### (Generalized) Differential Cohomology and Lie Infinity-Connections

#### Posted by Urs Schreiber

Over the years, Dan Freed, Michael Hopkins and I. M. Singer and others have been developing the theory of *Generalized differential cohomology* and applied it with great success to various problems appearing in the theory of charged $n$-particles, usually known as NS-branes and D-branes and M-branes.

The idea is:

Given a class $\omega$ in the (generalized) cohomology $\Gamma^\bullet(X)$ of a space $X$, regard it as classifying an $n$-bundle-like thing and then find a way to equip that with something like a connection $\nabla$, such that the curvature differential form $F_\nabla$ of that connection reproduces the image of $\omega$ in deRham cohomology (with coefficients in the ring $\Gamma(pt)$): $[\omega_\mathbb{R}] = [F_\nabla] \,.$

When $\Gamma^\bullet(-) = H^\bullet(-,\mathbb{Z})$ is ordinary integral cohomology, this reproduces the notion of Cheeger-Simons differential characters, which is a way of talking about equipping line $n$-bundles ((n-1) gerbes) with a connection.

The notion of generalized differential cohomology allows to go beyond that and equip any other kind of cohomology class with a corresponding notion of “connection and curvature”. This has notably been applied to the next interesting generalized cohomology theory after ordinary integral cohomology: K-theory. It turns out that the differential forms appearing in *differential K-theory* model the RR-fields appearing in string theory.

Here I try to review some basics, provide some links – and then start to relate all this to the theory of parallel $\infty$-transport and $L_\infty$-connections.

**Some literature.**

I am not sure I can give a complete historical account, but one of the early influential articles on generalized differential cohomology, whose first section provides a good introduction, is

D. S. Freed
*Dirac charge quantization and generalized differential cohomology*

(arXiv).

As far as I am aware, the most recent and sophisticated development of this idea is that described in section 4 of

M. J. Hopkins and I. M. Singer
*Quadratic functions in geometry, topology and M-theory*

(arXiv).

The title of that article points to what has been one of the main motivations for considering this theory, namely functions that pair two “$n$-things with connection” and serve as, in particular, symplectic forms on the space of all these, thus providing a way to perform *geometric quantization of background $n$-fields*.

For ordinary differential cohomology ($n$-gerbes with connection) this quantum theory of $n$-connections was developed in two articles by Freed, Moore and Segal, which I once tried to summarize here

Freed, Moore, Segal on p-Form Gauge Theory, I

Freed, Moore, Segal on p-Form Gauge Theory, II .

**The basic idea.**

Maybe the quickest and most elegant way to describe the idea is this:

For every generalized cohomology theory $\Gamma$, there is a natural homomorphism

$\Gamma^\bullet(X) \to H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt)$

which sends each generalized cohomology class of a space $X$ to a differential form representing it.

For ordinary cohomology this is just the ordinary image of integral cohomology classes in deRham cohomology.

The corresponding *differential* generalized cohomology now is the collection of pairs, consisting of a generalized cohomology class together with a specific representative differential form.

This can be expressed as saying that the differential cohomology theory $A^\bullet_\Gamma$ is the pullback of the diagram

$\array{ && \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt) \\ && \downarrow \\ \Gamma^\bullet(X) &\to& H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt) }$

namely

$\array{ A^\bullet_\Gamma(X) &\to& \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt) \\ \downarrow && \downarrow \\ \Gamma^\bullet(X) &\to& H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt) } \,.$

More precisely, this pullback
really has to be read as a *weak* pullback (*homotopy pullback*).

That means that as we chase a pair

$(\omega, F) \in \Gamma^\bullet(X) \times \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt)$

consisting of a cohomology class and a closed curvature form from the top left to the bottom right of the diagram, the result along the two different ways need not be equal

$F = \omega_{\mathbb{R}}$

but may be just *cohomologous*

$F - \omega_{\mathbb{R}} = \partial h \,.$

That $h$ is the connection. In the case that $\Gamma^\bullet(-) = H^\bullet(-,\mathbb{Z})$ is ordinary integral cohomology, that $h$ is the (Cheeger-Simons) *differential character* proper, with $\omega$ regarded as its image in cohomology, and $F$ its curvature.

By comparison with the familiar cases, where this is what they are:

for instance an ordinary line bundle with connection on $X$ with Chern class $\omega$ and curvature 2-form $F$ the parallel transport of a connection is an element $h \in C^1(X,\mathbb{R})$ and we have

$\partial h = \omega_{\mathbb{R}} - F$

(maybe up to a sign…) Similarly for higher abelian gerbes with connection aka higher Cheeger-Simons differential characters.

**Relation to $L_\infty$-connections and $n$-Transport**

As emphasized in section 4 of Quadratic functions, a class of a generalized cohomology theory here is best thought of in terms of a map

$X \to S_\Gamma$

from $X$ into the spectrum representing the cohomology theory.

Let’s think about this $n$-categorically, in order to make contact with our way of talking:

for our base space $X$, choose a good cover $Y \to X$ and the corresponding Čech groupoid $Y^\bullet$.

An $n$-bundle on $X$ is a morphism

$g : Y^\bullet \to \mathbf{B} G \,,$

where $G$ is some $n$-group.

For instance $n$th integral cohomology is obtained by setting

$G = B^{n-1}U(1)$

while K-theory is obtained by setting, essentially,

$G = U(\infty) \simeq U_K$

or similar.

Equipping such the $n$-bundle represented by such a cocycle with a connection amounts to picking a parallel transport $n$-functor

$\mathrm{tra} : \Pi_{n+1}(Y^\bullet) \to \mathbf{B} E G \,,$

where $E G = INN(G)$ denotes the inner automorphism $n+1$ group and where $\Pi_{n+1}(Y^\bullet)$ denotes the fundamental $(n+1)$-groupoid of $Y$ merged with jumps in the fibers.

This has to make the square

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G }$

commute. If the fibers are $(n+1)$-connected, we can reformulate this elegantly as

$\array{ \Pi_{n+1}^{vert}(Y) &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G }$

The differential version of that (compare the slides I provided here)

is a diagram

$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(g) \\ \uparrow& & \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(g) }$

as described at great length in *$L_\infty$-connections and applications* (pdf, blog, arXiv).

There these diagrams are completed to

$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(g) \\ \uparrow&& \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(g) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{\{P_i\}}{\leftarrow}& inv(g) }$

which precisely amounts to picking up the characteristic curvature classes $\{P_i\}$ of the chosen connection which give the “differential” realization of the cohomology class $g$.

Noticing that

$inv(g) = H^\bullet(B G, \mathbb{R})$
for $G$ an ordinary compact group, we see that, *rationally*, the sequence of forms on the universal $G$-bundle

$\array{ CE(g) \\ \uparrow \\ W(g) \\ \uparrow \\ inv(g) }$

integrates to

$\array{ \mathbf{B} G \\ \uparrow \\ \mathbf{B} E G \\ \uparrow \\ \mathbf{B} B G } \,,$

where the last item has to be read as the one-object $\infty$-groupoid obatined from replacing the space $B G$ (which in general fails to have a group structure) with its rational approximation

$B G \simeq \prod_k K(\mathbb{Q}, f(k)) \,,$ (as on the bottom of p. 4 of Freed, Hopkins, Teleman, Twisted equivariant K-theory with complex coefficients) which happens to be an (abelian) $\infty$-group.

We see this way how our $\infty$-transport realizes the notion of differential cohomology in that completing the $n$-bundle (cohomology class) $\Pi_{n+1}^{vert}(Y) \stackrel{g}{\to} \mathbf{B} G$

to an $n$-bundle with connection

$\array{ \Pi_{n+1}^{vert}(Y) &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G \\ \downarrow && \downarrow \\ \Pi_{\infty}(X) &\stackrel{\{P_i\}}{\to}& \mathbf{B} B G }$

picks up the differential characteristic forms $\{P_i\}$ representing our cohomology class in deRham cohomology.

**Ordinary differential cohomology**

Here is a more detailed description of the relation between $n$-transport/$L_\infty$-connections and “differential ordinary cohomology”, aka Cheeger-Simons differential cohomology

(this being essentially a review of what is described starting on slide 616)

When we restrict to ordinary differential cohomology ($n$-gerbes with connection) one nice side-effect is that we can work entirely with *strict* $n$-groupoids. Which makes many things more tractable.

For each $n$, there is the strict $n$-group

$B^{n-1} U(1)$

which is trivial in each degree except of its topmost one, where it has $U(1)$-worth of $(n-1)$-morphisms.

I write

$\mathbf{B} B^{n-1} U(1)$

for the corresponding one-object $n$-groupoid which is trivial everywhere except that it has $U(1)$ worth of $n$-morphisms.

For $X$ any smooth space and $\pi : Y \to X$ a good cover (or good surjective submersion, more generally), we can think of the Čech groupoid

$Y^\bullet = (Y \times_X Y \stackrel{s,t}{\to} Y)$

as a strict $n$-groupoid by throwing in all the higher $k$-simplices (passing to nerves this means that we look at the simplicial space induced by $Y \to X$ and truncate it beyond level $n$).

Then the descent data for a $B^{n-1} U(1)$-$n$-bundle (= abelian $(n-1)$-gerbe) is a strict $n$-functor

$g : Y^\bullet \to \mathbf{B} B^{n-1} U(1)$

and indeed equivalence classes of such $n$-functors realize the $n$-th integral cohomology of $X$:

$EquivClasses(Hom_{n-Grpd}(Y^\bullet,\mathbf{B} B^{n-1}U(1))) = H^{n+1}(X,\mathbb{Z}) \,.$

So far this is nothing but an $n$-functorial restatement of the standard fact about Čech cohomology.

Now we turn this into “differential” cohomology by allowing the functor to also act on $n$-dimensional volumes in $X$.

I write:

$\Pi_{n+1}(X)$

for the strict fundamental $(n+1)$-groupoid of $X$: $(k \leq n)$-morphisms are thin-homotopy classes of globular smooth $k$-volumes in $X$, $(n+1)$-morphisms are full homotopy classes of globular $(n+1)$-volumes.

Then it’s a theorem (the proof of which has appeared in the literature so far only for $n=1$ and $n=2$, but it’s clear how this continues) that smooth strict $(n+1)$-functors from this path groupoid to $\mathbf{B} B^n U(1)$ are the same as closed $(n+1)$-forms on $X$:

$n\mathrm{Funct}^\infty(\Pi_{n+1}(X), \mathbf{B}B^n U(1)) = \Omega^{n+1}_{closed}(X) \,.$

So the question of differential cohomology is how to relate a cocycle $n$-functor

$g : Y^\bullet \to \mathbf{B} B^{n-1}U(1)$

with a curvature $(n+1)$-functor

$F : \Pi_{n+1}(X) \to \mathbf{B} B^n U(1) \,.$

To relate these, we need a connection on an $n$-bundle whose integral class is given by $g$ and whose curvature is $F$. This works as follows:

There is a strict $(n+1)$-groupoid

$\Pi_{n+1}(Y^\bullet)$

whose $k$-morphisms are generated from

- globular $k$-paths in $Y$

- together with “jumps” between $k$-paths in the fiber of $Y$

modulo the obvious relations which say that it does not matter whether I first move smoothly in $Y$ and then jump in the fiber, or the other way round.

More technically, this is the weak pushout

$\array{ \Pi_{n+1}(Y \times_X Y) &\stackrel{\pi_1}{\to}& \Pi_{n+1}(Y) \\ \downarrow^{\pi_2} && \downarrow \\ \Pi_{n+1}(Y) &\to& \Pi_{n+1}(Y^\bullet) }$

(called $C(Y)$ in 0705.0452).

Then: $B^{n-1}U(1)$-$n$-bundles with connection are given by smooth $(n+1)$-functors

$\mathrm{tra} : \Pi_{n+1}(Y^\bullet) \to \mathbf{B} E B^{n-1} U(1)$

such that pulled back to the jumps in fibers they reproduce a $B^{n-1} U(1)$-cocycle, meaning that they can be completed to a square

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} B^{n-1}U(1) \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y^\bullet) &\stackrel{\mathrm{tra}}{\to}& \mathbf{B} E B^{n-1}U(1) \,. }$

Here $E B^{n-1} U(1) := INN(B^{n-1} U(1))$ is the inner automorphism $(n+1)$-group of $B^{n-1} U(1)$ and the vertical arrows are the canonical inclusions.

Given that square, it so happens that the shifted part of $\mathrm{tra}$ descends down to $X$ in that we can further complete to a double square

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} B^{n-1}U(1) && (integral class) \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y^\bullet) &\stackrel{\mathrm{tra}}{\to}& \mathbf{B} E B^{n-1}U(1) && (connection) \\ \downarrow && \downarrow \\ \Pi_{n+1}(X) &\stackrel{F}{\to}& \mathbf{B} B^n U(1) && (curvature) \,. }$

This way the $n$-connection

$[\mathrm{tra}] \in \bar H^{n+1}(X)$

refines the integral class

$[g] \in H^{n+1}(X,\mathbb{Z})$

to a differential class with $(n+1)$-form curvature

$F \in \Omega^{n+1}_{closed}(X) \,.$

**Differential K-theory**

After the more detailed discussion above about the appearance of ordinary differential cohomology from the point of view of $\infty$-transport/$L_\infty$-connections, I’ll now say something about differential K-theory from that point of view.

Fix a base space $X$ and a surjective submersion $\pi : Y \to X$ as before, and write, also as before, $Y^\bullet$ for the corresponding Čech groupoid.

For $K^0$-classes we can get away with thinking of that as just an ordinary 1-groupoid, since $K^0$ is just about ordinary (1-)bundles.

A $K^0$-class of $X$ is represented by a functor

$g : Y^{\bullet} \to (\mathbf{B} U) \times \mathbb{Z}$

where $U = U(\infty)$.

To get started, let’s look at ordinary $U(n)$ first and start with just the cocycle for a rank $n$ vector bundle

$g : Y^{\bullet} \to \mathbf{B} U(n) \,,$

where, as before, $\mathbf{B} U(n)$ denotes the one-object groupoid with $U(n)$ worth of morphisms

Again, equipping that with a connection amounts to extending to a diagram

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U(n) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra}{\to}& \mathbf{B} E U(n) }$

of smooth 2-functors, where, also as before, $E U(n) := INN(U(n)) = (U(n) \stackrel{Id}{\to} U(n))$ is the inner automorphism 2-group of $U(n)$.

By the theorem in Smooth functors vs. differential forms, this diagram represents precisely a $U(n)$-bundle with connection.

Recall from the Lie picture that we want to complete further to

$\array{ CE(u(n)) \\ \uparrow \\ W(u(n)) \\ \uparrow \\ inv(u(n)) }$

with

$inv(u(n)) = H^\bullet( B U(n),\mathbb{R}) = \wedge^\bullet( u_2, u_4, u_6, \cdots ) \,.$

Hence $inv(g)$ can be regarded as the Chevalley-Eilenberg algebra of the Lie $\infty$-algebra

$b^1 u(1) \oplus b^3 u(1) \oplus b^5 u(1) \oplus \cdots$

The $\infty$-group integrating that is

$\Pi_\infty(X_{inv(g)}) = B U(1) \times B^{3} U(1) \times B^5 U(1) \times \cdots$

and hence we complete to

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U(n) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra_\nabla}{\to}& \mathbf{B} E U(n) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{ch(\nabla)}{\to}& \mathbf{B} \prod_i B^{2i+1} U(1) } \,.$

The bottom morphism picks up the characteristic classes, as always, which here is the Chern character.

While the top square lives in the world of strict smooth 2-categories and strict smooth 2-functors between them, the lower square needs to be read in weak $\infty$ something. I won’t attempt to discuss that in more detail and just assume we trust that this makes sense and exists as the integration of our corresponding Lie $\infty$ diagram

$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(u(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(u(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{ch(F_A)}{\leftarrow}& inv(u(n)) } \,.$

Then as we pass $U(n) \to U(\infty) = U$ we should get the “$n$-transport incarnation” of differential $K^0$-theory in that

$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U \times \{k \in \mathbb{Z}\} && (K-class) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra_\nabla}{\to}& \mathbf{B} E U && (connection) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{ch(\nabla)}{\to}& \mathbf{B} \prod_i B^{2i+1} U(1) && (Chern character) } \,.$

And this implies that the connection form on $Y$ itself is a sum of a bunch of (higher) Chern-Simons forms.

Sorry, that’s maybe not too shocking a statement in a way, but I thought it deserves to be said.

Another interesting thing to think about is whether things would prettify here if we’d modeled differential K-classes more explicitly in terms of $\mathbb{Z}_2$-graded vector bundles with *super*-connections on them.

I once chatted about how there is a nice functorial (parallel transport-like) way to think of the required superconnections here in the entry

Quillen’s superconnections – Functorially .

Plugging the observations made there into the formalism discussed here might lead to pleasing results…

## Re: (Generalized) Differential Cohomology and Lie Infinity-Connections