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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?
Next, by the discussion at “Quantization via Linear homotopy types” we are to choose some linearization in the form of
$\mathbf{B}\mathbb{G} \longrightarrow \mathbf{B}GL_1(E) \,.$This here just to remark that this subsumes the approach of “all multiplicative cohomology theory at once” in Lurie’s A Survey of Elliptic Cohomology:
in the $E_\infty$-arithmetic $\infty$-topos there is a canonical spectrum object
$N \colon (A\in E_\infty Ring) \mapsto (underlying(A)\in Spectra)$and $GL_1(N) \simeq \mathbb{G}_m$. Thus the theta line bundle
$\mathcal{M}_G = [\Pi(\Sigma), \mathbf{B}G] \longrightarrow \mathbf{B}\mathbb{G}_m$in this case is the morphism of spectral stacks which to each $E_\infty$-ring $A$ assigns the $A$-line bundle
$\mathcal{M}_G(A) \longrightarrow \mathbf{B}GL_1(E)$as in the Survey (or in between the lines there).
In view of this, the main gap that still prevents me from seeing the full “2-equivariant” story in detail is that I still don’t know what, if anything, plays the role of the algebraic incarnation of the Chern-Simons 3-bundle
$\mathbf{B}G \longrightarrow \mathbf{B}^3\mathbb{G}_m$for reductive algebraic groups $G$.
Thinking a bit more, I am coming to the conclusion that refining the Chern-Simons 3-bundle to complex or arithmetic geometry is not actually the way to go. Instead, the polarization structure which that is supposed to induce is already all in the framed cobordism category.
Here is the story as I see it now:
Start with the smooth Chern-Simons 3-bundle $\mathbf{B}G \to \mathbf{B}^3 U(1)$. Regarded as a fully dualizable object in $Corr_n(\mathbf{H}_{/\mathbf{B}^3 U(1)})$ this induces a local prequantum field theory of the form
$Bord_n^{framed} \longrightarrow Corr_n(\mathbf{H}_{/\mathbf{B}^ 3 U(1)})$.
The value of this on the diffeomorphism type of a closed 2-dimensional manifold $\Sigma$ is the theta line bundle
$Loc_G(\Sigma) = [\Pi(\Sigma), \mathbf{B}G] \longrightarrow \mathbf{B} U(1)$.
By cohomological quantization we are to choose any multiplicative cohomology theory $E$ twisted by $\mathbf{B}U(1)$ and push that theta bundle in $E$-cohomology to the point to get the space of quantum states. The typical choice if $E = K U$ thought of as twisted via $\mathbf{B}U(1) \to B U(1) \to GL_1(K U)$. To push we need an $E$-orientation.
But in fact the above 3d TFT is defined on 3-framed cobordisms, so we are not in fact supposed to assign just one $E$-space of quantum states, but one to each framing of $\Sigma$, natural in the moduli space of framed surfaces of type $\Sigma$.
That works: at least for $G = T$ a torus (abelian Chern-Simons) then a choice of framing of $\Sigma$ naturally induces a framing of $Loc_G(\Sigma)$ and hence gives a trivialization of the J-homomorphism there, and this naturally gives an orientation in $E$-cohomology for any $E$. Hence we get a functor
$Framings(\Sigma) \to E Mod$.
from the homotopy type of the moduli space of framings.
By (Randal-Williams 10) this is pretty much the geometric realization of the conformal moduli space. Hence we get something that should deserve to be denoted as follows (need to give a more precise form of this statement)
$\Pi(\mathcal{M}_\Sigma) \to E Mod$.
This should be the Hitchin connection on the space of conformal blocks. Sections of this are the theta-functions and being functions on $\Pi(\mathcal{M}_\Sigma)$ these are some kind of automorphic functions.
(I should maybe say again that the point of all this is to find a way to speak about all these ingredients such that it seamlessly generalizes, next for instance to the 7d Chern-Simons theory and its conformal blocks over 6d manifolds, then to 11d CS and something over 10d manifolds.)
Now (for some mysterious reason…) I am trying to formulate some of these questions as a kind of research proposal. A first version is here:
To be fine-tuned. But I have to go offline for the moment.
Unfortunately term is beginning here, so I’ve little time to read this fascinating proposal. Good luck with it.
Anyway, some typos:
to be canonical equivalent
canonically
and so the proper higher analog of that are the…
analogs
have just 1-dimensional formal group
missing ’a’
which in suitable limit is a nonbabelian
missing ’a’ and ’nonabelian’
generlizing
to to put us
In the bibliography there are two [L09b]s.
Thanks! Have fixed these now.
Ttem 3
“‘fully local” (note triple apostrophe)
cobordism night just assigns
modulistack
unsing
that the the
Bibliography out of order from [39].
Thanks! Fixed now.
Need to continue tomorrow when I am more awake.
Thanks! Fixed now.
Need to continue tomorrow when I am more awake.
In terms of actual mathematics, this is nicely exhibited by the classification result [19] which actually constructs
too many ’actual’s.
attracting a much attention
No ’a’.
tradtional; structzure; cobordims
from the physics discussion if self-dual higher gauge theory and using…
’of’ rather than ’if’.
Thanks once more. Have fixed these, the new version is here: pdf.
But as I wrote to you by private email, I need to focus now on preparing a differently-formatted version of this. If only I could use all that energy not on proposing research, but on doing it :-/
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