nLab differential cohomology

Idea

Differential cohomology is a refinement of plain cohomology such that a differential cocycle is to its underlying ordinary cocycle as a bundle with connection is to its underlying bundle.

The best known version of differential cohomology is a differential refinement of generalized (Eilenberg-Steenrod) cohomology, hence of cohomology in stable homotopy theory (as opposed to more general nonabelian cohomology, cf. nonabelian differential cohomology). For ordinary cohomology the refinement to ordinary differential cohomology is modeled for instance by complex line bundles with connection on a bundle, bundle gerbes with connection, etc, and generally by Deligne cohomology. Similarly, differential refinements of topological K-theory to differential K-theory may be presented by vector bundles with connection (e.g Simons-Sullivan 08).

A systematic characterization and construction of differential generalized (Eilenberg-Steenrod) cohomology in terms of suitable homotopy fiber products of the mapping spectra representing the underlying cohomology theory with differential form data was then given in (Hopkins-Singer 02) (motivated by discussion of the quantization of the M5-brane via holographically dual 7d Chern-Simons theory by Edward Witten).

In this stable case one characteristic property of differential cohomology that was first observed empirically in ordinary differential cohomology (Simons-Sullivan 07) and differential K-theory (Simons-Sullivan 08) is that it is a kind of cohomology theory which fits into a differential cohomology diagram which is an interlocking system of two fiber sequences that expresses how the data of a geometric bundle with connection may be decomposed into the underlying bundle and the curvature of the connection and how connections on trivial bundles as well as bundles with flat connections are examples of the general concept. (The homotopy fiber product used in (Hopkins-Singer 02) provides one of the two commuting squares in this diagram.)

Schematically the characteristic differential cohomology hexagon (see there for details) looks as follows

$$\array{ && connection\;on\;trivial\;bundle && \stackrel{de\;Rham\;differential}{\longrightarrow} && differential\;forms \\ & \nearrow & & \searrow & & \nearrow_{curvature} && \searrow \\ flat\;differential\;forms && && geometric\;bundle\;with \;connection && && rational\;shape\;of\;bundle \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{}} && \nearrow_{\mathrlap{Chern\;character}} \\ && geometric\;bundle\;with\;flat\;connection && \underset{forget\;connection}{\longrightarrow} && shape\;of\;bundle }$$

It turns out that such differential cohomology diagrams are exactly what all stable homotopy types in cohesive (∞,1)-toposes $\mathbf{H}$ naturally sit in (Bunke-Nikolaus-Völkl 13). All the traditional differential cohomology theories and their joint generalization due to (Hopkins-Singer 02) are special cases of this for the case that $\mathbf{H} = T Smooth \infty Grpd$ is the tangent cohesive (∞,1)-topos of that of smooth ∞-groupoids (∞-stacks over the site of smooth manifolds).

It is therefore natural to define differential cohomology to be the intrinsic cohomology of cohesive (∞,1)-toposes. (Notice that while the differential cohomology diagram itself only involves the shape modality and the flat modality of cohesion, the sharp modality is needed to produce, via differential concretification, the correct moduli stacks of differential cocycles over a given base object from the mapping stack of that into the representing stable homotopy type).

Viewed in this generality, differential cohomology makes sense for instance also in supergeometry where it subsumes structures such as super line 2-bundles and more generally higher super gerbes with connection, which naturally appear as differential refinements of the geometric twists of (differential) K-theory and tmf.

Examples

• The ordinary differential cohomology $\bar H^\bullet(-,\mathbb{Z})$ is modeled by

  • Deligne cohomology;

  • Cheeger-Simons differential characters;

  • higher circle bundle gerbes with connection;

  • circle n-bundles with connection.

• apart from that people studied mainly differential K-theory.

In physics differential cocycles model gauge fields.

• Cocycles in ordinary differential cohomology (e.g. Deligne cohomology) model

  • in degree 2: the electromagnetic field

  • in degree 3: the Kalb-Ramond field

  • in degree 4: the supergravity C-field

• Cocycles in differential K-theory model the

  • RR-field .

For $c \in \bar \Gamma^\bullet(X)$ a differential cocycle representing a gauge field, one says that

• its image $F(c)$ in differential forms is the corresponding field strength;

• its image $cl(c)$ in non-differential cohomology is the “topological twist” of the gauge field. In special cases this can be identified with magnetic charge.

Other examples:

• differential algebraic K-theory

Differential stable cohomology

the following are ancient notes that need to be brushed up and polished.

Characterization following Hopkins-Singer

A standard definition of differential cohomology is in terms of a homotopy pullback of a generalized (Eilenberg-Steenrod) cohomology theory with the complex of differential forms over real cohomology:

Let $\Gamma^\bullet$ be a generalized cohomology theory in the sense of the generalized Eilenberg–Steenrod axioms, and let $\Gamma^\bullet \to H^\bullet(-,\mathbb{R}) \otimes \Gamma^\bullet(*)$ be a morphism to real singular cohomology with coefficients in the ring of $\Gamma$-cohomology of the point. Then the differential refinement of $\Gamma^q$, the degree $q$differential $\Gamma$-cohomology is the homotopy pullback $\bar \Gamma^\bullet$ in

$$\array{ \bar \Gamma^\bullet(-) &\stackrel{F}{\to}& \Omega^{\geq q}(-)\otimes \Gamma^\bullet(*) \\ \downarrow^{cl} && \downarrow \\ \Gamma^\bullet(-) &\stackrel{ch}{\to}& Z^\bullet(-, \mathbb{R}) \otimes \Gamma^\bullet(*) } \,,$$

where

• $Z^\bullet(-,\mathbb{R})$ is the complex underlying real singular cohomology $H^\bullet(-,\mathbb{R})$

• $\Omega^\bullet(-,\mathbb{R})$ is the complex underlying deRham cohomology

• $\Omega^\bullet(-) \to H^\bullet(-,\mathbb{R})$ is deRham theorem morphism

• $ch: \Gamma^\bullet(-) \to Z^\bullet(-, \mathbb{R}) \otimes \Gamma^\bullet(*)$ is the Chern character map.

For more details on this see at differential function complex.

Characterization following Bunke-Schick

The following are old notes once taken in a talk by Thomas Schick at Oberwolfach Workshop, June 2009 – Strings, Fields, Topology. Needs to be brushed-up, polished, improved, rewritten…

The following theory of differential cohomology (also called smooth cohomology) is developed and used in the work of Ulrich Bunke and Thomas Schick. Contrary to the above, it does not take the notion of homotopy limit as fundamental, but instead characterizes the universality of the above commuting diagram by other means. On the other hand, this means that their axiomatization at the moment only capture cohomology classes, not the representing cocycles. It is sort of known that for various applications specific cocycle representatives do play an important role, and one may imagining refining to discussion below eventually to accommodate for that.

• idea:

  • combine cohomology + differential forms

main diagram

$$\array{ \hat H(M) &\stackrel{I}{\to}& H^\bullet(M) \\ \downarrow^{R} && \downarrow \\ \Omega^\bullet_{d=0}(M) &\stackrel{}{\to}& H^\bullet_{dR}(M) \simeq H^\bullet(M,\mathbb{R}) }$$

so differential cohomology $\hat H^\bullet(M)$ combines the ordinary cohomology $H^\bullet(M)$ with a differential form representative of its image in real cohomology.

• $I$ projects a differential cohomology to its underlying ordinary cohomology class;

• $R$ send the differential cohomology class to its curvature differential form data

we want an exact sequence

$$\array{ H^{\bullet-1}(M) &\stackrel{ch}{\to}& \Omega^{\bullet-1}(M)/{im(d)} &\stackrel{d}{\to}& \hat H(M) &\stackrel{I}{\to}& H^\bullet(M) \to 0 \\ &&& {}_{d}\searrow & \downarrow^R \\ &&&& \Omega^\bullet_{d=0}(M) }$$

Definition

Given cohomology theory $E^\bullet$, a smooth refinement $\hat E^\bullet$ is a functor $\hat E : Diff \to Grps$ with transformations $I, R$ such that

$$\array{ \hat E(M) &\stackrel{I}{\to}& E^\bullet(M) \\ \downarrow^{R} && \downarrow \\ \Omega^\bullet_{d=0}(M, V) &\stackrel{}{\to}& E^\bullet_{dR}(M) \simeq E^\bullet(M,\mathbb{R}) }$$

where

$V = E^\bullet(pt)\otimes \mathbb{R}$

is the graded non-torsion cohomology of $E$ on the point (so now all the gradings above denote total grading)

and such that there is a transformation

$$a : \Omega^{\bullet -1}(M)/{im(d)} \to \hat E^*(M)$$

that gives the above kind of exact sequence.

Definition

If $E^*$ is multiplicative, we say $\hat E^*$ is multiplicative with product $\vee$ if $\hat E$ takes values in graded rings and the transformations are compatible with multiplicative structure, where

$$a(\omega) \vee x = a(\omega \wedge R(x))$$

Definition

$\hat E$ has $S^1$-integration if there is a natural (in $M$) transformation

$$\int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)$$

compatible with $\int$ of forms and for $E$ it is given by the suspension isomorphism

$$\int \circ p^* = 0$$

for $p : M \times S^1 \to M$ and

$$\int \circ ( id \times (z \mapsto \bar z) )^* = - \int$$

Remark

Ordinary cohomology theories are supposed to be homotopy invariant, but differential forms are not, so in general the differential cohomology is not.

Lemma

Given $\hat E$ a smooth cohomology theory. The homotopy formula:

given $h : M \times [0,1] \stackrel{smooth}{\to} N$ a smooth homotopy we have

$$h^*_1(X) - h^*_0(X) = a( \int_{M \times [0,1]/M} h^*(R(x)))$$

Corollary

$ker(R)$ (i.e. flat cohomology) is a homotopy invariant functor.

Definition

$\hat H{flat} := ker(R)$

Proof of lemma

It suffices to show

$$\iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))$$

for all $x \in \hat E(M \times [0,1])$.

Observe if $x = p^* y$ the left hand side vanishes, $\int R(p^* y) = 0$.

For general $x$ $\exists y \in \jhat E(M)$; $x - p^*(y) = a (\omega)$ $\omega \in \Omega(M \times [0,1])$.

Stokes’ theorem gives $i^*_1 \omega - i^*_0 \omega = \int_{[0,1]} d \omega$ $= \int R(a(\omega)) = \int R(x-p^* \omega) = \int R(x)$.

On the other hand

$$i^*_1(x) - i^*_0(x) = i^*_1(a(\omega)) - i^*_0(a(\omega)) = a(\int R(x))$$

A calculation: $\hat H^1_{flat}(pt) = \hat H^1(pt) = \mathbb{R}/\mathbb{Z} = \hat K^1(pt)$.

Theorem

(Hopkins–Singer)

For each generalized cohomology theory $E^*$ a differential version $\hat E^*$ as in the above definition does exist.

Moreover $\hat E_{flat}^* = E \mathbb{R}/\mathbb{Z}^{\bullet -1}$.

Remark

It’s not evident how to obtain more structure like multiplication.

Theorem

Using geometric models, multiplicative smooth extensions with $S^1$-integration are constructed for

• K-theory (Bunke–Schick)

• MU-bordisms (unitary bordisms)
  (Bunke–Schröder–Schick–Wiethaupts; and from there Landweber exact cohomology theories)

Theorem

(Uniqueness theorem due to Bunke–Schick. Simons–Sullivan proved this for ordinary integral cohomology.)

Assume $E^*$ is rationally even, meaning that

$$E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k$$

plus one further technical assumption.

Then any two smooth extensions $\hat E^*$, $\tilde E^*$ are naturally isomorphic.

If required to be compatible with integration the ismorphism is unique.

If $\hat E, \tilde E$ are multiplicative, then this isomorphism is, as well.

Example

If we don’t require compatibility with $S^1$-integration, then there are “exotic” abelian group structures on $\hat K^1$.

Examples

Differential cobordism cohomology theory

A model for a multiplicative differential refinement of complex cobordism cohomology theory, the theory represented by the Thom spectrum is in

• Ulrich Bunke, Thomas Schick, Ingo Schröder, Moritz Wiethaup, Landweber exact formal group laws and smooth cohomology theories (arXiv:0711.1134)

See differential cobordism cohomology theory

Related concepts

• nonabelian differential cohomology

• tangent cohesion, smooth spectrum

• differential orientation

• twisted differential cohomology

• Hodge-filtered differential cohomology

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

References

General

Differential cohomology was first developed for the special cases of ordinary differential cohomology and of differential K-theory (interest into which came from discussion of D-brane charge in type II superstring theory). See the references there. A survey is in

• Ulrich Bunke, Differential cohomology in geometry and analysis (pdf slides)

The suggestion that differential cohomology is or should be characterized by the differential cohomology diagram is due to

• James Simons, Dennis Sullivan, Axiomatic Characterization of Ordinary Differential Cohomology (arXiv:math/0701077)

  (discussed there for ordinary differential cohomology)

and

• James Simons, Dennis Sullivan, Structured vector bundles define differential K-theory (arXiv:0810.4935)

  (discussed there for differential K-theory).

A first systematic account characterizing and constructing stable differential cohomology in terms of homotopy fiber products of bare spectra with differential form data (which is really the homotopy-theoretic refinement of one of right square in the differential cohomology diagram) was given in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. Volume 70, Number 3 (2005), 329-452. (arXiv:math/0211216, euclid:1143642908)

motivated by applications in higher gauge theory and string theory, as explained further in

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv:hep-th/0011220)

with further reviews including

• Alessandro Valentino: Differential cohomology and quantum gauge fields (2008) [pdf]

• Richard Szabo, Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology, contribution to: 7th International Conference on Mathematical Methods in Physics (ICMP 2012) [arXiv:1209.2530, doi:10.22323/1.175.0009, inspire:1185286]

The homotopy-pullback definition of differential generalized cohomology from

• Hopkins-Singer 02, Section 4.1

is picked up in

• Ulrich Bunke, David Gepner, Section 2.2 of: Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory (arXiv:1306.0247)

• Ulrich Bunke, Section 4.3 of: Differential cohomology (arXiv:1208.3961)

• Bunke-Nikolaus-Völkl 13, Section 4.4

The suggestion that differential cohomology is naturally the intrinsic cohomology of those (∞,1)-toposes which are cohesive appeared in

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)

and the observation, working with sheaves of spectra, that indeed every stable homotopy type in a cohesive (∞,1)-topos canonically sits inside a differential cohomology diagram is due to

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, J. Homotopy Relat. Struct. 11 (2016) 1-66 [arXiv:1311.3188, doi:10.1007/s40062-014-0092-5]

Discussion in the generality of differential non-abelian cohomology:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalizaed, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]

Another book-length overview:

• Araminta Amabel, Arun Debray, Peter J. Haine (eds.), Differential Cohomology: Categories, Characteristic Classes, and Connections. Based on Fall 2019 talks at MIT’s Juvitop seminar by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. (arXiv:2109.12250)

On differential cohomology in mathematical physics:

• Arun Debray, Differential Cohomology, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.14338]

Lectures and talks

• Simons Center Workshop on Differential Cohomology January 10, 2011- January 14, 2011 (web)

• Bunke 2012

• Daniel Grady, Differential cohomology and Applications, talk at Geometry, Topology & Physics, NYU Abu Dhabi, April 2018 (pdf)