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 -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 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 -topos 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 -bundles and smooth -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 -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 -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.