our latest local version, which I have just touched-up a little further:
Beware that there the relevant page is now page 12, and the concluding formula of that discussion now sits in the first line of page 13.
Regarding your final comment: The logic here is that we take for granted the existence of the internal hom in the slice, but are motivating why it is the dependent product of that which should be regarded as the “stack of slice homs” (as opposed to the sliced stack of slice homs).
Anyway, thanks again. And if you or your student spots more typos, please let me know.
Is Remark 2.10.2 in the paper correct? It appears to rely on the concretification formula that was labeled as incorrect in this thread: https://nforum.ncatlab.org/discussion/11465/.
Alas, that’s the remark with the mistake, where Marco Benini and Alexander Schenkel spotted it.
The problem is that the evident component description of $\sharp_1 [X, \mathbf{B} G_{conn}] \to \sharp_1 [X, \mathbf{B}G]$ which I give in dcct…
(as a map of simplicial presheaves over $CartSp$: in degree 0 sending smooth forms to the point; and in degree 1 sending gauge transformations with potentially discontinuous components to these components)
…fails the Kan property in lowest degree, contrary what I made myself believe when I wrote this down.
Therefore the naive fiber product of the evident representing simplicial presheaves (which would give the desired answer) is not actually the homotopy fiber product. The correct homotopy fiber product comes out as intended in degree 1, but is too large in degree 0.
We should maybe post an erratum on this remark.
One way to correct the concretification formula is discussed in:
Section 3.2 (pp. 17) of:
The stack of Yang-Mills fields on Lorentzian manifolds
CMP 359 (2018) 765-820
I have now added an “errata” comment at least to the article’s home webpage here.
Yes, thanks for the reference.
Concerning the corrections: it would make sense at least to post the new (corrected) version on arXiv. This is what the 99.5% or so of the readers will read anyway.
If I may point out: their construction is specific to dimension d=1 (as I pointed out here: https://nforum.ncatlab.org/discussion/11465/concretification/?Focus=85528#Comment_85528) and requires as an input the additional data of the stack BG together with the map BG_conn→BG.
So at least speaking for myself, I am not entirely sure if I would refer to it as “differential concretification”, since the usual concretification functor only accepts a single argument as input.
On the other hand, additional data is common to all constructions in differential cohomology (e.g., the smooth Oka principle for differential cohomology also requires additional data), so maybe there is some sense to this terminology.
True. I remember that I talked to my coauthors about updating back then, though some argued that it’s “just a side remark”. But I’ll meet that coauthor in person soon and will discuss. Thanks for pushing me.
What I would really like to do is develop the reworked formula in generality. Myself, I don’t quite see how I will find time for this any time soon. But maybe it would make for a good thesis topic, if one finds somebody interested…
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