Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
@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.
@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.
1 to 6 of 6