nLab differential cohesion and idelic structure

This page originates in notes prepared for a talk “Fractures, Ideles and the Differential Hexagon” at CUNY workshop on differential cohomologies, New York, August 2014 (video recording)

Abstract We discuss how the synthetic differential geometry-like axiomatics of differential cohesion provides a theory of twisted generalized/nonabelian differential cohomology which has realizations not just in higher differential geometry but also notably in higher complex analytic geometry and moreover in “higher arithmetic geometry” (E-∞ arithmetic geometry) in a way that systematizes some of the analogies which motivate the geometric Langlands correspondence.

Motivation

A fruitful approach to mathematical theory is what might be called “inter-geometric”, meaning that definitions and theorems make sense and hold when interpreted in different flavors of geometry. Classical examples are the GAGA principle, the function field analogy, the geometric Langlands correspondence, more recent are various approaches to F1-geometry and global analytic geometry. While in these examples the analogy between different theories of geometry has been established case-by-case, there is by and large no meta-theory which would systematically imply the analogy.

This is of practical concern for instance in the Langlands program, where it is an open problem how the methods and insights which are deeply on the side of complex analytic geometry, such as involving mirror symmetry, might have incarnations on the arithmetic geometry-side. And vice-versa, the complex-analytic version of the conjecture was obtained by educated guessing via analogy from the arithmetic side in the first place, but this guesswork has been questioned (Langlands 14). Both of these issues would be resolved if one had an “inter-geometric” theory from which the correspondence both in arithmetic geometry and in complex-analytic geometry would both follow systematically.

More generally, inter-geometric theory is relevant in higher geometric quantization, where choices of polarizations correspond to descending to more rigid geometries (for introduction and review see Schreiber 14).

A noteworthy example in this context is the construction of the refined Witten genus in the guise of the string orientation of tmf: here what is initially a concept in complex analytic (super-)geometry is constructed by passage (via the construction of tmf) to the moduli stack of elliptic curves all the way down in arithmetic geometry, and in fact then via the fracture theorems by its base changes to p-adic geometry and to rational homotopy theory (and further to K(n)-local stable homotopy theory). In fact, the supersingular elliptic curves which are the ones that contribute at height 2 and hence make for the genuinely stringy (second chromatic level) contribution to the refined Witten genus exist only in positive characteristic, invisible to complex geometry. There is thus a kind of p-adic string theory (closed string theory!) appearing here, which is however not of the kind that existing literature with such title would shed any light on.f

The role of more “rigid” arithmetic geometry – closer to the bottom absolute geometry – in quantization might be summarized in parts by the following table:

quantization of 3d Chern-Simons theory and holographically of the WZW-model/the string

geometry	worldvolume	moduli of fields	moduli of polarizations	geometric quantization/theta functions
differential geometry	worldsheet	moduli stack of flat connections
complex analytic geometry	complex curve	Jacobian/moduli of (semi-)stable bundles	moduli stack of Riemann surfaces	modular functor, Witten genus, …
arithmetic geometry	arithmetic curve	Galois representations and automorphic forms (via Langlands correspondence) (in particular Tamagawa measures, etc.)	moduli stack of curves	equivariant elliptic cohomology, string orientation of tmf, …

The construction of the bottom right items here is a ground-breaking accomplishment in algebraic topology, but at least in view of the origin of the WZW-string and the Witten genus in string theory it maybe raises more questions than it solves: from the perspective of physics these are but the first example of a tower of higher dimensional brane phenomena, the next instance being the partition function of the M5-brane and then that of 10d string theory itself (see e.g. at self-dual higher gauge theory).

$k$	$(4k+3)d$ higher dimensional Chern-Simons theory	form theta characteristics $\to$	$(4k+2)d$ self-dual higher gauge theory
0	3d Chern-Simons theory		chiral WZW-string/modular functor/equivariant elliptic cohomology
1	7d Chern-Simons theory		self-2-2form on M5-brane
2	11d Chern-Simons theory		RR-fields in type II string theory

In all of these higher dimensional cases the inter-geometric aspect appears. Where one assigned an elliptic cohomology theory to worldsheets equipped with polarization structure, it is only the arithmetic geometry cases of supersingular elliptic curves which contribute; similarly in the higher dimensional cases it is the Artin-Mazur formal groups in positive characteristic which induce at the given height to the Calabi-Yau cohomology. Finding the higher analog of the string orientation of tmf for these higher dimensional cases is as desirable as it seems to be intractable without some more inter-geometric theory to guide one.

Here we will not present solutions to these rather deep questions. But we do want to discuss something that looks like steps in the right direction.

Notice that the idea of “inter-geometric theory” is ancient, it originates with the synthetic geometry of Euclid which, with the parallel axiom removed, subsumes Euclidean, elliptic and hyperbolic geometry:

synthetic geometry
Euclidean geometry
hyperbolic geometry
elliptic geometry

The idea of refining such a synthetic reasoning to differential geometry is not as ancient, but far from new, this is known as synthetic differential geometry. For the kinds of applications as mentioned above we need something a bit more expressive, we consider differentially cohesive homotopy theory:

higher synthetic differential geometry
higher differential geometry
higher complex analytic geometry
higher arithmetic geometry

This we review and discuss below. First we recall now the key motivating ingredients of the function field analogy and the Langlands correspondence.

1) Langlands correspondence and Weil uniformization

The central idea of the Langlands correspondence (see for instance (Frenkel 05) for a review of the basic aspects that we refer to) is that given a global field $K$, then $n$-dimensional linear representations of its Galois group are in correspondence with certain linear representations – called automorphic representations – of the general linear group $GL_n(\mathbb{A}_K)$ with coefficients in the ring of adeles $\mathbb{A}_K$ of $K$ on the linear space of functions on the double coset space

$$GL_n(K) \;\backslash\; GL_n(\mathbb{A}_{K}) \;/\; GL_n(\mathbb{O}_K) \,,$$

where $\mathbb{O}_K$ denotes the ring of integral adeles. In particular for the “absolute” case that $K = \mathbb{Q}$ is the rational numbers, this is

$$GL_n(\mathbb{Q}) \;\backslash\; GL_n(\mathbb{A}_{\mathbb{Q}}) \;/\; GL_n(\mathbb{A}_{\mathbb{Z}})$$

where

• $\mathbb{A}_{\mathbb{Z}} = \prod_p \mathbb{Z}_p$ is the product of all rings of p-adic integers – the ring of integral adeles (finite integral adeles);

• $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q}\otimes \mathbb{A}_{\mathbb{Z}} = \prod_p^{\prime} \mathbb{Q}_{p}$ is the restricted product of all rings of p-adic rational numbers – the ring of adeles (ring of finite adeles).

For the case that $n = 1$ then the left part of this quotient is the idele class group, for higher $n$ this is an object in some nonabelian generalization of class field theory.

The striking observation that leads to the conjecture of the geometric Langlands correspondence is that under the function field analogy this double quotient, as a stacky quotient, is analogous to the moduli stack of vector bundles $Bun_{\Sigma}(GL_n)$ over a complex curve $\Sigma$ in the specific presentation that is given by the Weil uniformization theorem. Namely, this says that choosing any point $x\in \Sigma$ and a formal disk $x \in D \subset \Sigma$ around it, then the formal disk $D$ around $x$ together with the complement $\Sigma-\{x\}$ of the point

$$\array{ && \Sigma-\{x\} \\ & && \searrow \\ && && \Sigma \\ & && \nearrow \\ && D }$$

is a “good enough cover”, hence by Cech cohomology we have

$$\begin{aligned} Bun_\Sigma(GL_n) & \simeq [\Sigma-\{x\}, GL_n] \;\backslash \;[D-\{x\}, GL_n] \;/\; [D,GL_n] \\ & = \mathbb{O}_{\Sigma}[(z-x)^{-1}] \;\backslash\; GL_n(\mathbb{C}((z-x))) \;/\; GL_n(\mathbb{C}[ [(z-x)] ]) \end{aligned} \,,$$

expressing the groupoid (stack) of $GL_n$-principal bundles on $\Sigma$ as the groupoid of $GL_n$-valued transitions functions on the space of double intersections of the cover, which is $D-\{x\}$, subject to gauge transformations given by $GL_n$-valued functions on the cover itself, hence on $\Sigma-\{x\}$ and on $D$. But (holomorphic)

• functions on the formal disk $D$ at $x$ are, essentially by definition, formal power series in $(z-x)$;

• functions on the punctured formal disk $D -\{x\}$ are formal power series which need not converge at the missing point, hence are Laurent series in $(z-x)$;

• functions on the complement $\Sigma-\{x\}$ are, similarly, meromorphic functions on $\Sigma$ with poles allowed at $x$.

The expression for $Bun_\Sigma(GL_n)$ obtained this way in the second line above is analogous to the idelic space appearing the Langlands program. This analogy proceeds via the function field analogy and the F1-geometry-analogy, which say that:

• the ring of p-adic integers

  $$\mathbb{Z}_p = \{ a_0 + a_1 p + a_2 p^2 + \cdots \}$$

  is analogous to the ring of functions on the formal disk $D$ at $x$, namely the power series ring

  $$\mathbb{C}[ [ (z-x) ] ] = \{ a_0 + a_1 (z-x) + a_2 (z-x)^2 + \cdots \}$$

• the ring of p-adic numbers

  $$\mathbb{Q}_p = \{ a_{-k} p^{-k} + \cdots + a_{-1} p^{-1} + a_0 + a_1 p + a_2 p^2 + \cdots \}$$

  is analogous to the ring of functions on the pointed formal disk $D - \{x\}$, namely the Laurent series ring

  $$\mathbb{C}((z-x)) = \{ a_{-k} (z-x)^{-k} + \cdots + a_{-1} (z-x)^{-1} + a_0 + a_1 (z-x) + a_2 (z-x)^2 + \cdots \};$$

• the subring $\mathbb{Z}[p^{-1}] \subset \mathbb{Q}$ of rational numbers with denominator a power of $p$ is analogous to the subring of meromorphic functions on $\Sigma$ with possible poles at $x$.

Notice that the cover of the complex curve $\Sigma$ involved in the Weil uniformization theorem is exhibited by the following fiber product diagram

$$\array{ && \Sigma-\{x\} \\ & \nearrow && \searrow \\ D-\{x\} && && \Sigma \\ & \searrow && \nearrow \\ && D }$$

Indeed, in further support of this analogy one may see that also the p-adic integers together with the rational functions form a “good enough cover” of the “F1-arithmetic curve” Spec(Z) of this form. This is the statement of the arithmetic fracture square:

Proposition

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\left(\underset{p\;prime}{\prod} \mathbb{Z}_p \right) && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$$

Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,,$$

Since the ring of adeles is the rationalization of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also (by the discussion here) a pushout diagram in CRing, and in fact in topological commutative rings (for $\mathbb{Q}$ with the discrete topology and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/completion topology).

The statement of prop. immediately lifts to flat finitely generated torsion free modules, involving now the rationalization and the completion of modules. It then naturally lifts futher to stable homotopy theory, now with spectra regarded as ∞-modules over the sphere spectrum $\mathbb{S}$:

Proposition

(Sullivan arithmetic fracture square)

For every spectrum $X$ the canonical square diagram

$$\array{ && X_{\mathbb{Q}} \\ & \swarrow && \nwarrow \\ \left( \prod_p X_p^{\wedge} \right)_{\mathbb{Q}} && && X \\ & \nwarrow && \swarrow \\ && \prod_p X_p^\wedge }$$

formed by p-completion and rationalization of spectra is a homotopy pullback square (hence a homotopy pushout square).

Moreover, the geometric Langlands correspondence eventually relates the moduli stack of bundles $Bun_{\Sigma}(G)$ realized via Weil uniformization with respect to such “fracture cover” to a moduli stack of local systems (flat connections) by identifying quasicoherent sheaves on one side with D-modules on the other

“$\mathcal{O}Mod(Loc_{\Sigma}({{}^L G})) \stackrel{\simeq}{\longrightarrow} \mathcal{D} Mod( Bun_{\Sigma}(G))$”

Hence in order to systematize the implementation of such consideration in various flavors of geometry, we need an “inter-geometric” axiomatics that incorporates these ingredients, notably

1. differential cohomology;

2. D-geometry.

Such an axiomatics we turn to now, recalling how it has implementations in higher differential geometry and higher complex analytic geometry. Then at the end below we discuss how the axioms also have implementation in a E-∞ arithmetic geometry, where moreover they reproduce precisely the classical number-theoretic fracture squares and hence an “automorphic” incarnation of all moduli ∞-stacks of higher gauge fields.

2) Axiomatic twisted differential generalized cohomology

The above analogy calls for being formalized. We need an axiomatics that allows to implement differential geometry in systematic analogy. Just as back in the old days there was established a systematic analogy

synthetic geometry
Euclidean geometry
hyperbolic geometry
elliptic geometry

for modern applications we need a systematic dictionary of the form

higher synthetic differential geometry
higher differential geometry
higher complex analytic geometry
higher arithmetic geometry

We will discuss here how this may be done via the axiomatics called cohesive homotopy theory and differential cohesion in (Schreiber 13).

Differential generalized cohomology

To that end, first consider the following flavors of geometry.

Example

Let $S$ denote either of the following sites:

• $SmoothMfd$ smooth manifolds;

• $ComplexAnalyticMfd$ complex analytic manifolds;

• $SuperMfd$ supermanifolds;

• $FormalSmoothhMfd$, $FormalComplexAnalyticMfd$, $FormalSuperMfd$ formal manifolds;

• any site such that

  1. it has finite products;

  2. every object is locally of contractible etale homotopy type;

  3. $Hom(\ast, -)$ preserves split hypercovers.

Write

$$\mathbf{H} \coloneqq Sh_\infty(S) \simeq L_{lwhe} sPSh(S)$$

for the homotopy theory obtained from the category of simplicial presheaves on $S$ by universally turning local (stalkwise) weak homotopy equivalences into actual homotopy equivalences (i.e. the hypercomplete (∞,1)-category of (∞,1)-sheaves over this site.

Write specifically

• $Smooth \infty Grpd\coloneqq Sh_\infty(SmoothMfd)$ – smooth ∞-groupoids;

• $ComplexAnalytic \infty Grpd\coloneqq Sh_\infty(ComplexAnalyticMfd)$ – complex analytic ∞-groupoids;

• $SmoothSuper \infty Grpd\coloneqq Sh_\infty(SmoothSuperMfd)$ – smooth super ∞-groupoids;

• $FormalSmooth\infty Grpd \coloneqq Sh_\infty(FormalSmoothMfd)$ – formal smooth ∞-groupoids.

Proposition

The homotopy theories $\mathbf{H}$ from example have the property that there is an adjoint quadruple of derived functors ((∞,1)-functors)

$$\mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} \infty Grpd \simeq L_{whe} Top$$

such that the top left adjoint preserves finite products and the bottom right adjoint is a fully faithful embedding.

By going back and forth this induces an adjoint triple of (∞,1)-comonads on $\mathbf{H}$ which we write

$$(\Pi \dashv \flat \dashv \sharp) \colon \mathbf{H} \to \mathbf{H}$$

and call, respectively: shape modality $\dashv$ flat modality $\dashv$ sharp modality.

Following 1-categorical terminology proposed by William Lawvere (see at cohesive topos) we say:

Definition

Homotopy theories with the properties as in prop. we call cohesive homotopy theories? (cohesive (∞,1)-toposes).

It is commonplace that a single idempotent (∞,1)-monad such as $\Pi$ is equivalently a localization of a homotopy theory, and that a sincle idempotent co-monad such as $\flat$ is equivalently a co-localization.

Lawvere argued since the 1990s (see here) is that the presence of adjoint pairs and more so of adjoint triples of these on a category – “adjoint modalities” – is a remarkably expressive structure for axiomatizing synthetic differential geometry. What (Schreiber 13) observes is that in homotopy theory this is considerably more so the case:

Claim

• the shape modality $\Pi$ is naturally thought of as sending each geometric homotopy type $X \in \mathbf{H}$ to its fundamental ∞-groupoid $\Pi X$ of geometric paths inside it, equivalently to its geometric realization;

• the flat modality $\flat$ is naturally thought of as sending each moduli ∞-stack $\mathbf{B}G$ of $G$-principal ∞-bundles to the moduli ∞-stack $\flat\mathbf{B}G$ of $G$-principal flat ∞-connections;

• the homotopy fiber $\flat_{\mathrm{dR}}G$ of the counit $\flat \mathbf{B}G \to \mathbf{B}G$ is naturally thought of as sending the moduli $\infty$-stack of $\mathfrak{g}$-L-∞ algebra valued differential forms;

• the canonical map $\theta \colon G \longrightarrow \flat_{dR} G$ is naturally thought of as the $G$-Maurer-Cartan form;

• for braided ∞-groups $G$ the various homotopy fibers of $\theta \colon G \longrightarrow \flat_{dR} G$ are moduli $\infty$-stacks $\mathbf{B}G_{conn}$ of non-flat $G$-principal ∞-connections;

• the sharp modality $\sharp$ serves to produce various more suble moduli ∞-stacks, for instance:

  • given a Hodge filtration on $\flat_{dR} G$ then $BunConn_\Sigma(G)$ – the moduli $\infty$-stack of all principal ∞-connections on a given base space $\Sigma$ – is obtained by “differential concretification” (see Schreiber 14 for more);

  • for $Type_{\mathbf{H}}$ the object classifier of $\mathbf{H}$, then for any bare homotopy type $\Sigma \in \infty Grpd \hookrightarrow \mathbf{H}$ the homotopy pullback $\mathcal{M}_{\Sigma}$ in

    $$\array{ \mathcal{M}_{\Sigma} &\longrightarrow& &\longrightarrow& \ast \\ \downarrow && && \downarrow^{\mathrlap{\dashv \Sigma}} \\ Type_{\mathbf{H}} &\stackrel{}{\to}& \sharp Type_{\mathbf{H}} & \stackrel{\Pi}{\longrightarrow} & \sharp Type_{\mathbf{H}} }$$

    For instance for $\mathbf{H} = ComplexAnalytic\infty Gprd$ then this is a moduli stack of complex structures on $\Sigma$.

This is quite a bit of structure, concisely axiomatized by the presence of the adjoint modalities $\Pi \dashv \flat \dashv \sharp$. And more is implied:

Proposition

For any cohesive (∞,1)-topos $\mathbf{H}$ over ∞Grpd, then its Goodwillie tangent space, the tangent (∞,1)-category $T \mathbf{H}$ of parameterized spectrum objects in $\mathbf{H}$ is itself a cohesive $(\infty,1)$-topos over bare parameterized spectra $T \infty Grpd$ – the tangent cohesive (∞,1)-topos:

$$T \mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} T \infty Grpd \,.$$

This is an extension of $\mathbf{H}$ by stable homotopy theory

$$\array{ Spectra(\mathbf{H}) &\hookrightarrow& T \mathbf{H} \\ && \downarrow \\ && \mathbf{H} } \,.$$

In (Bunke-Nikolaus-Völkl 13) it was observed that:

Proposition

For $\hat E \in Spectra(\mathbf{H}) \hookrightarrow T \mathbf{H}$ a stable cohesive homotopy type, then the canonical diagram formed from the unit of the shape modality $\Pi$ and the counit of the flat modality $\flat$

$$\array{ && \Pi_{dR} {\hat E} && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}{\hat E} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_{\hat E}}} && \searrow \\ \Pi_{dR} \flat {\hat E} && && {\hat E} && && \Pi \flat_{dR} \hat E \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_E}} \\ && \flat {\hat E} && \longrightarrow && \Pi \hat E }$$

is homotopy exact in that

1. both squares are homotopy pullback (and hence homotopy pushout) squares;

2. the diagonals are homotopy fiber sequences (and hence homotopy cofiber sequences);

3. also the long top and bottom sequences are homotopy fiber sequences (and hence homotopy cofiber sequences).

Remark

In view of claim the differential cohomology hexagon of prop. has the following interpretation:

$$\array{ && connection\;forms\;on\;trivial\;bundles && \stackrel{de\;Rham\;differential}{\longrightarrow} && curvature\;forms \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ flat\;differential\;forms && && geometric\;bundles\;with \;connection && && rationalized\;bundle \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && geometric\;bundles\;with\;flat\;connection && \underset{comparison}{\longrightarrow} && shape\;of\;bundle }$$

In particular, when applied to sheaves of spectra of the form considered in (Bunke-Gepner 13), which effectively embody the construction of generalized differential cohomology that was proposed in (Hopkins-Singer 02), then the right part of the hexagon reproduces their defining decomposition as homotopy pullbacks of L-∞ algebra valued differential form along the Chern character map $E \to E \wedge H \mathbb{R}$ of plain spectra $E$ (see at differential cohomology diagram – Hopkins-Singer coefficients).

In view of this it is natural to ask if there are more general sheaves of spectra than those proposed in (Hopkins-Singer 02, Bunke-Gepner 13) which could still be sensibly regarded as encoding a kind of differential cohomology. Proposition in view of claim answers this in the most encompassing way: every sheaf of spectra on smooth manifolds, and in fact more generally every stable cohesive homotopy type is meaningfully regarded as a generalized differential cohomology theory, in that the axiomatics of cohesion provides a detailed decomposition of any such into data which behaves just right.

It may therefore be useful to regard prop. as the differential refinement of the Brown representability theorem:

Brown representability theorem	$\;\;\;$ proposition
cohomology theory = spectrum	differential cohomology theory = cohesive spectrum

Twisted differential generalized cohomology

More is true, also twisted cohomology is naturally encoded by the axiomatics of cohesive homotopy theory, as we pass from the fiber $Spectra(\mathbf{H})$ of the tangent cohesive (∞,1)-topos $T\mathbf{H}$ over the point to general cohesive parameterized spectrum objects:

Proposition

For $\hat E \in Spectra(\mathbf{H}) \hookrightarrow T\mathbf{H}$ a spectrum object, the canonical ∞-action of its automorphism ∞-group is exhibited by the universal $\hat E$-fiber ∞-bundle

$$\left[ \array{ \hat E &\to& \hat E//Aut(\hat E) \\ && \downarrow \\ && \mathbf{B}Aut(\hat E) } \right] \in T \mathbf{H} \,.$$

NSS 12

Proposition

For any unstable cohesive homotopy type $X \in \mathbf{H} \hookrightarrow T \mathbf{H}$ the mapping stack

$$[X,\hat E//Aut(\hat E)] \in T \mathbf{H}$$

is the bundle of spectra which over a twist $\tau \colon X \to Pic(\hat E)$ is the $\tau$-twisted $\hat E$-cohomology of $X$.

See here for details and further discussion.

So cohesion faithfully axiomatizes “inter-geometric” twisted differential generalized cohomology. In order to find also an “inter-geometric” Weil uniformization theorem for this we need however to add another axiom, one that makes infinitesimal objects such as formal disks appear explicitly.

Weil uniformization for twisted differential generalized cohomology

To that end, consider again first an example

Example

Let $S_{reduced} \longleftarrow S \longleftarrow S_{infinitesimal}$ be one of the following fiber sequence of sites

• $SmoothMfd \longleftarrow FormalSmoothMfd \hookleftarrow FormalPts$

• $ComplexAnalyticMfd \longleftarrow FormalComplexAnalyticMfd \hookleftarrow FormalPts$

where $FormalPts$ is the site of infinitesimally thickened points with the trivial topology;

Under forming hypercomplete (∞,1)-sheaf (∞,1)-topos this yields

$$\array{ \mathbf{H}_{reduced} &\hookrightarrow& \mathbf{H} &\longrightarrow& \mathbf{H}_{infinitesimal} \\ ComplexAnalytic\infty Grpd & \longrightarrow & FormalComplexAnalytic\infty Grpds & \longrightarrow & Infinitesimal\infty Grpd }$$

Here the last item is essentially formal moduli problems but without the condition of $\Gamma(-) = \ast$ and without the condition of Lurie-infinitesimal cohesion (beware the terminology clash), see at differential cohesion – Lie theory for more on this.

Proposition

In example we have

$$\mathbf{H}_{reduced} \stackrel{\hookrightarrow}{\stackrel{\longleftarrow}{\stackrel{\overset{}{\hookrightarrow}}{\longleftarrow}}} \mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} \mathbf{H}_{infinitesimal}$$

By going back and forth, the adjoint quadruple on the left induces a further adjoint triple of adjoint modalities which we write

$$(\Re \dashv \Im \dashv \&) \colon \mathbf{H} \to \mathbf{H}$$

which we call reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality .

Moreover, $\mathbf{H}_{infinitesimal}$ satisfies infinitesimal cohesion in that for all objects in here the points-to-pieces transform $\flat \to \Pi$ is an equivalence.

$\mathbf{H}_{reduced}$	$\hookrightarrow$	$\mathbf{H}$	$\longrightarrow$	$\mathbf{H}_{infinitesimal}$
cohesion		differential cohesion		infinitesimal cohesion
moduli ∞-stacks		formal smooth ∞-groupoids		formal moduli problems

Claim

The infinitesimal shape modality $\Im$ is naturally thought of as producing de Rham space objects. In particular:

1. for $G \in Grp(\mathbf{H})$ an ∞-group object then the mapping stack

   $$Loc_\Sigma(G) \coloneqq [\Im \Sigma, \mathbf{B}G]$$

   is the moduli ∞-stack of $G$-local systems on any $\Sigma \in \mathbf{H}$;

2. quasicoherent sheaves on $\Im X$ are D-modules on $X$.

Remark

In terms of claim then the statement of the geometric Langlands correspondence is that there is a natural correspondence between $\Im[\Sigma, \mathbf{B}G]$ and $[\Im\Sigma, \mathbf{B}{}^L G]$.

Definition

Since by prop. $\mathbf{H}$ is cohesive also over $\mathbf{H}_{infinitesimal}$, this gives relative modalities

$$(\Pi^{rel} \dashv \flat^{rel} \dashv \sharp^{rel}) \;\colon\; \mathbf{H} \to \mathbf{H}_{infinitesimal} \to \mathbf{H}$$

which we call the relative shape modality, relative flat modality and relative sharp modality, respectively.

See (Schreiber 13, 3.10.10).

Proposition

For $\Sigma\in ComplexAnalyticMfd \hookrightarrow ComplexAnalytic\infty Grpd$ then the relative flat modality, def. , is given by forming the disjoint union

$$\flat^{rel} \Sigma \simeq \underset{x \in \Sigma}{\coprod} D_{x}$$

of all formal disks $D_x \hookrightarrow \Sigma$ around points $x \in \Sigma$.

See (Schreiber 13, 5.6.1.4).

Remark

In summary, the differential cohesive structure is reflected in the existence of a triple of triples of operations that naturally exist on all objects in $\mathbf{H}$:

1. cohesion

   • shape modality$\Pi$ gives geometric realization/classifying spaces;

   • flat modality$\flat$ gives underlying point sets and moduli for flat ∞-connections/local systems;

   • sharp modality$\sharp$ induces moduli stacks for non-flat ∞-connections (via differential concretification of the naive mapping stacks);

2. infinitesimal cohesion

   • relative shape modality $\Pi^{rel}$ has as homotopy fibers over $X$ spaces $\Pi^{rel}_{dR}(X)$ whose function spaces are rationalizations of function spaces on $X$.

   • relative flat modality $\flat^{rel}$ creates collections of formal disks;

   • relative sharp modality $\sharp^{rel}$ induces synthetic differential moduli stacks of non-flat ∞-connections

3. relative differential cohesion

   • reduction modality$\Re$ removes infinitesimal dimensions;

   • infinitesimal shape modality$\Im$ produces de Rham spaces, detects formally etale morphisms and induces etale toposes;

   • infinitesimal flat modality$\&$

Example

Every $X$ in $\mathbf{H}$ sits in a canonical square

$$\array{ && \Pi^{rel}_{dR} X && & && rationalization\;of\;X \\ & \nearrow && \searrow && & \\ \Pi^{rel}_{dR} \flat^{rel} X && && X & \\ & \searrow && \nearrow && & \\ && \flat^{rel} X && & && formal\;disks\;in\;X }$$

and the stabilization of this, equivalently the result of passing to $\hat E$-spectrum-valued functions on this yields

$$\array{ && [\Pi^{rel}_{dR} X, \hat E] && & && rational\;\hat E-functions \\ & \swarrow && \nwarrow && & \\ [\Pi^{rel}_{dR}\flat^{rel} X, \hat E] && && [X,\hat E] & \\ & \nwarrow && \swarrow && & \\ && [\flat^{rel} X, \hat E] && & && \hat E-adeles }$$

which is homotopy cartesian.

3) $E_\infty$-Arithmetic differential cohomology

Above we found general synthetic axioms for differential cohomology and realization of these axioms in higher differential geometry and higher complex analytic geometry. Both turned out to exhibit also relative differential cohesion, def. , over “formal moduli problems”.

While higher arithmetic geometry (i.e. E-∞ arithmetic geometry) is not cohesive over the standard base (∞,1)-topos ∞Grpd, it does turn out to exhibit this kind of relative differential cohesion in a way that the corresponding relative differential cohomology hexagon subsumes the traditional arithmetic fracture square of prop. :

Proposition

Let $A$ be an E-∞ ring and let $\mathfrak{a} \subset \pi_0 A$ be a finitely generated ideal in its underlying commutative ring.

Then there is an adjoint triple of adjoint (∞,1)-functors

$$\array{ \underoverset{ A Mod_{\mathfrak{a}comp}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} }$$

where

• $A Mod$ is the stable (∞,1)-category of modules, i.e. of ∞-modules over $A$;

• $A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the full sub-(∞,1)-categories of $\mathfrak{a}$-torsion and of $\mathfrak{a}$-complete $A$-∞-modules, respectively;

• $(-)^{op}$ denotes the opposite (∞,1)-category;

• the equivalence of (∞,1)-categories on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.

This is effectively the content of (Lurie “Completions”, section 4), a refinement to stable homotopy theory of what in homological algebra is sometimes known as Greenlees-May duality.

For our purposes we notice the following immediate consequence.

Corollary

The traditional arithmetic fracture square of prop. is the left part of the “differential cohomology diagram” induced by the adjoint modality $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. , for the special case that $A = \mathbb{S}$ is the sphere spectrum and $\mathfrak{a} = (p)$ a prime ideal

$$\array{ && \Pi_{\mathfrak{a}dR} X && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{\mathfrak{a}dR} X \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && \flat_{\mathfrak{a}} X && \longrightarrow && \Pi_{\mathfrak{a}} X } \,,$$

Remark

By the discussion at completion of modules in the section Monoidalness, the adjoint modality of prop. is a monoidal (∞,1)-functor without, possibly, respect the tensor unit in $A Mod$. This means that $(\Pi_{\mathfrak{a}}\dashv \flat_{\mathfrak{a}})$ passes to “commutative ∞-monoids-without unit” in $A Mod$, hence to (formal duals of) nonunital E-∞ algebras. By this proposition (Lurie “Algebra”, prop. 5.2.3.15) nonunital E-∞ rings are equivalent to augmented E-∞ rings over the sphere spectrum, hence this is E-∞ arithmetic geometry under $Spec(\mathbb{S})$.

Notice that in addition $\Pi_{\mathfrak{a}}$ here should preserve finite products (because by the discussion at completion of a module – monoidalness the underlying $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$ preserves all small (∞,1)-colimits and because by this proposition finite coproducts in $CRng(A Mod)$ are computed in the underlying $A Mod$.

Therefore we may think of $\Pi_{\mathfrak{a}}$ as a shape modality and of $\flat_{\mathfrak{a}}$ as a sharp modality on affine E-∞-arithmetic geometry under $Spec(\mathbb{S})$ – namely on formal duals of nonunital E-∞ rings .

(It may be entertaining to note that on the level of ∞-groups of units then E-∞ arithmetic geometry under $Spec(\mathbb{S})$ translates to abelian ∞-groups of twists over the sphere spectrum – which has been argued to be the homotopy-theoretic incarnation of superalgebra, see at superalgebra – abstract idea for more on this.)

In conclusion, the situation is summarized by the following table.

cohesion in E-∞ arithmetic geometry:

cohesion modality	symbol	interpretation
flat modality	$\flat$	formal completion at
shape modality	$ʃ$	torsion approximation
dR-shape modality	$ʃ_{dR}$	localization away
dR-flat modality	$\flat_{dR}$	adic residual

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

References

• Ulrich Bunke, David Gepner, Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory (arXiv:1306.0247)

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)

• Jacob Lurie, section 4 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem

• Jacob Lurie, Higher Algebra

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – General theory, Journal of Homotopy and Related Structures, June 2014 (arXiv:1207.0248)

• Urs Schreiber, Differential cohomology in a cohesive ∞-topos, based on Habilitation thesis, Hamburg 2011 (arXiv:1310.7930)

• Urs Schreiber, What, and for what is Higher geometric quantization, notes for a talk given at Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends, Oxford, July 5-8 2014

Related talk notes:

• Urs Schreiber, Differential cohesion and Arithmetic geometry, talk at Karslruher Weihnachts-Workshop 2015

Formalization in cohesive modal homotopy type theory homotopy type theory:

• David Jaz Myers, Modal Fracture of Higher Groups, talk at CMU-HoTT Seminar, 2021 (pdf, pdf)