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In two recent threads [1, 2] I had started to look into elementary formalization of the following obstruction problem in higher geometry:
given
a Klein geometry $H \to G$,
a WZW term $\mathbf{L}_{WZW} : G/H \longrightarrow \mathbf{B}^{p+1} \mathbb{G}_{conn}$;
a Cartan geometry $X$ modeled on $G/H$
then:
Here are first concrete observations, holding in any elementary $\infty$-topos (meaning: this may be proven using HoTT, not needing simplicial or other infinite diagrams):
First, a lemma that turns the datum of a global WZW term $Fr(X) \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}$ on the frame bundle of $X$ (each of whose fibers looks like the formal disk $\mathbb{D}$ around the base point, or any other point, in $G/H$) into something closer to cohomological data on $X$. In the following $Fr(X)$ may be any fiber bundle $E$ and $\mathbf{B}^{p+1}\mathbb{G}_{conn}$ may be any coefficient object $A$.
Lemma. Let $E \to X$ be an $F$-fiber bundle associated to an $Aut(F)$-principal bundle $P \to X$. Then $A$-valued functions on $E$ are equivalent to sections of the $[F,A]$-fiber bundle canonically associated to $P$.
Proof. By the discussion at infinity-action, the universal $[F,A]$-fiber bundle $[F,A]/Aut(F)\to \mathbf{B} Aut(F)$ is simply the function space $[F,A]_{\mathbf{B} Aut(F)}$ formed in the slice over $\mathbf{B} Aut(F)$, with $F$ regarded with its canonical $Aut(F)$-action and $A$ regarded with the trivial $Aut(F)$-action.
Now, by universality, sections of $P \underset{Aut(F)}{\times} [F,A] \to X$ are equivalently diagonal maps in
$\array{ && [F,A]/Aut(F) \\ & \nearrow & \downarrow \\ X & \longrightarrow & \mathbf{B} Aut(F) }$But by Cartesian closure in the slice and using the above, these are equivalent to horizontal maps in
$\array{ E & = & P \underset{Aut(F)}{\times} F && \longrightarrow && A \times \mathbf{B}Aut(F) \\ && & \searrow && \swarrow \\ && && \mathbf{B}Aut(F) }$Finally by $(\underset{\mathbf{B}Aut(F)}{\sum} \dashv \mathbf{B}Aut(F)^\ast)$ this is equivalent to maps $E \to A$. $\Box$
$\,$
[ continued in next comment ]
For the problem at hand, we are interested in specifically those such sections which are fiberwise fixed to be $\mathbf{L}_{WZW}^{formal} : \mathbb{D}\hookrightarrow G/H \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}^{p+1}\mathbb{G}_{conn}$.
Observation/Definition Given a $G$-action on any $H$, and given a point $x \colon \ast \to H$, then the stabilizer $\infty$-group of that point, $Stab(x)$, is the looping of the 1-image factorization of $\ast \stackrel{x}{\longrightarrow} H \to H/G$.
Hence we have
$\array{ \ast &\stackrel{x}{\longrightarrow}& H &\longrightarrow& H/G \\ \downarrow && &\nearrow & \downarrow \\ \mathbf{B} Stab(x) &=& \mathbf{B} Stab(x) &\longrightarrow& \mathbf{B}Aut(F) }$where the diagonal morphism is a 1-monomorphism. Specifically for $H$ of the form $[F,A]$ then a cartoon of the stabilizer subgroup of any $\nabla$ in there is this:
$Stab(\nabla) = \left\{ \array{ F && \stackrel{\simeq}{\longrightarrow} && F \\ & {}_{\mathllap{\nabla}} \searrow & \swArrow_{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\nabla}} \\ && A } \right\}$If here $\nabla = \mathbf{L}_{WZW}$ and after applying differential concretification, this is the “quantomorphism n-group” of the WZW term regarded as a higher prequantum bundle. This is going to be the generalization of the String-group for GS super $p$-branes.
[ continued in next comment ]
The globalized WZW terms that we are after are maps $Fr(X) \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}$ such that over a Cartan cover of $X$ by $\coprod_i G/H$ – and identifying the frame bundle on each chart with the trivial bundle $(G/H)\times \mathbb{D}$ whose fiber is the formal disk in $G/H$ – are constant, in that they factor through the projection to $\mathbb{D}$ followed by the restriction $\mathbf{L}_{WZW}^{formal} : \mathbb{D}\to \mathbf{B}^{p+1}\mathbb{G}_{conn}$.
Turning again to the more generic notation of the previous comment this means that given a section $X \to P \underset{Aut(F)}{\times} [F,A] \to X$ we are now to consider a 1-epimorphism $U \to X$ over which this section factors through a given $\ast \stackrel{\nabla}{\longrightarrow} [F,A]$.
Proposition. Sections of $P \underset{Aut(F)}{\times} [F,A] \to X$ (hence functions $E \to A$) which are locally constant on $\nabla$ in this sense are equivalent to lifts of the structure group of $E$ through $Stab(\nabla) \to Aut(F)$.
Proof. The assumption is a diagram of the form
$\array{ U &\stackrel{\nabla}{\longrightarrow}& \ast \\ \downarrow && \downarrow \\ && \mathbf{B} Stab(\nabla) \\ \downarrow && \downarrow & \searrow \\ X &\longrightarrow& [F,A]/Aut(F) &\longrightarrow& \mathbf{B} Aut(F) }$where the left morphism is a 1-epimorphism. By the previous comment, the lower right morphism is a 1-monomorphism. Hence we have an essentially unique diagonal lift
$\array{ U &\stackrel{\nabla}{\longrightarrow}& \ast \\ \downarrow && \downarrow \\ && \mathbf{B} Stab(\nabla) \\ \downarrow &\nearrow& \downarrow & \searrow \\ X &\longrightarrow& [F,A]/Aut(F) &\longrightarrow& \mathbf{B} Aut(F) }$$\Box$
[ continued in next comment ]
Now finally specifying $E$ to the frame bundle $Fr(X)$ formalized in differential cohesion in the sense discussed here, this is the pullback
$\array{ Fr(X) &\longrightarrow& X \\ \downarrow && \downarrow \\ X &\longrightarrow& \int_{inf} X }$We are really interested in a WZW term on $X$
$\mathbf{L}_{WZW}^{global} \colon X \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}$but by composition this gives a WZW term on the frame bundle
$\array{ Fr(X) &\longrightarrow& X &\stackrel{\mathbf{L}_{WZW}^{global}}{\longrightarrow}& \mathbf{B}^{p+1}\mathbb{G}_{conn} \\ \downarrow && \downarrow \\ X &\longrightarrow& \int_{inf} X }$Its value on the fiber of the frame bundle over a point $x \in X$ is just the restriction of $\mathbf{L}_{WZW}^{global}$ to the formal disk around that point. And if you imagine applying differential concretification to $[F,A]$ in all of the above discussion, then only this fiberwise contribution remains, i.e. we are looking at a global WZW term incarnated in its cohesive joint restriction to all formal disks at once (that might remind one of differential cohesion and idelic structure).
The above says that for this case the obstruction for such a global WZW term to exist is a lift of the structure group of the frame bundle of the given Cartan geometry $X$ to $Stab(\mathbf{L}_{WZW}^{formal})$.
I think when applied in the model of smooth supergeometric differential cohesion, then this yields the classical anomaly cancellation for all the Green-Schwarz super-$p$-branes of the completed brane scan of superstring/M-theory (something that does not seem to have been discussed before).
One step remains to be formalized: the respect of differential concretification for the $Aut(F)$-action. This I don’t see presently how to do “elementarily”, i.e. without invoking simplicial objects.
To recap, the idea is to connect a global WZW term $\mathbf{L}_{WZW}^{global} : X \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}$ to a lift of the structure group of $X$ by
pulling it back to the frame bundle $Fr(X) \to X$ whose fibers are the formal disks $\mathbb{D}$ inside $X$;
identifying the result with a section of $P \underset{Aut(\mathbb{D})}{\times} [\mathbb{D}, \mathbf{B}^{p+1}\mathbb{G}_{conn}]$;
applying differential concretification to this section to make it a section of $P \underset{Aut(\mathbb{D})}{\times} (\mathbf{B}^p \mathbb{G})Conn(\mathbb{D})$
and then using that this last section is essentially constant on a cover by $G/H$s, and so forth.
now this last step in the above list requires that differential concretification
$[\mathbb{D}, \mathbf{B}^{p+1}\mathbb{G}_{conn}] \longrightarrow (\mathbf{B}^p \mathbb{G})Conn(\mathbb{D})$has the structure of a homomorphism of $Aut(\mathbb{D})$-$\infty$-actions. I think this may be shown (at least if $\mathbb{D}$ itself is concrete, hm…) in terms of simplicial diagrams of simplicial sheaves by observing that the actions on the concretified stack are just the actions on it after embedding into the unconcretified stack and followed at the end by concretification again.
I still need to find a nicer formalization of this step…
Re #4,
that might remind one of differential cohesion and idelic structure,
how important is it that in arithmetic circumstances, there are generally different kinds of point in the same space?
Right, the present discussion revolves all around things looking locally the same. This is really the core of Caratan geometry, and here we are building on this by adding to the “rolling without sliding” of the homogeneous model geometry also a higher WZW cocycle being “rolled along”.
Also, even while we see appear here “functions” on the space of all formal disks in a given space, they conceptually play a very different role than in the Langlands context. Here we are looking at the target space geometry of a (higher) pre-quantum WZW model, while (higher) Langlands theory looks at (or: there are arguments that this is what it secretly does) the transgression of that to phase space.
Still, when writing the above it was hard not to notice that once more functions on the space of all formal disks inside a given space play a central role.
one remark for completeness:
we used to define the quantomorphism group $QuantMorph(\mathbf{L})$ of $\mathbf{L} : F \to \mathbf{B}\mathbb{G}_{conn}$ as the homotopy fiber of
$Aut(F) \stackrel{\mathbf{L}\circ(-)}{\longrightarrow} \mathbb{G} Conn(F)$over $\mathbf{L}$.
That this is equivalent to the above definition of $QuantMorph(\mathbf{L})$ as the looping of the 1-image of $\ast \stackrel{\vdash \mathbf{L}/Aut(F)}{\longrightarrow} \mathbb{G} Conn(F)/Aut(F)$, hence as $\Omega_{\mathbf{L}}(\mathbb{G} Conn(F)/Aut(F))$ follows by the Cartesian pasting diagram
$\array{ QuantMorph(\mathbf{L}) &\longrightarrow& Aut(F) &\longrightarrow& \ast \\ \downarrow && \downarrow^{\mathrlap{\mathbf{L}\circ(-)}} && \downarrow^{\mathrlap{\mathbf{L}}} \\ \ast &\stackrel{\vdash \mathbf{L}}{\longrightarrow}& \mathbb{G}Conn(F) &\longrightarrow& \mathbb{G}Conn(F)/Aut(F) \\ && \downarrow && \downarrow \\ && \ast &\longrightarrow& \mathbf{B} Aut(F) }$I have thought more about the best form of the general statement here. I am thinking now that my focus above on the infinitesimal frame bundle is not the way the problem should be stated. Rather, I think now the following is the right way to look at it all, and then it all comes together nicely:
Consider again a homomorphism of groups $H \to G$ and given a WZW term
$\mathbf{L}_{WZW} : G/H \to \mathbf{B}\mathbb{G}_{conn}$
“on a Klein geometry”.
We want to globalize this to a Cartan geometry.
I think (by what follows below) that we just need what might be called a “semi-Cartan geometry”, namely: a space $X$ equipped with a $G$-principal bundle $P\to X$ and equipped with a section $\sigma \colon X \to P \underset{G} {\times} (G/H)$ of the asociated Klein geometry bundle such that there is a trivializing cover $U \to X$ over which the induced maps $\sigma|_U : U \to G/H$ are formally étale.
(In the model of smooth $\infty$-groups and $H \to G$ an inclusion of Lie groups then an actual Cartan connection is this and more data. But for the present purpose only the above part of the data should matter. Also, this part has the advantage that its abstract formalization is “more robust” in that it does not require a bunch of technical conditions. So it may be good to give the above a name such as “semi-Cartan geometry”. )
[continued in next comment]
Now the obstruction problem to be considered is clear: we are asking for the obstruction to a global WZW term on that Klein geometry bundle
$\mathbf{L}_{WZW}^{global} : P \underset{G}{\times} (G/H) \longrightarrow \mathbf{B}\mathbb{G}_{conn}$
such that it reduces fiberwise to $\mathbf{L}_{WZW}$ and in fact locally constantly so, in that there is a trivializing cover $U \to X$ over which $\mathbf{L}_{WZW}^{global}$ restricts to the projection followed by $\mathbf{L}_{WZW}$:
$\mathbf{L}_{WZW}^{global}|_U \;:\; U \times (G/H) \stackrel{p_2}{\longrightarrow} G/H \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}\mathbb{G}_{conn} \,.$The obstruction to this existing will be given via the above arguments as the lift of the structure group through
$QuantMorph(\mathbf{L}_{WZW}) \to G \,.$
(And this now finally looks right in the applications, too, in the for the GS super $p$-brane sigma-models their obstructions this way are going to be lifts to higher String-like extensions of the relevant extended super-Poincaré group.)
Given this, then the actual WZW term on $X$ whose construction is the goal of the whole exercise is of course the pullback of this global term along the section $\sigma$
$\mathbf{L}_{WZW}^X \coloneqq \sigma^\ast \mathbf{L}_{WZW}^{global} \,.$
This is clearly the “right answer” for the intended applications in that it follows now that for a sigma-model with target space $X$ and WZW term $\mathbf{L}_{WZW}^X$, then whenever we look at the perturbation theory in that we consider fields $\phi : \Sigma \to X$ which perturb about the constant map onto some point $x$, then their action functional $\exp(\tfrac{i}{\hbar} \int_\Sigma [\Sigma, \mathbf{L}_{WZW}^X])$ reduces to that of the origianal local WZW model $\exp(\tfrac{i}{\hbar} \int_\Sigma [\Sigma, \mathbf{L}_{WZW}])$ on the Klein geometry.
Finally, also the discussion at parameterized WZW model will be a special case, in fact the degenerate special case where $H = 1$ and where we don’t even consider a section of $P \underset{G}{\times} (G/H) = P$.
Is there something important then in Cartan’s approach that allows these constructions above which is absent in the more general Ehresmann approach?
Here I am not sure what you mean by “Cartan’s approach” and “Ehresmann’s approach”.
Historically, Cartan wrote a semi-rigorous text Cartan 23 and then Ehresmann gave a formalization in Ehresmann 50 and invented the terminology “Cartan connection” for it. Later other people in turn called what he had used as formalization “Ehresmann connections”, one model for principal connection. A Cartan connection is a principal connection (realized as an Ehresmann connection if desired) euipped with a reduction of its structure group and with a certain compatibility condition, the “Cartan condition”, with the tangent space of the base manifold.
Above I am using part of that definition: the reduction of the structure group of just a principal bundle and a weaker form of the resulting Cartan condition.
I guess I meant then the contrast between Cartan connections and the more general principal connections. But you tell me you’re weakening the former. So maybe I’m wondering now what is it in “the reduction of the structure group of just a principal bundle and a weaker form of the resulting Cartan condition” that’s doing the useful work beyond mere principal connections.
I seem to remember the idea going about that we’d lost something by subsuming Cartan’s ideas within principal bundle theory. Maybe John Baez expressed that idea some time ago. Ah yes
Cartan geometry has been sadly neglected after his student Ehresmann came up with the more general but more abstract approach to connections on principal bundles that we use today. From what I gather, Cartan wanted Ehresmann to formalize the notion of Cartan connection, and Ehresmann did, but then he went further…
So, it’s a pleasant revenge that to understand MacDowell and Mansouri’s approach to gravity, you really need Cartan connections — as the picture in my blog entry hints!
Are you then saying that for certain purposes a weaker Cartan condition is helpful?
Added: I see you also talked about Ehresmann and Cartan back then. Do you look back at old writings and find that you’ve wholly surpassed them, or do they sometimes contain surprises?
I wouldn’t say that principal connections are more general than Cartan connections, as the concepts don’t subsume each other. Instead a Cartan connection is a principal connection ($\simeq$ Ehresmann connection) equipped with extra structure (reduction of the structure group) and equipped with an extra property (compatibility with the tangent bundle).
Also, I keep going around disagreeing with the claim that Cartan connections have been neglected. Physicists use it all the time, they just don’t use the exact words “Cartan connection”, instead they say “vielbein with spin connection in Cartan formalism” or something like this. As soon as physicists consider spinors in gravity and in particular when they consider supergravity, then this is their default way of proceeding. Open any textbook on general relativity and the latest when fermions are coupled to gravity does the author also discuss Cartan theory. So certainly physicists know “Cartan formalism”. On the other hand I have never ever seen a physicist say “Ehresmann connection”! And rarely “principal connection”, for that matter. When I was a student I was taught Cartan frames in GR and nothing else. None of my professors would have known what a principal connection is, not what an Ehresmann connection is, not even what a principal bundle is. They would proceed pretty much exactly as Cartan did back then in 1923.
So, saying that Cartan connections have been neglected as compared to Ehresmann connections seems the wrong way around to me, for the above reasons. Moreover, I would suggest to distinguish between the way the space of fields is formalized (via metric tensors or Cartan connections or something else) and which exact Lagrangian is used on this space of fields, these are two somewhat independent issues. While those MacDowell-Mansouri Lagrangians are not widely considered in physics, the formulation of gravity in terms of Cartan vielbeins and “spin connection” is entirely standard. The textbook Supergravity and Superstrings - A Geometric Perspective is all written in terms of Cartan theory, for instance.
But of course if there is disagreement on this historical point then that’s not a disaster and we may just proceed, I will not have a fight over this.
Finally on your technical question:
Are you then saying that for certain purposes a weaker Cartan condition is helpful?
Yes, that’s what I said above. One may consider just reduction of a structure group along $H \to G$ without any connections on the bundles, and that already gives a way to coherently identify each tangent space $T_x X$ with $T_{e H} (G/H) = \mathfrak{g}/\mathfrak{h}$. I suppose in that picture of model spaces rolling along, this would be “rolling with possible sliding”. Adding the connection allows to enforce the “no sliding”-part. But it may be useful to consider both separately.
I have written a section pre-Cartan geometry.
Thanks. “Rolling with possible sliding” and “rolling without sliding” certainly helps the intuition.
Here is an exposition of the story as I see it so far: pdf (just the story, no details yet).
Re #14, the authors of ’Parabolic Geometries’, Andreas Cap and Jan Slovak have an introductory article (ps) (probably the beginning of their book in fact), where they write:
For structures like Riemannian metrics, the point of view of Cartan connections, while conceptually nice, is not really necessary, since due to the simplicity of the situation, the normalized Cartan connection is essentially equivalent to a principal connection which corresponds to the Levi-Civita connection on the tangent bundle. Probably because of this reason and the enormous success of principal connections in the subsequent period, Cartan’s ideas did not receive as much attention as they deserve in our opinion. Moreover, if one studies more complicated structures then the point of view of Cartan connections is really a strong simplification.
Maybe the difference is a mathematics/physics matter. Anyway, far more important is understanding what can be done now. I still haven’t got a clear sense of what’s so special about parabolic subgroups. It seems to be that they make the quotient into a decent space, a generalized flag manifold. In Two constructions with parabolic geometries Cap writes
The most interesting examples of Cartan geometries are those, in which the Cartan geometry is equivalent to some simpler underlying structure. Obtaining the Cartan geometry from the underlying structure usually is a highly nontrivial process which often involves prolongation. Cartan himself found many examples of this situation, ranging from conformal and projective structures via 3–dimensional CR structures to generic rank two distributions in manifolds of dimension five.
Parabolic geometries are Cartan geometries of type $(G, P)$, where $G$ is a semisimple Lie group and $P \subset G$ is a parabolic subgroup. The corresponding homogeneous spaces $G/P$ are the so–called generalized flag manifolds which are among the most important examples of homogeneous spaces. Under the conditions of regularity and normality, parabolic geometries always are equivalent to underlying structures. This basically goes back to the pioneering works of N. Tanaka, see e.g. [29].
In section 2 of this article we give a precise description of the underlying structures which are equivalent to regular normal parabolic geometries. In this underlying picture, the structures are very diverse, including in particular the four examples of structures listed above. From that point of view, parabolic geometries offer a unified approach to a broad variety of geometric structures.
Thanks for the quote. I haven’t seen the article yet. I find no file behind your link (even after adding the missing “http://” which confuses the forum software) and it does not seem to appear in the book on Parabolic geometries as far as I see. Could you maybe check?
By the way, the Levi-Civita connection on the tangent bundle is of course in wide use in physics, but not the underlying principal connection. Not alone at least. Passing from the tangent bundle to the frame bundle is essentially the introduction of the vielbein, and that together with adding the principal connection on the frame bundle is the corresponding Cartan connection. This is physicist’s $\Gamma \mapsto (e,\omega)$.
Maybe you remember those lecture notes Zanelli 05 which we once talked about? A math-oriented physicist speaking about gravity. Cartan is mentioned all over the place. On p. 12 “Here we adopt Cartan’s point of view.” Ehresmann is never mentioned. Principal connections are mentioned but once in a footnote.
Odd. I can see it there first if I google
gs2 slovak parabolic
But now it doesn’t seem to want to download. And I can only see his homepage cached.
http://webcache.googleusercontent.com/search?q=cache:56jbh0oV6NMJ:www.math.muni.cz/~slovak/index.eng.html+&cd=2&hl=en&ct=clnk&gl=uk
I’m sure you’re right about the relative attention paid to Cartan. Is there maybe a difference between physicists and mathematicians (where does that leave mathematical physicists)?
Yes, there is certainly an enormous difference here between physicists and mathematicians. It is of course true that mathematicians who do not relate to Riemannian geoemtry etc. do not speak about Cartan connections as much as about principal connections. One sees this reflected in the curious MO discussion “What is torsion” which would have a super-simple answer in terms of Cartan connections. The mathematical physicist José Figueroa-O’Farrill gets in this direction in his reply, and then Mathieu Anel is the one to go back and read out Cartan.
But I was was concentrating on physicists, because the quote you gave in #13 is about application of Cartan geometry in gravity, and it seems to me that is precisely the realm where Cartan needs no extra advertisement.
I have now written out what I (presently) think is the full proof of the central obstruction theorem:
Theorem Given a map of $\infty$-stacks $\phi : F \to A$ and given an $F$-fiber $\infty$-bundle $E \to X$, then there exists an extension of $\phi$ to $E$ (in the form of a map $E \to A$ that locally constantly restricts to $\phi$ on the fibers) precisely if the structure $\infty$-group of $E$ lifts to the stabilizer $\infty$-group of $\phi$ under the canonical $Aut(F)$-$\infty$-action on $[F,A]$.
(Recall that the idea is that taking here $F$ the Lie integration of an extended super-Minkowski spacetime and $\phi$ a higher WZW term, then this gives the obstruction theory to globalizing the $p$-brane sigma models to curved extended super-spacetime. In this case the stabilizer $\infty$-groups here are, after differential concretification, the higher quantomorphism/Heisenberg groups of the WZW term, which we know reduce for the ordinary WZW term to the string 2-group. Hence the above obstruction theorem gives a generalization of Green-Schwarz anomaly cancellation, I think.)
This, and what I think is its proof, is currently in theorem 3.6.270 on p. 342 in dcct pdf.
I think I have now also full proof of one direction of the globalization obstruction (details in section 3.2 of this pdf):
Theorem Given $V$ a differentially cohesive $\infty$-group, $X$ a $V$-manifold, and $\mathbf{L}_{WZW}$ an equivariant WZW-term on $V$, then an obstruction to a globalization of the WZW term over $X$ to exist is the existence of an integrable $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V})$-structure on $X$,
(i.e. a lift of the structure group of the frame bundle to the quantomorphism n-group of the restriction $\mathbf{L}_{WZW}^{\mathbb{D}^V_e}$ of the WZW term to the infinitesimal neighbourhood of the neutral element in $V$, such that this lift restricts over a $V$-cover $U$ to the canonical $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V_e})$-structure on $V$).
I still need to prove that this is not just a necessary but also a sufficient condition. This is harder…
Finally coming back to this. I realize the following is an evident open question that I haven’t sorted out:
given a local model space $V$ (a group object) and a WZW term $\mathbf{L}_{WZW}^V \colon V \longrightarrow \mathbf{B}^n \mathbb{G}_{conn}$ and an $V$-manifold $X$ and a definite globalization $\mathbf{L}_{WZW}^X$ of $\mathbf{L}_{WZW}^V$ over $X$
$\array{ && \flat^{rel} U \\ & \swarrow && \searrow \\ V && \swArrow && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}^V}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}^n \mathbb{G}_{conn} }$then consider forgetting the connections, regarding the total spaces $P^V\to V$ and $P^X \to X$ of the underlying $n$-bundles.
Question: Is it true that $P^X$ is a $P^V$-manifold?
In parts this is immediate. Take the above diagram after forgetting ${}_{conn}$, then regard the map $\mathbf{B}^n \mathbb{G} \longleftarrow \ast$ and then base change the whole diagram along this map. This gives a correspondence of the form
$\array{ && \tilde U \\ & \swarrow && \searrow \\ P^V && && P^X }$and by pullback stability both maps are local diffeos and the right one is 1-epi.
But is $\tilde U$ of the form $\flat^{rel} (something)$?
Need to rush off now…
I need to think out loud for a bit more here.
The point that I am still not fully decided on is what the right way is to define Cartan geometry for differentially extended groups.
So the issue is that given a group cocycle $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+1} \mathbb{G}$ on a higher cohesive group $G$, then its WZW term, if it exists, is in general not defined on $G$, but on $\tilde G$, given by the homotopy fiber product
$\array{ && \tilde G \\ & \swarrow && \searrow^{\mathrlap{\theta_{\tilde G}}} \\ G && && \Omega^1_{flat}(-,G) \\ & {}_{\mathllap{\theta_G}}\searrow && \swarrow \\ && \flat_{dR}\mathbf{B}G }$(With that and given a compatible map $\Omega^1_{flat}(-,G) \to \Omega^{p+2}_{cl}(-,\mathbb{G})$ of the global curvature form data, then the WZW term is the map $\mathbf{L}_{WZW} : \tilde G \to \mathbf{B}^{p+1}\mathbb{G}_{conn}$ induced by the map from this fiber product diagram into the defining one for differential cohomology. The fact that $\tilde G$ in general has a differential component to it (so that maps $\Sigma \to \tilde G$ are compatible pairs of maps $\Sigma \to G$ and of differential form data on $\Sigma$) may seem unexpected but is exactly right in that it gives the “tensor multiplet” fields for the higher WZW models. Anyway.)
So the general theory now allows me to speak of Cartan geometries $\tilde X$ locally modeled on $\tilde G$ and hence of WZW terms on $\tilde X$ that locally restrict to $\mathbf{L}_{WZW}$ (“definite globalizations”).
But there must be some more compatibility conditions which I suppose one should impose. About this I am not sure yet. Because $\tilde X$ would be the right target for the globalized WZW model, but for purposes of computing for instance isometries, we will need to specify the underlying actual spacetime $X$ which shouldn’t be locally modeled on $\tilde G$, but on $G$.
So I suppose I should consider bundles of Cartan geometries
$\array{ \tilde X \\ \downarrow \\ X }$where $X$ is locally modeled on $G$ and such that on infinitesimal disks this bundle restricts to $\tilde G \to G$
$\array{ U \times \mathbb{D}^{\tilde G} &\longrightarrow& T_{inf} \tilde X &\stackrel{ev}{\longrightarrow}& \tilde X \\ \downarrow && \downarrow && \downarrow \\ U \times \mathbb{D}^{G} &\longrightarrow& T_{inf} X &\stackrel{ev}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{p}} \\ U &\stackrel{}{\longrightarrow}& X }$This connects to the point that I was going on about elsewhere: I suppose I want to require that the $\mathbf{Aut}(\mathbb{D}^{G})$-structure of $X$ reduces to an $\mathbf{Aut}_{Grp}(\mathbb{D}^{G}) = \mathbf{Aut}^{\ast/}(\mathbf{B}\mathbb{D}^{G})$-structure (i.e. that the tangent bundle becomes a bundle of Lie algebras, instead of just of vector spaces underlying Lie algebras). Because with this and assuming/requesting that $\theta_{\tilde G}$ is suitably left-invariant then I suppose we have association $T_{inf} \tilde X \simeq (T_{inf} X) \underset{\mathbb{D}^G}{\times} \mathbb{D}^{\tilde G}$. Or something like this, I am still a bit stuck here.
Ah, I think I’ve got it.
In addition to the requirement that the group cocycle $\mathbf{c}$ is equipped with a compatible refinement of the Hodge filtration to yield
$\array{ \Omega^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \Omega^{p+2}_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{G} }$we should make this also compatible with the homotopy fiber of the cocycle, hence with the extension $\hat G \to G$ that the cocycle classifies, by demanding there to be
$\array{ \Omega^{p+1,p+2}_{cl}(-,\mathbb{G}) &\longrightarrow& \ast &\longleftarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \Omega^{p+2}_{cl}(-,\mathbb{G}) &\longrightarrow& \flat_{dR}\mathbf{B}^{p+2}\mathbb{G} &\longleftarrow& \mathbf{B}^{p+1}\mathbb{G} }$such that the pullback of the left vertical morphism along $\mu$ is $\Omega^1_{flat}(-,\hat G)$.
All this is structure that is naturally available when the cocycle $\mu$ arises from integration of an $L_\infty$-cocycle $\mathfrak{g} \longrightarrow b^p+1 \mathbb{R}$, for then we may take $\Omega^1_{flat}(-,G) = Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^\bullet(-) )$ and $\Omega^{p+2}_{cl}(-,\mathbb{G}) = Hom_{dgAlg}( \wedge^\bullet (\mathbb{R}[p+1]), \Omega^\bullet(-) )$ and $\Omega^{p+1,p+2}_{cl}(-,\mathbb{G}) = Hom_{dgAlg}( \wedge^\bullet (\mathbb{R}[p]\to\mathbb{R}[p+1]), \Omega^\bullet(-) )$.
But then we have that for $\mathbf{L}_{WZW}^X : X \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}$ a definite globalization of the WZW term for $\mathbf{c}$ over a $G$-Cartan geometry $X$, then the homotopy pullback of that along the induced map $\Omega^{p+1,p+2}_{cl}(-,\mathbb{G}) \to \mathbf{B}^{p+1}\mathbb{G}_{conn}$ is a space $\hat X$ which is locally modeled on $\tilde {\hat G}$, as it should be.
What’s $\mathbb{H}$ here? I’m also finding your second diagram a little confusing. Is this just a vanilla diagram, or cartesian squares or what? What’s $\Omega^{p+1,p+2}_{cl}$?
As you know, I’m fond of taking the everyday formulated in type theory and then see what happens in general contexts, so was interested to see $G$-principal infinity-bundles as a form of “Some B are E” in Science of Logic:
$\array{ P &\longrightarrow& E \\ \downarrow && \downarrow \\ B &\stackrel{c}{\longrightarrow} & \mathbf{B}G }$Is there any merit in seeing the first diagram of #26 that way “Some $G$ are $\Omega^1_{flat}(-,G)$?
Then the morphism $\Omega^1_{flat}(-,G) \to \Omega^{p+2}_{cl}(-,\mathbb{G})$ is “All $\Omega^1_{flat}(-,G)$ are $\Omega^{p+2}_{cl}(-,\mathbb{G})$”.
DavidR, thanks for taking note, and sorry for the silly typo. (I was on a train and in a rush, as usual. Am longing to find a place to rest.) The $\mathbb{H}$s were meant to be $\mathbb{G}$s.
The diagrams are indeed just any old diagrams, the condition is only that the fiber of the leftmost morphism is $\Omega^1_{flat}(-,\hat G)$. The object $\Omega^{p+1,p+2}_{cl}(-,\mathbb{G})$ is to be thought of as the 1-sheaf of closed $p+2$-forms $\omega$ equipped with trivializing potentials $\alpha$, i.e. $\mathbf{d}\alpha = \omega$. On the one hand that gives indeed the comutativty of those diagrams, on the other it gives indeed the right pullback.
Sorry if all this seems opaque, it’s really rather straightforward: the WZW term is a differential refinement of the group cocycle, and I am just identifying the data needed to make this compatible with passing to the extension classified by the group cocycle. If you still have the wish and energy to follow this, you may find a more pretty-printed account in section 5.2.16 of dcct, currently pages 404-405 of dcct pdf.
DavidC,
yes, I think indeed it works out this way, one just needs to make sure to carry the “constructivism” around, i.e. to remember that the object interpreting “some $B$ are $E$” is really the space of all witnesses of a $B$ being equal to an $E$, via an chosen equivalence.
If this is done, then indeed any principal bundle, being the space of local trivializations of the cocycle that classifies it, is given “some points in the base are equivalent to the abstract point, when regarded in the context $\mathbf{B}G$”.
The analogous statement is true for differential cohomology, which is the homotopy fiber product
$\array{ && \mathbf{B} \mathbb{G}_{conn} \\ & \swarrow && \searrow \\ \mathbf{B} \mathbb{G} && && \Omega^{2}_{cl}(-,\mathbb{G}) \\ & \searrow && \swarrow \\ && \flat_{dR}\mathbf{B}^2 \mathbb{G} }$and hence is the realization of “some $\mathbb{G}$-bundles are closed forms, when regarded in the context of de Rham cohomolgy, via the Chern character”. Namely the choice of equivalence making this true is precisely the choice of connection/differential cocycle.
These statements may seem strange when viewed through the lense of of traditional theory. But being serious about “$\infty$-logic” means realizing that these are entirely valid statements, and possibly there is something to be gained by adopting the perspective which they suggest.
So to explain more of what I am after.
The goal is to start with the input of a “brane bouquet” in the form of an iterative tower of $L_\infty$-cocycles and extensions, and to “fully integrate” this to a story where to each $L_\infty$-algebra we assign a class of super-Cartan geometies locally modeled on it, to each cocycle a concept of definite globalizations of WZW terms on these spacetimes and to each extension a corresponding extension of these super-spacetimes.
So a stage in the “brane bouqet” is a diagram of super $L_\infty$-algebras of the form
$\array{ \mathfrak{g}_2 &\stackrel{\mu_2}{\longrightarrow}& b^{p_2 +1} \mathbb{R} \\ \downarrow \\ \mathfrak{g}_1 &\stackrel{\mu_1}{\longrightarrow}& b^{p_1 + 1} \mathbb{R} }$where $\mathfrak{g}_2 \to \mathfrak{g}_1$ is the homotopy fiber of $\mu_1$.
Now we may fix an actual representative of $\mathfrak{g}_1$ such that $\mathfrak{g}_2$ is the actual 1-pullback
$\array{ \mathfrak{g}_2 &\longrightarrow& (b^{p_1} \mathbb{R}\to b^{p_1+1}\mathbb{R}) \\ \downarrow && \downarrow \\ \mathfrak{g}_1 &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1} \mathbb{R} }$and write $\Omega^1_{flat}(-,\mathfrak{g}) \coloneqq Hom_{dgAlg}(CE(\mathfrak{g}),\Omega^\bullet(-))$ etc. for the 1-sheaves of flat differential forms with coefficients in $\mathfrak{g}$.
Then first of all the $L_\infty$-integration process by which
$\mathbf{B}G \coloneqq \tau_{p+1}\exp(\mathfrak{g})$and
$\mathbf{c}_1 = \exp(\mu_1) \colon \mathbf{B}G \longrightarrow \mathbf{B}^{p_1+2} (\mathbb{R}/\Gamma)$etc. gives that these choices of 1-sheaves of flat forms are compatible with the abstract de Rham image of the cocycle in that we have the commuting diagrams
$\array{ \Omega^1_{flat}(-,\mathfrak{g}) &\longrightarrow& \Omega^{p+2}_{cl} \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}_1}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p_1+2} (\mathbb{R}/\Gamma) } \,.$Forming the homotopy pullback of this along the naturality square of the $\flat_{dR}$-unit gives the morphism
$\mathbf{L}_{WZW_1} \;\colon\; \tilde G_1 \longrightarrow \mathbf{B}^{p_1+1} (\mathbb{R}/\Gamma)_{conn} \,.$With this we may now ask for manifolds $X_1$ locally modeled on $\tilde G_1$ and carrying a map $\mathbf{L}_{WZW_1}^{X_1} \colon X_1 \to \mathbf{B}^{p_1+1} (\mathbb{R}/\Gamma)_{conn}$ such that restricted to infinitesimal neighbourhoods it is equivalent, suitably coherently to $\mathbf{L}_{WZW_1}$.
That’s what I had studied so far. But now I want to answer the question: suppose we went through the above both for both $\mu_1$ and $\mu_2$. Then we want to combine this by having a $\mathbf{L}_{WZW_2}$ defined not on any old $\tilde G_2$-manifold $X_2$, but on one that is to an $X_1$ as $\mathfrak{g}_2$ is to $\mathfrak{g}_1$.
The naive idea would be to take $X_2$ to be the homotopy fiber of $\mathbf{L}^{X_1}_{WZW_1}$. But as always with differential coefficients, if one takes the naive homotopy fiber of a differential cocycle one gets just the flat version of the proper differential refinement of the homotopy fiber of the underlying cocycle.
So here then we fix this by adding in the required differential twist. Instead of pulling back $\{0\} \to \mathbf{B}^{p_1+1} (\mathbb{R}/\Gamma)$ we pull back $\Omega(-,b^{p_1}\mathbb{R} \to b^{p_1+1}\mathbb{R}) \to \mathbf{B}^{p_1+1} (\mathbb{R}/\Gamma)$. Crucially, these forms we are including here are nontrivial when regarded as a 1-sheaf, but become trivial after included into differential cohomology (since by construction they are exact). Reflection on the diagrams in #27 shows that this is precisely the way it needs to be for the homotopy pullback of this inclusion along $\mathbf{L}_{WZW_1}^{X_1}$ to be $\tilde G_2$.
Hence then in conclusion we end up with systems of higher Cartan geometries of the form
$\array{ X_2 &\stackrel{\mathbf{L}_{WZW_2}^{X_2}}{\longrightarrow}& \mathbf{B}^{p_2+1} (\mathbb{R}/\Gamma_2)_{conn} \\ \downarrow \\ X_1 &\stackrel{\mathbf{L}_{WZW_1}^{X_1}}{\longrightarrow}& \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} }$which are indeed globalizations of integrations of the $L_\infty$-systems that we started out with.
Well, I need to write out proof now that the $X_2$ being given as the homotopy fiber product of $\mathbf{L}_{WZW_1}^{X_1}$ with $\Omega(-,b^{p_1}\mathbb{R} \to b^{p_1+1}\mathbb{R}) \to \mathbf{B}^{p_1+1} (\mathbb{R}/\Gamma)_{conn}$ is really a $\tilde G_2$-manifold. But that should be straightforward now.
I have now written that out more readably (I hope) at geometry of physics – WZW-terms - Consecutive WZW terms and twists.
Have produced some talk slides for my talk at the AMS-EMS-SPM meeting 2015 this week.
On the slide titled ’Higher WZW terms’ there is “SmothMfd”.
On the ’Thank you’ slide there is “Course notes at…” Did you mean for this to carry over?
On the slide titled ’Higher WZW terms’ there is “SmothMfd”.
Thanks! Fixed now.
On the ’Thank you’ slide there is “Course notes at…” Did you mean for this to carry over?
Yes, this was intended. I thought in closing I offer a pointer to more details. Does it seem unsuitable for some reason? Then I should change it.But right now I don’t see what’s wrong with it? [edit: have made it more explicit now by adding “For more details see course notes at…”]
Regarding the edit, yes I think that makes more sense. The old version made it look like the slides were part of a course you were giving.
parameteriaztion (slide 11); vaccum (slide 15)
Thanks, DavidR and DavidC, now fixed. Thanks for your help, I really appreciate it.
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