@Fred,

it arose from wondering whether there was an algebraic version of the basic gerbe on an algebraic group of an appropriate type (corresponding to what is in the real case the generator of H^3 for compact simple simply connected Lie groups). I asked on MO, and was told there is no such thing, since all gerbes for the etale site are torsion, at least over such schemes that algebraic reductive groups are examples of. However, the basic gerbe *is* holomorphic, by work of Brylinski. So log geometry is the thing to try. Urs mentioned that really what’s going on is that there is a 2-gerbe on the classifying stack of the group (in the real case), and in fact this 2-gerbe is holomorphic as well (I don’t think this has been previously noted: it’s joint between myself and with Raymond Vozzo otherwise). Of course, then one would hope to get an log-geometric 2-gerbe on the classifying stack.

There was also a bunch of other stuff about how $Hom(\pi_1(\mathbb{C}^\times),G) = G//G^{ad}$, and the basic gerbe on $G$, which is equivariant for the adjoint action (hence lives on $G//G^{ad}$), can be generalised to other moduli stacks of flat bundles over Riemann surfaces/algebraic curves, using spaces of flat connections and central extensions of gauge groups - things that should make sense in Urs’ “intergeometric” setup.

]]>I see... This is interesting. I am actually trying to define a global analytic version of DB-cohomology. I need for this to use logarithmic geometry (because it seems necessary in the global analytic setting). Since DB cohomology is defined using logarithmic forms, i agree with you that its natural setting is log-geometry...

Could you explain me more precisely this discussion with Urs?

Analytic motivic cohomology gives a nice global replacement of Betti cohomology, and the problem to define a global version of DB cohomology (which may be called motivic-Arakelov cohomology) is to combine it properly with the Hodge filtration (using a king of global regulator, which is actually a Chern character with values in Hodge-filtered de Rham cohomology). ]]>

@Fred

In discussion with Urs in Edinburgh I mentioned that Brylinski’s construction of the basic gerbe on SL_2(C) looks manifestly like it should be defined in logarithmic geometry. It’s not given by a class in étale cohomology, since that cohomology group is torsion. More generally I believe that Deligne-Beilinson cohomology should be able to be defined for any log-scheme.

The regulator map from algebraic K_2 lands in DB cohomology. It is known that the class of the 2-gerbe on BG lifts to a class in K_2, so there should be *something* over Z, just in log-geometry.

There is a natural morphism $BU(n)\to BGL_n^{an}$. If the $2$-gerbe may be defined without using denominators, you should be able to define it on $BGL_n$ over $Z$ in the algebraic setting (i think this is something you told me about Hopkins' and others approach in algebraic geometry over $Z$, in that parisian coffee at place de la Nation; is this related to the $K_2$-valued cohomology of Brylinski-Deligne?). You can then take the associated non-strict global analytic gerbe, and pull it back to $BU(n)$. The problem would be to show that one may define the obtained $2$-gerbe in a strict way over $BU(n)$. I have difficulties to read Brylinski in the text because it is very explicit... I would prefer a viewpoint in an $\infty$-topos on the construction of this gerbe. How is the same thing done by Hopkins in algebraic geometry?

I think the right track to see if things work in global analytic geometry is first to understand how things work in algebraic geometry over the base ring $Z$. If we know that, it is easy to add the archimedean norm, normally, whenever this is possible. Otherwise, we will also see easily that it doesn't work.

So how do things work algebraically over $Z$? ]]>

Thanks for this update! I see, hm, will still have to think about this.

That reminds me finally that I promised to give you the links on the complex analytic version of the Chern-Simons 2-gerbe:

the relevant articles by Brylinski are listed here. In particular in arXiv:0002158 he gives an explicit Cech cocycle for the holomorphic gerbe on a complex reductive Lie group $G$.

David Roberts points out that this holomoprhic gerbe on $G$ should still be multiplicative (as is well known for its differential geometric incarnation) and would hence give a 2-gerbe on $\mathbf{B}G$. That’s what we’d be interested in for Chern-Simons theore. But for the moment the first step would be to see if the gerbe on $G$ has any global analytic incarnation, on a suitable subgroup of $G$. Maybe for $G = GL_n$ the “subgroup” $U(n)$ that you discuss now might do. That would be nice.

]]>One may thus define $BU(n)$ but the naive space $U(n)$ is not a group.

In any case, one can still work with $BU(n)$ as if it were a classifying stack, i guess...

It would be interesting to see if some of the ideas of geometric Langlands still make sense in this setting. Any comments on this?

And on the possibility of doing higher Chern-Simons using this strange $BU(n)$, Urs? ]]>