### 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 $\mathbf{H}$ 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 $\mathbf{H} =$ 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.

## 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.