### Twisted Differential Nonabelian Cohomology

#### Posted by Urs Schreiber

This is something we are currently working on, various aspects of which have been the subject of recent discussion here:

Hisham Sati, U. S., Zoran Škoda, Danny Stevenson
*Twisted differential nonabelian cohomology*
*Twisted $(n-1)$-brane $n$-bundles and their Chern-Simons $(n+1)$-bundles with characteristic $(n+2)$-classes*

(pdf,
60 pages theory, 40 pages application currently (but still incomplete))

**Abstract.** We introduce nonabelian differential cohomology classifying
$\infty$-bundles with smooth connection
and their higher gerbes of sections, generalizing [SWIII].
We construct classes of examples of these from lifts,
twisted lifts and obstructions to lifts through shifted central
extensions of groups by the shifted abelian $n$-group $\mathbf{B}^{n-1}U(1)$.
Notable examples are String 2-bundles [BaSt]
and Fivebrane 6-bundles [SSS2].
The obstructions to lifting ordinary principal bundles
to these, hence in particular the obstructions to lifting $\mathrm{Spin}$-structures
to $\mathrm{String}$-structures
and further to
$\mathrm{Fivebrane}$-structures [SSS2, DHH],
are abelian Chern-Simons 3- and 7-bundles
with characteristic class the first and second fractional Pontryagin class,
whose abelian cocycles have been constructed explicitly by
Brylinski and McLaughlin [BML].
We realize their construction as an
abelian component of obstruction theory in nonabelian cohomology by
$\infty$-Lie-integrating the $L_\infty$-algebraic data in [SSS1].
As a result, even if the lift fails,
we obtain *twisted* String 2- and *twisted* Fivebrane 6-bundles
classified in twisted nonabelian (differential) cohomology and
generalizing the twisted bundles appearing in twisted K-theory.
We explain the Green-Schwarz mechanism in heterotic string theory in terms of
twisted String 2-bundles and its magnetic dual version in terms of twisted Fivebrane
6-bundles.
We close by transgressing differential cocycles to mapping spaces, thereby
obtaining their volume holonomies, and show that for Chern-Simons cocycles
this yields the action functionals for Chern-Simons theory and its higher
dimensional generalizations, regarded as extended quantum field theories.

This is work in progress, but maybe already of interest, given the recent discussions here. Comments, especially critical comments, are very welcome and there is room for further collaboration, clearly. There are various aspects here which deserve to be developed further for their own sake.

Here is a quick summary of the global idea from p. 6:

We proceed as follows.

To set ourselves up in a suitably general context of differential geometry we model ${Spaces}$ as sheaves on ${CartesianSpaces}$, with ${DiffeologicalSpaces}$ [BaHo] and ${SmoothManifolds}$ contained as subcategories of tame objects.

To have manifest close contact to familiar constructions in
homological algebra, differential
geometry and physics which we want to reproduce and generalize,
the model for $\infty$-categories which we choose
is *strict* $\infty$-categories, known as $\omega{Categories}$.
This turns out to be not only convenient, admitting all tools of
*nonabelian algebraic topology* [BrHiSi],
but also sufficient.

To handle the homotopy theoretic context of $\omega$-categories internal
to ${Spaces}$, equivalently: $\omega$-category valued sheaves,
$\omega {Categories}({Spaces}) \simeq
{Sheaves}({CartesianSpaces}, \omega{Categories})
\,,$
we obtain from the known homotopy model category structure on
$\omega{Categories}$ [BrGo, LaMeWo]
by stalkwise refinement the structure of a
*category of fibrant objects* [K.-S. Brown].
This yields a homotopy (bi-)category
$\mathbf{Ho}(\omega{Categories}({Spaces}))$ whose Hom-spaces
realize cohomology in this context, analogous to [Jardine:CocycleCats].

To establish contact with ordinary abelian Čech cohomology with coefficients in complexes of sheaves of abelian groups we consider descent for $\omega$-category valued presheaves [Street] and the corresponding notion of $\omega$-stacks. Dually this leads to a notion of codescent for $\omega$-category valued co-presheaves, which serves to translate from cohomology in terms of descent to cohomology in terms of the homotopy category.

In this context we set up our central definition of twisted differential nonabelian cohomology:

– **nonabelian cohomology** for structure $\omega$-group $G$
is cohomology with coefficients in
$\mathrm{hom}(\Pi(-),\mathbf{B}G)$,
for $\Pi$ an $\omega$-category valued copresheaf;

– **twisted cohomology**
is a refinement of the obstruction to lifting
of cocycles
through extensions $\mathbf{B} \hat G \to \mathbf{B}G$

– **differential cohomology**
is a refinement of the obstruction to
the extension of cocycles
along the inclusion of copresheaves
$\mathcal{P}_0(-) \hookrightarrow \Pi_\omega(-)$
of discrete
$\omega$-groupoids into
fundamental $\omega$-groupoids

As a tool for explicitly constructing
(twisted, differential) nonabelian cocycles
we describe $\infty$-Lie theory of smooth $\omega$-groupoids and
Lie $\infty$-algebroids,
following [Severa, Getzler, Henriques]
as the theory of two consecutive adjunctions
relating $\omega{Groupoids}({Spaces})$
to ${Spaces}$ and ${Spaces}$
to $L_\infty {Algebroids}$. Both adjunctions are examples of Stone
dualities induced by ambimorphic objects.
The first adjunction is induced by the
*object of finite paths*, $\Pi_\omega(-)$, while the second is induced by the
*object of infinitesimal paths*, $\Omega^\bullet(-)$

Using these adjunctions we $\infty$-Lie integrate the $L_\infty$-algebraic cocycles from [SSS1] from $L_\infty{Algebroids}$ to $\omega {Groupoids}({Spaces})$ to obtain nonabelian differential cocycles when certain integrability conditions are met .

Applied to the String- and Fivebrane- $L_\infty$-algebraic cocycles and their Chern-Simons obstructions of [SSS1] this yields an explicit construction of twisted differential cocycles representing twisted String 2- and twisted Fivebrane 6-bundles with connection. The twist itself is the obstruction to obtaining untwisted such bundles and lives in abelian Deligne cohomology where it represents Chern-Simons connections and their higher analogs.

Finally we transgress the differential cocycles thus obtained to mapping spaces and show that the transgressed differential cocycles exhibit the \underline{holonomy} [SWII, SWIII] and can be interpreted as the action functionals for extended Chern-Simons quantum field theories.

## Re: Twisted Differential Nonabelian Cohomology

Hi Urs,

That article looks like a very self-contained didatic book, with subjects properly listed on the index and solved exercises at the end. Can I understand it as such?