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February 27, 2008

(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 nn-particles, usually known as NS-branes and D-branes and M-branes.

The idea is:

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

When Γ ()=H (,)\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 nn-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 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 “nn-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 nn-fields.

For ordinary differential cohomology (nn-gerbes with connection) this quantum theory of nn-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 Γ\Gamma, there is a natural homomorphism

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

which sends each generalized cohomology class of a space XX 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 Γ A^\bullet_\Gamma is the pullback of the diagram

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

namely

A Γ (X) Ω closed (X)Γ (pt) Γ (X) H (X,)Γ (pt). \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

(ω,F)Γ (X)×Ω closed (X)Γ (pt) (\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=ω F = \omega_{\mathbb{R}}

but may be just cohomologous

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

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

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

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

h=ω F \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 L_\infty-connections and nn-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

XS Γ X \to S_\Gamma

from XX into the spectrum representing the cohomology theory.

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

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

An nn-bundle on XX is a morphism

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

where GG is some nn-group.

For instance nnth integral cohomology is obtained by setting

G=B n1U(1) G = B^{n-1}U(1)

while K-theory is obtained by setting, essentially,

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

or similar.

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

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

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

This has to make the square

Y g BG Π n+1(Y) tra BEG \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)(n+1)-connected, we can reformulate this elegantly as

Π n+1 vert(Y) g BG Π n+1(Y) tra BEG \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

Ω vert (Y) A vert CE(g) Ω (Y) (A,F A) W(g) \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 L_\infty-connections and applications (pdf, blog, arXiv).

There these diagrams are completed to

Ω vert (Y) A vert CE(g) Ω (Y) (A,F A) W(g) Ω (X) {P i} inv(g) \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}\{P_i\} of the chosen connection which give the “differential” realization of the cohomology class gg.

Noticing that

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

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

integrates to

BG BEG BBG, \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 BGB G (which in general fails to have a group structure) with its rational approximation

BG kK(,f(k)), 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 nn-bundle (cohomology class) Π n+1 vert(Y)gBG \Pi_{n+1}^{vert}(Y) \stackrel{g}{\to} \mathbf{B} G

to an nn-bundle with connection

Π n+1 vert(Y) g BG Π n+1(Y) tra BEG Π (X) {P i} BBG \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}\{P_i\} representing our cohomology class in deRham cohomology.


Ordinary differential cohomology

Here is a more detailed description of the relation between nn-transport/L 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 (nn-gerbes with connection) one nice side-effect is that we can work entirely with strict nn-groupoids. Which makes many things more tractable.

For each nn, there is the strict nn-group

B n1U(1) B^{n-1} U(1)

which is trivial in each degree except of its topmost one, where it has U(1)U(1)-worth of (n1)(n-1)-morphisms.

I write

BB n1U(1) \mathbf{B} B^{n-1} U(1)

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

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

Y =(Y× XYs,tY) Y^\bullet = (Y \times_X Y \stackrel{s,t}{\to} Y)

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

Then the descent data for a B n1U(1)B^{n-1} U(1)-nn-bundle (= abelian (n1)(n-1)-gerbe) is a strict nn-functor

g:Y BB n1U(1) g : Y^\bullet \to \mathbf{B} B^{n-1} U(1)

and indeed equivalence classes of such nn-functors realize the nn-th integral cohomology of XX:

EquivClasses(Hom nGrpd(Y ,BB n1U(1)))=H n+1(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 nn-functorial restatement of the standard fact about Čech cohomology.

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

I write:

Π n+1(X) \Pi_{n+1}(X)

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

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

nFunct (Π n+1(X),BB nU(1))=Ω closed n+1(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 nn-functor

g:Y BB n1U(1) g : Y^\bullet \to \mathbf{B} B^{n-1}U(1)

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

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

To relate these, we need a connection on an nn-bundle whose integral class is given by gg and whose curvature is FF. This works as follows:

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

Π n+1(Y ) \Pi_{n+1}(Y^\bullet)

whose kk-morphisms are generated from

- globular kk-paths in YY

- together with “jumps” between kk-paths in the fiber of YY

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

More technically, this is the weak pushout

Π n+1(Y× XY) π 1 Π n+1(Y) π 2 Π n+1(Y) Π n+1(Y ) \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)C(Y) in 0705.0452).

Then: B n1U(1)B^{n-1}U(1)-nn-bundles with connection are given by smooth (n+1)(n+1)-functors

tra:Π n+1(Y )BEB n1U(1) \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 n1U(1)B^{n-1} U(1)-cocycle, meaning that they can be completed to a square

Y g BB n1U(1) Π n+1(Y ) tra BEB n1U(1). \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 EB n1U(1):=INN(B n1U(1))E B^{n-1} U(1) := INN(B^{n-1} U(1)) is the inner automorphism (n+1)(n+1)-group of B n1U(1)B^{n-1} U(1) and the vertical arrows are the canonical inclusions.

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

Y g BB n1U(1) (integralclass) Π n+1(Y ) tra BEB n1U(1) (connection) Π n+1(X) F BB nU(1) (curvature). \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 nn-connection

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

refines the integral class

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

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

FΩ closed n+1(X). 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 L_\infty-connections, I’ll now say something about differential K-theory from that point of view.

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

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

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

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

where U=U()U = U(\infty).

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

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

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

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

Y g BU(n) Π 2(Y) tra BEU(n) \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, EU(n):=INN(U(n))=(U(n)IdU(n))E U(n) := INN(U(n)) = (U(n) \stackrel{Id}{\to} U(n)) is the inner automorphism 2-group of U(n)U(n).

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

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

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

with

inv(u(n))=H (BU(n),)= (u 2,u 4,u 6,). inv(u(n)) = H^\bullet( B U(n),\mathbb{R}) = \wedge^\bullet( u_2, u_4, u_6, \cdots ) \,.

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

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

The \infty-group integrating that is

Π (X inv(g))=BU(1)×B 3U(1)×B 5U(1)× \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

Y g BU(n) Π 2(Y) tra BEU(n) Π (X) ch() B iB 2i+1U(1). \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

Ω vert (Y) A vert CE(u(n)) Ω (Y) (A,F A) W(u(n)) Ω (X) ch(F A) inv(u(n)). \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)U()=UU(n) \to U(\infty) = U we should get the “nn-transport incarnation” of differential K 0K^0-theory in that

Y g BU×{k} (Kclass) Π 2(Y) tra BEU (connection) Π (X) ch() B iB 2i+1U(1) (Cherncharacter). \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 YY 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 2\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…

Posted at February 27, 2008 7:24 PM UTC

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22 Comments & 9 Trackbacks

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

Have a look at http://arxiv.org/abs/0710.4340
Posted by: Eugene on February 27, 2008 9:48 PM | Permalink | Reply to this

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

Have a look at http://arxiv.org/abs/0710.4340

Interesting. Thanks a lot!

I just had a quick look at the beginning. Do I understand correctly that you are essentially saying:

it is not quite right to think of geometric quantization as something that involves just integral closed 2-forms and hence line bundles “with curvature”. Rather, it really involves line bundles with connection and its important to keep track of that extra bit of information.

Is that right? That would be a statement very close to my heart. I’ll try to have a closer look at your article later when I have more time.

Posted by: Urs Schreiber on February 28, 2008 9:46 AM | Permalink | Reply to this

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

You may consider my paper with Malkin as a long comment on Example 2.7 in Hopkins-Singer with an application to (pre)quantization of Deligne-Mumford stacks. Hopkins-Singer categorify Cheeger-Simons characters on a fixed manifold and argue that the resulting category of cocycles is equivalent to the category of principal bundles with connections. It’s a bit hard to write down this equivalence of categories explicitly for an arbitrary manifold, so they don’t. It is easy to write it when the manifold is a disk. So one can explicitly write down this equivalence on a good cover {U i}\{U_i\} of your manifold MM and all these functors should assemble into a functor between the two categories over MM. When can we pass in such a way from local to global equivalences of categories? When the two presheaves of categories are stacks. And this is indeed the case. So example 2.7 of Hopkins-Singer is saying:

prequantization is an isomorphism from the stack of Cheeger-Simon degree 2-cocycles to the the stack of principle S 1S^1 bundles with connections.

I expect that this point of view is useful in general. For example in thinking about nn-gerbs with connetions. But to turn this into mathematics one has to come to terms with descent for nn-stacks, which is something I am not comfortable with.

Posted by: Eugene on February 28, 2008 5:58 PM | Permalink | Reply to this

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

Hopkins-Singer categorify Cheeger-Simons characters on a fixed manifold

Not that it matters much in absolute terms, but I should say that this use of the word “categorify” is different from the way we usually use it here. What you mean is: they find the right category that a certain object lives in, I think.

and argue that the resulting category of cocycles is equivalent to the category of principal bundles with connections.

I’d thought the point of the example is to exhibit concretely the relation. It is a known theorem that Cheeger-Simons differential characters in degree (n+1)(n+1) are the same as Deligne cocycles in the same degree, hence the same as n1n-1-gerbes with connection. That includes ordinary line bundles with connectionfor n=1n=1.

It’s a bit hard to write down this equivalence of categories explicitly for an arbitrary manifold, so they don’t.

A good proof of the equivalence of Cheeger-Simons characters and Deligne cohomology is a little hard to find. The relation of Deligne cohomology to abelian nn-gerbes, though, is not so mysterious, though. But of course I guess one could argue that the literature on that can still be improved on.

But to turn this into mathematics one has to come to terms with descent for n-stacks, which is something I am not comfortable with.

With Konrad I am writing essentially this up for general (nonabelian) 2-bundles with connection. The general idea I have described in The first edge of the cube.

Konrad also has a nice article on the 2-category of gerbes with connection.

(I could have said the word “2-stack” here, but I have that tendency not to.)

Posted by: Urs Schreiber on February 28, 2008 6:17 PM | Permalink | Reply to this

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

Cheeger-Simons differential characters are cohomology classes, so Hopkins-Singer do categorify them by exhibiting a category whose equivalence classes of objects are the characters. Isn’t this what you would call a categorification?

I think this categorification is nice, because equivalence classes of bundles with connections are not as nice as actual bundles with connections. For instance, you can’t glue equivalence classes, but you can glue bundles

Posted by: Eugene on February 28, 2008 6:40 PM | Permalink | Reply to this

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

Cheeger-Simons differential characters are cohomology classes, so Hopkins-Singer do categorify them by exhibiting a category whose equivalence classes of objects are the characters.

Oh, now I see what you mean. I had probably been under the impression that this had been known before. But quite possibly I am wrong. At least, certainly for the equivalent Deligne cohomology the description has always been in terms of cocycles and coboundaries.

I think this categorification is nice,

Yes, certainly (now that I understand what you mean). It’s always a bad idea to pass to equivalence classes and forgetting where they came from.

Over on a previous recent entry here, Alessandro and me were talking about the \infty-groupoid of generalized differential characters.

Posted by: Urs Schreiber on February 29, 2008 9:52 AM | Permalink | Reply to this

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

Indeed, that’s the spirit of one approach to homotopy theory, esp. rational homotopy theory. there’s lot so info at the chain level that cohomology, even as a ring, can’t see, viz. Steenrod ops and Massey products.
A neat think about the rational case is that there is an A_infty ring structure on the cohomology making it quasi-iso the underlying dga.

Posted by: jim stasheff on February 29, 2008 12:27 PM | Permalink | Reply to this

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

Indeed, that’s the spirit of one approach to homotopy theory, esp. rational homotopy theory.

Sorry, could you remind me what you are referring to with this comment?

there’s lot so info at the chain level that cohomology, even as a ring, can’t see, viz. Steenrod ops and Massey products.

Am I right that you are referring to that step

BG iK(,f(i)) B G \mapsto \prod_i K(\mathbb{Q},f(i))

?

A neat think about the rational case is that there is an A A_\infty ring structure on the cohomology making it quasi-iso the underlying dga.

Oh, interesting. Could you say that again with more details? Which dga precisely do you have in mind here?

Posted by: Urs Schreiber on February 29, 2008 12:48 PM | Permalink | Reply to this

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

Hi Urs,

very interesting!
Let me add to the references

U.Bunke, T.Shick
Smooth K-theory
(Arxiv)

They have a different, yet equivalent formulation of differential K-theory, whose main ingredients are ‘geometric families’, which are the technical tools one uses in studying the index of families of Dirac operators.
Moreover, they show that the Chern character morphism between K-theory and rational cohomology lifts at the differential level.

Posted by: Alessandro on February 28, 2008 10:06 AM | Permalink | Reply to this

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

Thanks! I wasn’t aware of that.

Moreover, they show that the Chern character morphism between K-theory and rational cohomology lifts at the differential level.

Oh, I see. So that hasn’t been done in the Freed-Hopkins-Singer context?

Posted by: Urs Schreiber on February 28, 2008 10:27 AM | Permalink | Reply to this

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

Oh, I see. So that hasn’t been done in the Freed-Hopkins-Singer context?

No, as far as I know, but the result has been morally used in various contexts.

Posted by: Alessandro on February 28, 2008 10:44 AM | Permalink | Reply to this

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

Here is a more detailed description of the relation between nn-transport/L 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 (nn-gerbes with connection) one nice side-effect is that we can work entirely with strict nn-groupoids. Which makes many things more tractable.

For each nn, there is the strict nn-group

B n1U(1) B^{n-1} U(1)

which is trivial in each degree except of its topmost one, where it has U(1)U(1)-worth of (n1)(n-1)-morphisms.

I write

BB n1U(1) \mathbf{B} B^{n-1} U(1)

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

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

Y =(Y× XYs,tY) Y^\bullet = (Y \times_X Y \stackrel{s,t}{\to} Y)

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

Then the descent data for a B n1U(1)B^{n-1} U(1)-nn-bundle (= abelian (n1)(n-1)-gerbe) is a strict nn-functor

g:Y BB n1U(1) g : Y^\bullet \to \mathbf{B} B^{n-1} U(1)

and indeed equivalence classes of such nn-functors realize the nn-th integral cohomology of XX:

EquivClasses(Hom nGrpd(Y ,BB n1U(1)))=H n+1(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 nn-functorial restatement of the standard fact about Čech cohomology.

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

I write:

Π n+1(X) \Pi_{n+1}(X)

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

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

nFunct (Π n+1(X),BB nU(1))=Ω closed n+1(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 nn-functor

g:Y BB n1U(1) g : Y^\bullet \to \mathbf{B} B^{n-1}U(1)

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

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

To relate these, we need a connection on an nn-bundle whose integral class is given by gg and whose curvature is FF. This works as follows:

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

Π n+1(Y ) \Pi_{n+1}(Y^\bullet)

whose kk-morphisms are generated from

- globular kk-paths in YY

- together with “jumps” between kk-paths in the fiber of YY

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

More technically, this is the weak pushout

Π n+1(Y× XY) π 1 Π n+1(Y) π 2 Π n+1(Y) Π n+1(Y ) \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)C(Y) in 0705.0452).

Then: B n1U(1)B^{n-1}U(1)-nn-bundles with connection are given by smooth (n+1)(n+1)-functors

tra:Π n+1(Y )BEB n1U(1) \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 n1U(1)B^{n-1} U(1)-cocycle, meaning that they can be completed to a square

Y g BB n1U(1) Π n+1(Y ) tra BEB n1U(1). \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 EB n1U(1):=INN(B n1U(1))E B^{n-1} U(1) := INN(B^{n-1} U(1)) is the inner automorphism (n+1)(n+1)-group of B n1U(1)B^{n-1} U(1) and the vertical arrows are the canonical inclusions.

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

Y g BB n1U(1) (integralclass) Π n+1(Y ) tra BEB n1U(1) (connection) Π n+1(X) F BB nU(1) (curvature). \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 nn-connection

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

refines the integral class

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

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

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

Using other structure nn-groups G nG_n instead of the simple B n1U(1)B^{n-1}U(1) leads to other differential cohomologies, I think.

Posted by: Urs Schreiber on February 28, 2008 11:13 AM | Permalink | Reply to this

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

Slow down! Most readers probably follow anyway but notice:
which sends each generalized cohomology class of a space X to a differential form representing it.

patently should be to a class which need not know anything about differentials

but three lines later you say exactly what the idea is

then
(This is slightly oversimplifying, since it does not explicitly involve the connection itself, just its curvature form.)

but that’s a perfectly good, complete translation of the idea in words


getting there via connections may be one way to do it
is it what Freed Hopkins et al do? or do
they do something which can be translated into that language?

Posted by: jim stasheff on February 28, 2008 2:56 PM | Permalink | Reply to this

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

getting there via connections may be one way to do it is it what Freed Hopkins et al do?

Yes.

The pullback diagram which I recalled

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

really has to be read as a weak pullback (I suppose you would say homotopy pullback or the like (?)).

That means that as we chase a pair

(ω,F)Γ (X)×Ω closed (X)Γ (pt) (\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=ω F = \omega_{\mathbb{R}}

but may be just cohomologous

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

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

Posted by: Urs Schreiber on February 28, 2008 3:30 PM | Permalink | Reply to this

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


really has to be read as a weak pullback (I suppose you would say homotopy pullback or the like (?)).


That means that as we chase a pair


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

yes, they are =
notice the lower right corner

also you indeed want an h as you say
but why call it a connection
and why call F a curvature

jsut a resemblance?


Posted by: jim stasheff on February 28, 2008 5:57 PM | Permalink | Reply to this

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

but why call it a connection and why call FF a curvature

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

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

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

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

Posted by: Urs Schreiber on February 28, 2008 6:28 PM | Permalink | Reply to this

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

If Ω closed \Omega^\bullet_{closed} was a cohomology theory we could of course take a homotopy pullback of spectra, then apply Hom(X,)Hom(X,-). I suspect that the homotopy pullback occurs in a category where we have forgotten something about spectra then find the resulting object is in the image of the forgetful functor U:SpectrathingsU: \mathbf{Spectra} \to things. Transgression of forms (from XX to ΩX\Omega X) would play a role in this interpretation.

Posted by: David Roberts on February 29, 2008 12:17 AM | Permalink | Reply to this

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

I suspect that the homotopy pullback occurs in a category where we have forgotten something

Good point. I need to see if either

a) I can track down a more explicit description of this pullback idea in the literature

or

b) (what I’d rather like to do) figure out how the diagram

Π vert(Y) g BG Π(Y) BEG =BINN 0(G) Π(X) F B iB f(i)U(1), \array{ \Pi^{vert}(Y) &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{B} E G & = \mathbf{B} INN_0(G) \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{F}{\to}& \mathbf{B} \prod_i B^{f(i)} U(1) } \,,

which I keep claiming also exhibits the differential refinement of the cocycle

gg by differential form data FF via a connection \nabla can be thought of as arising from a pullback.

By the way: it’s fun to see how INN(G)INN(G) is the home of the coboundary relation “gF=g-F = \partial \nabla”.

When you have time, we should finally write up the second part of those notes we started with, showing that inserting G=AUT(H)G = AUT(H) into the above diagram reproduces the Breen-Messing differential cocycles. It’s true. It just needs to be written up cleanly and comprehensively. (And maybe some other gems can be found along the way.)

Posted by: Urs Schreiber on February 29, 2008 10:04 AM | Permalink | Reply to this

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

When you have time, we should finally write up the second part of those notes we started with,

for sure. I’ve been meaning to ask you about that, as perhaps some of the anomolous terms, qq (Proposition 3 of the original notes) and f 1,f˜f_1,\tilde{f} (Proposition 4 loc. cit.) will go away when we polish this up. With L L_\infty-wisdom under your belt I’m sure things will fit more smoothly to Breen-Messing.

Posted by: David Roberts on February 29, 2008 12:10 PM | Permalink | Reply to this

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

as perhaps some of the anomolous terms […] will go away

Oh, I should have said that more explicitly: yes, these terms disappear.

When I wrote these notes originally, I hadn’t understood yet that the INN(AUT(G))INN(AUT(G))-valued 3-functor

tra:Π 3(Y)BINN(AUT(G)) tra : \Pi_3(Y) \to \mathbf{B} INN(AUT(G))

has to be regarded as the generalized curvature 3-functor of a 2-transport. Instead I regarded it as a 3-transport itself.

Then later I realized that it has to be fit into that diagram

Y g BAUT(G) Π 3(Y ) tra BINN(AUT(G)) \array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} AUT(G) \\ \downarrow && \downarrow \\ \Pi_{3}(Y^\bullet) &\stackrel{tra}{\to}& \mathbf{B} INN(AUT(G)) }

This says that while the 3-functor evaluated on 3-paths takes values in all of INN(AUT(G))INN(AUT(G)), its gluing data on multiple overlaps has be just an ordinary AUT(G)AUT(G) 2-cocycle. It’s precisely those “anomalous terms” in my original notes which vanish by this requirement (this I hadn’t understood back then).

So the statement really is: diagrams as above are equivalent to Breen-Messing differential cocycles.

In other words, the statement is that any G (n)G_{(n)}-bundle with connection is a pullback of the universal G nG_n-bundle

G (n) EG (n)=INN(G (n)) BG (n) \array{ G_{(n)} \\ \downarrow \\ E G_{(n)} = INN(G_{(n)}) \\ \downarrow \\ \mathbf{B} G_{(n)} }

in its nn-groupoid incarnation, and that this can be thought of as the pullback of the universal G (n)G_{(n)}-bundle with universal connection, due to the fact that the universal connection on that thing is the tautological diagram

G (n) Id G (n) EG (n) Id EG (n) BG (n) Id BG (n) \array{ G_{(n)} &\stackrel{Id}{\to}& G_{(n)} \\ \downarrow && \downarrow \\ E G_{(n)} &\stackrel{Id}{\to}& E G_{(n)} \\ \downarrow && \downarrow \\ \mathbf{B} G_{(n)} &\stackrel{Id}{\to}& \mathbf{B} G_{(n)} }

(as in figure 11, p. 68)

With L L_\infty-wisdom under your belt I’m sure things will fit more smoothly to Breen-Messing.

Yes, the claim in the the L L_\infty version of all this is much more easily checked, as the need for evaluating these huge diagrams of pseudonatural transformations of 3-functor at the end of those notes is replaced by an easy DGCA computation.

I am hoping that eventually we understand the Lie \infty-integration theory well enough to be able to simply say that a Breen-Messing cocycle is simply the integration of the corresponding aut(g)aut(g)-connection

(Ω vert (Y) A vert CE(aut(g)) Ω (Y) (A,F A) W(aut(g)) =CE(inn(aut(g))) Ω (X) inv(aut(g))) \int \left( \array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(aut(g)) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(aut(g)) & = CE(inn(aut(g))) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{}{\leftarrow}& inv(aut(g)) } \right)

It’s essentially clear how this works and that it is true, but care needs to be exercised with a coupld of technicalities here.

As I mentioned, I think this \int-sign here is the composite of first applying the functor

Hom(,Ω ()):DGCAssmoothspaces Hom(--, \Omega^\bullet(-)) : DGCAs \to smooth spaces

and then the functor

Π 3:smoothspacessmoothGraygroupoids \Pi_3 : smooth spaces \to smooth Gray groupoids

(Ω vert (Y) A vert CE(aut(g)) Ω (Y) (A,F A) W(aut(g)) Ω (X) inv(aut(g)))=Π 3Hom(,Ω ())(Ω vert (Y) A vert CE(aut(g)) Ω (Y) (A,F A) W(aut(g)) Ω (X) inv(aut(g))) \int \left( \array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(aut(g)) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(aut(g)) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{}{\leftarrow}& inv(aut(g)) } \right) = \Pi_3 \circ Hom(-,\Omega^\bullet(-)) \left( \array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(aut(g)) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(aut(g)) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{}{\leftarrow}& inv(aut(g)) } \right)

What I need to figure out is where that integration procedure sees the need for Gray groupoids arise, where naively it would seem that we can even integrate to strict 3-groupoids (which however can’t be right without further qualification, since we know that our INN(AUT(G))INN(AUT(G)) is a Gray group, not a strict 3-group).

This is the issue I mentioned here in the discussion of ωCat\omega Cat-enriched categories.

Posted by: Urs Schreiber on February 29, 2008 12:40 PM | Permalink | Reply to this

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

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

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

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

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

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

where U=U()U = U(\infty).

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

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

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

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

Y g BU(n) Π 2(Y) tra BEU(n) \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, EU(n):=INN(U(n))=(U(n)IdU(n))E U(n) := INN(U(n)) = (U(n) \stackrel{Id}{\to} U(n)) is the inner automorphism 2-group of U(n)U(n).

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

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

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

with

inv(u(n))=H (BU(n),)= (u 2,u 4,u 6,). inv(u(n)) = H^\bullet( B U(n),\mathbb{R}) = \wedge^\bullet( u_2, u_4, u_6, \cdots ) \,.

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

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

The \infty-group integrating that is

Π (X inv(g))=BU(1)×B 3U(1)×B 5U(1)× \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

Y g BU(n) Π 2(Y) tra BEU(n) Π (X) ch() B iB 2i+1U(1). \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

Ω vert (Y) A vert CE(u(n)) Ω (Y) (A,F A) W(u(n)) Ω (X) ch(F A) inv(u(n)). \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)U()=UU(n) \to U(\infty) = U we should get the “nn-transport incarnation” of differential K 0K^0-theory in that

Y g BU×{k} (Kclass) Π 2(Y) tra BEU (connection) Π (X) ch() B iB 2i+1U(1) (Cherncharacter). \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 YY 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 2\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…

Posted by: Urs Schreiber on February 28, 2008 5:04 PM | Permalink | Reply to this
Read the post (Generalized) Differential Cohomology and Lie Infinity-Connections
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Excerpt: On generalized differential cohomology and its relation to infinity-parallel transport and Lie-infinity connections.
Tracked: February 28, 2008 5:17 PM
Read the post Charges and Twisted n-Bundles, I
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Excerpt: Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
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Re: (Generalized) Differential Cohomology and Lie Infinity-Connections

I am preparing some preliminary slides:

Nonabelian differential cohomology (31 pdf slides)

Posted by: Urs Schreiber on March 11, 2008 11:02 PM | Permalink | Reply to this
Read the post Chern-Simons Actions for (Super)-Gravities
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Excerpt: On Chern-Simons actions for (super-)gravity.
Tracked: March 13, 2008 11:18 AM
Read the post Slides: On Nonabelian Differential Cohomology
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Excerpt: On the notion of nonabelian differential cohomology.
Tracked: March 13, 2008 3:37 PM
Read the post Nonabelian Differential Cohomology in Street's Descent Theory
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Excerpt: A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
Tracked: March 22, 2008 7:55 PM
Read the post HIM Trimester on Geometry and Physics, Week 4
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Excerpt: Talk in Stanford on nonabelian differential cohomology.
Tracked: May 29, 2008 11:30 PM
Read the post Twisted Differential String- and Fivebrane-Structures
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Excerpt: An article on twisted differential nonabelian cohomology and its application to anomaly cancellation in string theory.
Tracked: March 20, 2009 11:20 PM

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