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February 28, 2011

Differential Cohomology in a Cohesive Topos

Posted by Urs Schreiber

The last months I have been busy with writing up a kind of thesis on the topic that I have been thinking about since a long time. This here is to present what should roughly be a version 1.0, up to proof-reading.

A pdf with the current version is behind the first link on this page, which is also the title of the opus:

Differential cohomology in a cohesive \infty-topos.

Below the fold is the abstract. I’d be grateful for whatever comments you might have.

I haven’t written an acknowledgement yet, because it is always hard to decide where to stop thanking people. But two names must be mentioned: I greatly profited from discussion with Mike Shulman on all general abstract aspects and from discussion with Domenico Fiorenza on the decisive aspects of concrete implementation.

Lengthy Abstract

We formulate differential cohomology and Chern-Weil theory – the theory of connections on bundles and of gauge fields – abstractly in the context of a certain class of ∞-toposes H\mathbf{H} that we call cohesive . Cocycles in this differential cohomology classify principal ∞-bundles equipped with cohesive structure (topological, smooth, synthetic differential etc.) and equipped with ∞-connections.

We construct the cohesive \infty-topos H=\mathbf{H} = Smooth∞Grpd of smooth ∞-groupoids and ∞-Lie algebroids and show that in this context the abstract theory reproduces ordinary differential cohomology (Deligne cohomology/differential characters), ordinary Chern-Weil theory, the traditional notions of smooth principal bundles with connection, abelian and nonabelian gerbes/bundle gerbes with connection, principal 2-bundles with 2-connection, connections on 3-bundles, etc., and generalizes these to base spaces that are orbifolds and generally smooth ∞-groupoids, such as smooth realizations of classifying spaces/moduli stacks for principal \infty-bundles and smooth \infty-groupoids of configuration spaces of higher gauge theories.

We exhibit a general abstract ∞-Chern-Weil homomorphism and observe that this generalizes the Lagrangian of Chern-Simons theory to ∞-Chern-Simons theory in that for every transgressive invariant polynomial on an ∞-Lie algebroid it sends principal ∞-connections to Chern-Simons nLab:circle n-bundles with connection whose higher parallel transport is the corresponding higher Chern-Simons action functional. There is a general abstract formulation of the higher holonomy of this parallel transport which realizes the action functional of ∞-Chern-Simons theory as a morphism on its cohesive configuration \infty-groupoid.

We show that in Smooth∞Grpd this construction reproduces the ordinary Chern-Weil homomorphism and refines it to cases such as the following. For the ordinary Killing form on a semisimple Lie algebra the \infty-Chern-Weil homomorphism yields the Chern-Simons circle 3-bundle controlling ordinary Chern-Simons theory. For the Killing form on the string Lie 2-algebra and the fivebrane Lie 6-algebra it yields differential fractional refinements of the first two Pontryagin classes whose homotopy fibers define twisted differential string structures and twisted differential fivebrane structures, respectively, that control the Green-Schwarz mechanism in heterotic string theory and dual heterotic string theory. For the canonical invariant polynomial on a strict Lie 2-algebra over a semisimple Lie algebra it yields the action functional of BF-theory coupled to topological Yang-Mills theory with cosmological constant. For the the canonical polynomial on the supergravity Lie 6-algebra it yields 11-dimensional Chern-Simons supergravity. For any symplectic Lie n-algebroid it yields the corresponding AKSZ theory action functional, such as in lowest degree the Poisson sigma model and the Courant sigma model.

Posted at February 28, 2011 11:04 PM UTC

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Re: Differential Cohomology in a Cohesive Topos

I have the most trivial comment in the world. It concerns the second sentence of the introduction: “leisurely” is an adjective, not an adverb, despite the -ly ending.

Posted by: Tom Leinster on March 1, 2011 4:03 AM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

“leisurely” is an adjective, not an adverb,

Oh, okay, I am changinging it to “We give a leisurely survey…”, thanks.

But this means my esteemed online dictionary is playing tricks on me: here it does claim that “leisurely” serves both as an adjective and an adverb.

What is a bit weird is the result of googling “leisurely survey”. On my system (is that just my cookies?) the second hit is differential cohomology in a cohesive topos .

Then every second hit for a total of a few dozens advertizes apartments that allow one to “leisurely survey the skyline”.

Hm, looking further through the Google hits, there is something like an estimated 4:1 ratio of usage as adjective versus usage as adverb. Not sure what that means. I’ll be changing it, thanks again.

Posted by: Urs Schreiber on March 1, 2011 8:00 AM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

I vote for both adjective and adverb
but I’m on the opposite side of the pond
from Tom.

Posted by: jim stasheff on March 1, 2011 1:55 PM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

My mistake then; sorry. Even the Oxford English Dictionary agrees that it can be an adverb. It just sounded wrong to my ear.

Anyway, here’s hoping someone makes a more substantial contribution to this thread…

Posted by: Tom Leinster on March 1, 2011 8:04 PM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

If we see typos, e.g.,

p. 4 of pdf

Lawvere’s proposals [Law07] for axiomatic characteriztions…

missing ‘a’,

should we just edit the webpage ourselves?

And then what of comments about style? E.g., my historian of physics colleagues would shudder with the anachronism of

Around 1850 Maxwell realized that the field strength of the electromagnetic field is modeled by a closed differential 2-form on spacetime.

The addition of ‘Anachronistically speaking,…’ or even ‘From our current perspective,…’ might placate them.

Posted by: David Corfield on March 3, 2011 10:14 AM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

If we see typos, e.g.,

Thanks, David. A colleague remarked that he would be willing to provide a list of trivial typos, but not at the present stage, with there still being so many of them… For which I apologize. But all help is appreciated.

should we just edit the webpage ourselves?

That would in any case be a great service, not only for me but for everyone exposed to these webpages. But since I need to fix the typos also in my LaTeX-mirror file of these webpages, please tell me about typos.

And then what of comments about style?

I’d be most interested in hearing them. And don’t mind if you go ahead and edit the relevant web-pages accordingly. If I would mind, I should put these pages on a write-protected wiki. But please do alert me of whatever changes you make on nnLab pages, best by using the Latest Changes on the nnForum

my historian of physics colleagues would shudder with the anachronism

I see. This appears at Higher category theory and physics – Gauge Theory – Introduction here. I have changed it to

Around 1850 Maxwell realized that the field strength of the electromagnetic field is modeled by what today we call a closed differential 2-form on spacetime.

Thanks again!

Posted by: Urs Schreiber on March 3, 2011 1:26 PM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

Freeman Dyson’s Missed Opportunities
has some pointed comments about the (non)-appreciation of Maxwell’s work by mathematicians of his time.

Posted by: jim stasheff on March 4, 2011 2:10 PM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

I can’t remember who, but some physicist said that by 1900, most physicists said Maxwell’s equations were well-understood. But only later did people discover

1) special relativity

and

2) gauge theory

both of which were sitting concealed in these equations.

This suggests that the things we already know about physics probably have a lot left to teach us. And indeed, you could say Urs is extracting some of those lessons here.

Is this stuff going into that book you were talking about, Urs?

Posted by: John Baez on March 11, 2011 11:18 AM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

Is this stuff going into that book you were talking about, Urs?

Do you mean the book

Mathematical Foundations of Quantum Field and Perturbative String Theory

?

If so, no, I am a co-editor of this book (with Brano and Hisham), but not an author. Maybe something like the text that I wrote at that webpage will become a foreword of that book, I don’t know yet.

By the way, on short notice recently, Frédéric Paugam agreed to contribute parts of his

Towards the mathematics of quantum field theory (pdf)

as far as I know the part about variational calculus and BV-theory with Beilinson-Drinfeld-style D-module technology.

That’s the next on my to-do list: BV-theory for the \infty-Chern-Simons action functionals in a cohesive \infty-topos. But apart from a first rough note on derived critical loci I am not there yet.

Trouble is I am still busy with proof-reading and polishing the document in its current state. I am now half-way through, at least…

Posted by: Urs Schreiber on March 29, 2011 5:01 PM | Permalink | Reply to this

Re: Differential Cohomology in a Cohesive Topos

Prompted by a question that David Spivak once asked here on MO I have tried to write an exegesis of Lawvere’s 1990 Como-conference preface:

which is secretly all about the notion of cohesive toposes. See behind the above link.

Posted by: Urs Schreiber on April 28, 2011 9:26 PM | Permalink | Reply to this

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