Schreiber next

This is a kind of “research statement”, where I formulate, for my own sake, a big question (with various sub-questions) in the general context of differential cohomology in a cohesive topos which should be interesting to answer next.

These are preliminary notes meant to help me organize my thoughts, and not meant to be claiming anything, and not meant to necessarily make sense to anyone except myself. Of course, if you see what I am aiming at, successfully or not, or where I go astray, I’d be thankful for whatever comments you might have.

This text is a very first version of what it may once become. If you come back later, it may have improved, may have been expanded. Or may have disappeared.

To set the stage, in Known I briefly recollect what we already have. This will suggest some evident questions, discussed in To be done. To see that this should be going in the right direction, I close by mentioning some consistency checks.

Known

Higher geometric

The central axiomatic assumption that all of the following is based on is that we place ourselves in the ambient context of an (∞,1)-topos which is, specifically, a cohesive (∞,1)-topos.

As discussed there, in any such context we have a canonical notion of ∞-Chern-Weil theory. Of this theory, we are here interested in the following phenomenon:

for $G$ an internal ∞-group in $\mathbf{H}$, and for $A$ an internal abelian $\infty$-group, for $n \in \mathbb{N}$ and for

$$\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n A$$

a representative of a universal characteristic class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$, we may ask for differential refinements of this to a morphism

$$\hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n A_{conn} \,,$$

where now on right we have the moduli object of A-principal n-bundles with connection, and on the left one of G-principal ∞-connections.

At ∞-Chern-Simons theory and ∞-geometric prequantization is discussed that such morphisms may be regarded as extended action functionals of generalized ∞-model?-type topological quantum field theories. Here “extended” is in the sense of extended TQFT: the transgression of these morphisms to mapping spaces $[\Sigma, \mathbf{B}G_{conn}]$ for $\Sigma$ of cohomological dimension $n$ are ordinary action functionals

$$\exp(i S_{\hat \mathbf{c}}) : [\Sigma, \mathbf{B}G_{conn}] \to A \,,$$

but also for $0 \leq dim \Sigma \leq n$ the transgression “to higher codimension” exists

$$[\Sigma, \mathbf{B}G_{conn}] \to \mathbf{B}^{n - dim \Sigma} A_{conn}$$

and is to be interpreted as providing natural higher prequantum line bundles of the system.

Higher algebraic

In the special case that in the above the object $\mathbf{B}G$ happens to be a discrete object, where the differential refinement above becomes trivial – the special case of “∞-Dijkgraaf-Witten theory” –, a fair bit is known, by now, about how to proceed from this input via some general abstract formulation of quantization to the corresponding extended TQFT:

1. Choose a representation in the form of an (∞,n)-functor

   $$\rho : \mathbf{B}^n A_{disc} \to \mathcal{C} \,,$$

   to some symmetric monoidal (∞,n)-category $\mathcal{C}$ with all duals.

   More in detail, choose a ring spectrum $R$, let

   $$\mathcal{C} := n Mod_R$$

   be the $(\infty,n)$-category of (infinity,n)-vector spaces over $R$.

2. Consider the adjunct

   $$Z_{cl} : Bord_n(\mathbf{B}G) \to \mathcal{C}$$

   of the composite

   $$\rho \circ \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathcal{C}$$

   under the free functor/forgetful functor adjunction

   $$(F \dashv U) : SymmMonCat^{dual}_{(\infty,n)} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Cat_{(\infty,n)}$$

   between (∞,n)-categories and symmetric monoidal (∞,n)-categories with all duals, where the cobordism hypothesis-theorem identifies

   $$Bord_n(\mathbf{B}G) \simeq F(\mathbf{B}G)$$

   with the (∞,n)-category of cobordisms that are equipped with maps into $\mathbf{B}G$.

   This $Z_{cl}$ is the extended “HQFT”/“classical QFT” of field configurations defined by $\hat \mathbf{c}$.

3. Universally extend $Z_{ck}$, suitably (open question: how exactly??), along the forgetful functor $Bord_n(\mathbf{B}G) \to Bord_n(*) \simeq Bord_n$ to a symmetric monoidal $(\infty,n)$-functor

   $$Z : Bord_n \to \mathcal{C} \,.$$

   This $Z$ is the extended TQFT that quantizes the original extended action functional, given by the universal characteristic class $\hat \mathbf{c}$.

To be done

In the above we had the action functionals in a cohesive context $\mathbf{H}$, but their quantization only in the context of discrete ∞-groupoids $\infty Grpd \hookrightarrow \mathbf{H}$. It seems clear that the way to proceed is by internalization of the algebraic part into $\mathbf{H}$.

Internal $\mathcal{O}$-monoidal $(\infty,n)$-categories

There is already is a good notion of internal category in an (∞,1)-category. This allows us to make sense of $Cat_{(\infty,n)}(\mathbf{H})$ internal to the cohesive context $\mathbf{H}$.

What is needed is a refinement of this to internal monoidal (∞,1)-categories, and more generally to internal $\mathcal{O}$-monoidal $(\infty,1)$-categories, and still more generally to internal (∞,1)-operads.

Since $(\infty,1)$-categories internal to $\mathbf{H}$ are presented by the model structure on complete Segal objects in a model structure on simplicial presheaves, it is natural to expext that internal $\mathcal{O}$-monoidal $(\infty,1)$-categories, and generally internal $(\infty,1)$-operads will be given by the model structure for dendroidal complete Segal spaces in simplicial presheaves.

The externally Quillen equivalent model structure on dendroidal sets is already known to have left Bousfield localizations to the model structure for dendroidal Cartesian fibrations and the model structure for dendroidal left fibrations, which model, externally, $\mathcal{O}$-monoidal (∞,1)-categories (in particular: symmetric monoidal (∞,1)-categories) and ∞-algebras over an (∞,1)-operad in ∞Grpd. Looking at the table - models for (infinity,1)-operads it is pretty clear what one needs to do and check to lift these latter localizations to the “complete Segal space”-type models and hence obtain model structure for internal such localizations.

This needs to be worked out in detail.

Internal cobordism hypothesis

Once one has this, the evident next question is: can we construct an adjunction

$$SymMonCat_{(\infty,n)}(\mathbf{H}) \stackrel{\overset{F_{\mathbf{H}}}{\leftarrow}}{\underset{U_{\mathbf{H}}}{\to}} Cat_{(\infty,n)}(\mathbf{H}) \,.$$

If there is any justice, then this ought to exists. But I have not much of an idea for how to approach this, apart from noticing that some step in this direction has probably already been taken, see below.

Available checks and insights

Here are some partial insights that may be taken as indications that the above program is going in the right direction.

Moduli of field configurations

The $\mathbf{H}$-internal adjunction above

$$SymMonCat_{(\infty,n)}(\mathbf{H}) \stackrel{\overset{F_{\mathbf{H}}}{\leftarrow}}{\underset{U_{\mathbf{H}}}{\to}} Cat_{(\infty,n)}(\mathbf{H})$$

should send $\mathbf{B}G_{conn} \in \mathbf{H}$ to an internal $(\infty,n)$-category

$$Bord_n(\mathbf{B}G_{conn})$$

whose ∞-stack of $n$-morphisms is that of cohesive families of $n$-dimensional cobordisms $\Sigma$ equipped with maps into $\mathbf{B}G_{conn}$.

By the discussion at ∞-Chern-Simons theory, such morphisms indeed classify the corresponding field configurations, so that $Bord_n(\mathbf{B}G_{conn})$ should indeed be the $(\infty,n)$-category of classical field configurations.

The extension along $Bord_n(\mathbf{B}G_{conn}) \to Bord_n(*)$ should therefore involve an “integration over all field configurations” and hence be the correct path integral.

Cobordisms with geometric structure

The idea to consider cobordisms equipped not just with maps into a topological space, but with maps into an ∞-stack (such as $\mathbf{B}G_{conn}$) has been considered for the 1-category of cobordisms in the thesis by David Ayala. (See the references there.) He discusses a generalization to this context of the Galatius-Madsen-Tillmann-Weiss theorem. Since the ordinary version of this theorem is a central ingredient in the proof of the external cobordism hypothesis, we may read this as supportive of the idea that the internal $Bord_n(\mathbf{B}G_{conn})$ naturally exists.

Higher geometric quantization – Higher Heisenberg algebras

The quantization step should factor through

$$Bord_n(\mathbf{B}G_{conn}) \stackrel{\hat \mathbf{c}_*}{\to} Bord_n(\mathbf{B}^n A_{conn}) \stackrel{\rho_*}{\to} Bord_n(\mathcal{C}) \,.$$

Among the morphisms in $Bord_n(\mathbf{B}^n A_{conn})$ will be equivalences between cobordisms equipped with maps to $\mathbf{B}^n A$, given in $\mathbf{H}$ by diagrams of the form

$$\array{ \Sigma &&\stackrel{}{\to}&& \Sigma \\ & {}_{\mathllap{\hat \mathbf{c}}}\searrow & \swArrow_{\alpha} & \swarrow_{\mathrlap{\hat \mathbf{c}}} \\ && \mathbf{B}^n A_{conn} } \,.$$

This automorphism ∞-group we know to be the higher analog of the group that integrates the Poisson bracket. It contains in particular the Heisenberg group integrating the Heisenberg Lie algebra of the system. (See section 3.3.17 here).

This is exactly what we expect to act on the vector spaces of states of the system… which it does, via postcomposition with $\rho$.

(…)