For $X$ a manifold of dimension $n$, there is a canonical map $\tau_X \colon X \longrightarrow \mathbf{B}\mathrm{GL}(n)$ to the moduli stack of the smooth general linear group, modulating the bundle to which the tangent bundle is associated.

I would like to formulate this construction axiomatically in differential cohesion.

It is clear to me how the formalization ought to proceed, but there are some gaps where I know what to do “informally”, but not yet how to do it fully axiomatically.

Here is how I imagine to start:

First, we fix some types that are to play the role of the local model spaces. One option is to ask for a type $\mathbb{A}^1$ that exhibits the cohesion in that the shape modality is $\mathbb{A}^1$-localization. Then take the $n$-dimensional model space to be $\mathbb{A}^n \coloneqq (\mathbb{A}^1)^{\times_n}$.

Then an $n$-dimensional manifold is a type $X$ such that there exists a morphism (the atlas)

$\coprod_i \mathbb{A}^n \longrightarrow X$which is 1) a 1-epimorphism and 2) is formally étale in that it is modal for the infinitsimal shape modality $\Pi_{inf}$.

Next we need to axiomatize the group $\mathrm{GL}(n)$. In the standard model of differential cohesion this group is the following:

First, the formal disk $D^n$ around the origin in $\mathbb{A}^n$ is the homotopy fiber of the unit $\mathbb{A}^n \longrightarrow \Pi_{inf}(\mathbb{A}^n)$ of the infinitesimal shape modality

$D^n \coloneqq \mathrm{fib}(\mathbb{A}^n \longrightarrow \mathbb{A}^n)$Then, $\mathrm{GL}(n)$ is just the automorphism group of that:

$\mathrm{GL}(n) \coloneqq \mathbf{Aut}(D^n) \,.$To produce the canonical morphism

$X \longrightarrow \mathbf{B} \mathrm{GL}(n)$we should proceed by producing a canonical morphism from the Cech nerve of $\coprod\mathbb{A}^n\to X$ to the Cech nerve of $\ast \to \mathbf{B}\mathrm{GL}(n)$, hence a map of simplicial types that starts out looking like this:

$\array{ \vdots && \vdots \\ \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ (\coprod_i \mathbb{A}^n) \underset{X}{\times} (\coprod_i \mathbb{A}^n) &\stackrel{(\tau_X)_1}{\longrightarrow}& \mathrm{GL}(n) \\ \downarrow\downarrow && \downarrow\downarrow \\ \coprod_i \mathbb{A}^n &\longrightarrow& \ast }$This is where I am presently stuck. I know how to proceed rigorously, but not how to proceed fully formally, if you see what I mean.

Rigorously, what I am supposed to say is that $(\tau_X)_1$ is the map which to a point

$\hat x \;\colon\; \ast \longrightarrow (\coprod_i \mathbb{A}^n) \underset{X}{\times} (\coprod_i \mathbb{A}^n)$assigns the element in $\mathrm{GL}(n)$ obtained by first forming the formal disk $D^n_{\hat X}$ around that point, as above, and then using the induced diagram

$\array{ && D^n_{\hat x} \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ \mathbb{A}^n && && \mathbb{A}^n \\ & \searrow && \swarrow \\ && X }$together with the fact that all maps here are formally étale to form something that one would denote

$(i_1^{-1}) \circ i_2 \in \mathbf{Aut}(D^n_{\hat x}) \simeq \mathrm{GL}(n) \,.$where “$i_1^{-1}$” means the inverse of the corestriction of $p_1$ onto its image, and where some kind of rigid translation $\mathbb{A}^n \stackrel{\simeq}{\longrightarrow} \mathbb{A}^n$ making the base points coincide is implicit.

But here I am a little stuck with the formalization. How would one formalize this “obvious” construction of $(\tau_X)_\bullet$?

]]>I’d like to formalize in cohesive homotopy theory the Penrose-Ward transform for higher bundles as it appears for 3-bundles in section 4 of

- Christian Saemann, Martin Wolf,
*Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space*(arXiv:1305.4870)

I think I know in principle how it goes. But at one step I need a *fiberwise* $\flat$-modality. This is something that Mike has kept asking about recently and I didn’t give an answer. I still don’t have the answer, but at least now I have a good motivating example.

So we consider a correspondence

$\array{ && Z \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 }$Let $G$ be an $\infty$-group and consider a $G$-principal $\infty$-bundle $P \to X_1$ equipped with a trivialization of its pullback along $p_1$.

The transform in question is supposed to do the following: from the chosen trivialization on $Z$ we are to produce a $p_1$-fiberwise $G$-valued function on $Z$. Associated with this is a $p_1$-fiberwise Maurer-Cartan form on $Z$ which glues to a global form on $Z$ and this we are to push down along $p_2$. (This is an abstract rephrasing of what appears in components in section 4 of the above article.)

Here I’ll ignore the push-forward for the moment and consider only the first step of producing that fiberwise Maurer-Cartan form.

On a single fiber $x \colon \ast \to X$ it works like this: let $g$ be the map that modulates $P \to X_1$. Then the assumed trivialization of the pullback of this $G$-bundle to $Z$ gives the diagram

$\array{ p_1^{-1}(x) &\longrightarrow& Z &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{x}{\longrightarrow}& X_1 &\stackrel{g}{\longrightarrow}& \mathbf{B}G }$and, by looping, this induces a map $p_1^{-1}(x) \longrightarrow G$. This is the $G$-valued function on the fiber. Its fiber MC form is given by the composite

$p_1^{-1}(x) \longrightarrow G \longrightarrow \flat_{dR} \mathbf{B}G \,.$Now we need to generalize this to a construction that does this for all fibers at once.

Need to think about that.

]]>This here is to vent some thoughts on how to formalize theta functions and, if possible, bundles of conformal blocks, in cohesive homotopy theory. It is related to the note *Local prequantum field theory (schreiber)* and, just as that note, is based on discussion with Domenico Fiorenza.

One basic idea here is that

theta functions are transgressions of Chern-Simons-type functionals to codimension 1.

along the lines of example 3.2.17 in

*Local prequantum field theory (schreiber)*that transgression is universally provided by the cobordism hypothesis for coefficients being the $n$-category of $n$-fold correspondences in the slice of the given cohesive $\infty$-topos over the “$n$-group of phases”.

More concretely, let $\mathbf{Fields} \in \mathbf{H}$ be any cohesive homotopy type, let $\mathbb{G}$ a cohesive abelian $\infty$-group object, then a “Chern-Simons $n$-bundle” is a map

$\exp(\tfrac{i}{\hbar} S_{CS}) \colon \mathbf{Fields}\longrightarrow \mathbf{B}^n \mathbb{G} \,.$Regard this as an object in the slice

$\exp(\tfrac{i}{\hbar} S_{CS}) \in \mathbf{H}_{/\mathbf{B}^n \mathbb{G}} \,.$Consider the $(\infty,n)$-category of $n$-fold correspondences in this slice

$\mathcal{C}\coloneqq Corr_n(\mathbf{H}_{/\mathbf{B}^n \mathbb{G}}) \,.$Every object here is supposed to be fully dualizable, hence $\exp(\tfrac{i}{\hbar} S_{CS})$ defines the local action functional of a local prequantum field theory of dimension $n$

$\exp(\tfrac{i}{\hbar} S_{CS}) \colon Bord_n^{fr} \longrightarrow Corr_n(\mathbf{H}_{/\mathbf{B}^n \mathbb{G}}) \,.$The claim is that to a closed $n$-framed $(n-1)$-manifold $\Sigma_{n-1}$ this monoidal $n$-functor assigns a map

$[\Pi(\Sigma), \mathbf{Fields}] \longrightarrow \mathbf{B}\mathbb{G}$regarded as an $(n-1)$-fold homotopy between trivial homotopies between the 0-map $\ast \to \mathbf{B}^n \mathbb{G}$.

This should be the theta bundle. Here $[\Pi(\Sigma),\mathbf{Fields}]$ is the mapping stack from the fundamental $\infty$-groupoid $\Pi(\Sigma)$ to $\mathbf{Fields}$, hence is the moduli stack of flat $\mathbf{Fields}$-valued $\infty$-connections on $\Sigma$, hence is the covariant phase space of the Chern-Simons theory. The “theta bundle” is equivalently the prequantum line bundle of the CS-theory on $\Sigma$.

One question to be thought about is this:

to turn $\exp(\tfrac{i}{\hbar} S_{CS})$ into a local prequantum field theory on cobordisms which are not framed but are equipped with $(G\to O(n))$-structure it needs to be equipped with the structure of a $G$-homotopy fixed point in $Core(\mathcal{C})$. What is this structure more explicitly?

]]>started a table-for-inclusion *arithmetic cohesion – table* and included it into relevant entries.

In the course of this I started a minimum at *adic residual*.