Contents

Contents

Context

In recent years, various notions and flavours of cohomology, of generalized cohomology and of nonabelian cohomology have been understood to play a fundamental role in theoretical fundamental physics. Notably, the refined notion of differential cohomology serves to model gauge fields (such as the electromagnetic field) that appear in the standard model of particle physics, and to model higher gauge fields that appear in string theory, which is a conjectured UV-completion of the standard model of particle physics and of Einstein’s theory of gravity.

As the name suggests, differential cohomology is a concept that is situated not in bare topology/homotopy theory as ordinary cohomology is, but that pairs homotopy theory with geometry and specifically with differential geometry. The combination of these two major strands of mathematics has in recent years found a long-expected working formalization in terms of what is alternatively called (∞,1)-topos theory or higher geometry or derived geometry.

Using this, there is now a natural mathematical formulation for many concepts that physicists in gauge theory have been developing for several decades already. For instance, what in physics is called a BRST complex can now naturally be understood as being the ∞-Lie algebroid underlying the smooth ∞-groupoid of cocycles in a suitable (∞,1)-topos with coefficients in the moduli ∞-stack of higher connections. Moreover, what in physics is called BV-BRST formalism is now understood to be the theory of derived critical loci in such BRST complexes.

Along these lines, some central aspects of string theory have usefully been illuminated by such geometric homotopy-type theory. A survey of some aspects is at:

Stabilization

However, the geometric refinements of classifying spaces to moduli ∞-stacks used in these applications have so far only been provided “non-stably”, that means: from the point of view of nonabelian cohomology as opposed to the stable context of generalized (Eilenberg-Steenrod) cohomology, which by the Brown representability theorem is represented by spectra instead of by just plains spaces.

For instance, type II string theory over D-branes is controled by a smooth refinement of K-theory. But while the traditional homotopy-theoretic classifying space of topological K-theory in degree 0 has an evident refinement to a smooth ∞-groupoid/smooth $\infty$-stack (essentially because vector bundles naturally form a stack) there is not to date a construction in the literature that would lift the entire K-theory spectrum to smooth higher geometry / derived differential geometry.

On the other hand, the general mathematical framework in which such a construction should take place has in recent been nicely been laid out: if a given geometric moduli ∞-stack is an object of some ambient (∞,1)-topos $\mathbf{H}$, then the corresponding geometric spectrum should be an object of the stabilization of that (∞,1)-topos to a stable (∞,1)-category. Moreover, by general results on these it is known that if $\mathbf{H}$ is the (∞,1)-category of (∞,1)-sheaves over some (∞,1)-site $C$, then its stabilization $Stab(\mathbf{H})$ arises by forming spectrum-valued $(\infty,1)$-sheaves.

Presentations

The problem now is that, while there is a large repository of known tools available for handling ∞-stacks = space-valued $(\infty,1)$-sheaves, not much is known about producing machinery for handling spectrum-valued sheaves.

One reason for this may be the following: while ∞-stacks are most conveniently presented by presheaves of simplicial sets, the available presentations for spectra used to all be very different in nature from simplicial sets: there was no good cellular model for spectra (like combinatorial spectra, for which however no good model category theory is known). This, however, has changed recently:

A cellular structure that generalizes simplices in a natural way to trees – called dendroidal sets – has been established as a good presentation for (∞,1)-operads in direct analogy to how simplicial sets present (∞,1)-categories. Moreover, there are now available left Bousfield localizations of the model structure on dendroidal sets and the model structure for dendroidal left fibrations which are presentations of the collection of connective spectra (these operadic localizations regard a connective spectrum essentially as its associated endomorphism operad).

Since the homotopy theory of dendroidal sets proceeds so very much in analogy with the presentation of higher category theory by simplicial sets, it seems natural to expect that the stabilization (and furthermore the full operadic generalization) of (∞,1)-topos theory has a similarly natural presentation by a model structure on homotopical presheaves with values in dendroidal sets.

Applications

The application of cohomology to string theory mentioned at the beginning crucially involves twisted cohomology. As discussed there, a coefficient object for twisted cohomology is given by a bundle in the ambient (∞,1)-topos. In the context of stable bare homotopy theory (no geometry involved), a nice formalization of this has been given in terms of (∞,1)-vector bundles: bundles of module spectra over E-∞ ring spectra – see the references listed here.

It seems clear that this approach is the natural and right one for stable twisted cohomology – in the discrete-geometric case. A pressing question therefore is:

What is a good generalization of this approach that can handle (∞,1)-sheaves of E-∞ ring spectra, or equivalently E-∞ ring spectra internal to the ambient (∞,1)-topos, as well as their sheaves of/internal bundles of module spectra? And does it, and if so, how does it have a nice presentation in terms of presheaves of dendroidal sets?

It seems that this should be useful for further investigation of the twisted differential K-theory that plays a central role in string theory, and also of the twisted tmf-theory that is expected to play a big role.

Last revised on June 2, 2012 at 22:42:12. See the history of this page for a list of all contributions to it.