Ulrich Bunke talks about joint work with David Gepner.
The claim here is: a deep theorem by Beilinson about the “Borel regulator” for the cyclotomic field which may look a bit mysterious, apparently has a natural interpretation in the differential cohomology refinement of the corresponding algebraic Ktheory.
Let $R$ be any number ring. The running example to keep in mind is the the cyclotomic field
$R = \mathbb{Z}/ (1 + \xi + \cdots + \xi^{p1})
\,.$
Write then
$B GL(R)$
for the classifying space for $R$module bundles and
$B GL(R) \stackrel{Quillen constr}{\hookrightarrow}
(B GL(R))^+$
for the Quillen construction, such that for
$K R$ the
algebraic Ktheory
spectrum, the one whose homotopy groups are the algebraic Ktheory groups $K_i(R)$
$K_i(R) \simeq \pi_i (B GL(R))^*$
we have
$\Omega^\infty K R \simeq B GL(R)^+ \times K_0(R)
\,.$
Now for any
choice $\sigma : R \hookrightarrow \mathbb{C}$, which for the running example: would be given by $\xi \mapsto \mu \in U(1)$ a $p$th root of unity, and a bundle $V \otimes_\sigma \mathbb{C} \to B GL(R)$, an observation by S. Götte gives a corresponding bundle $\mathcal{V}$ over the +construction fitting into
$\array{
V \otimes_\sigma \mathbb{C} &\to& \mathcal{V}
\\
\downarrow && \downarrow
\\
B GL(R) &\hookrightarrow& (B GL(R))^+
}$
and such that $\mathcal{V}$ is locally filtered.
The Borel regulator construction starts with a Kclass
$x \in K_{2n1}(R)$
classified by a map
$x : S^{2n1} \to B GL(R)^+$
and considers on the pullback bundle
$\array{
x^* \mathcal{V}
\\
\downarrow
\\
S^{2n1}
}$
a choice of connection $\nabla$ that
preserves the locally filtered structure.
Choose also a hermitean structure and let $\nabla^*$ be the corresponding adjoint connection
Then the Borel regulator of $x$ is the expression
$r_\sigma(x) = \int_{S^{2n1}} ch_{2n1}(\nabla^* , \nabla) \in
\mathbb{R}
\,,$
where
$ch_{2n1}(\nabla^*, \nabla) \in \Omega^{2n1}(S^{2n1})$
is the relative ChernSimons form between the two connections.
This defines a map
$r: K_{2n1}(R) \to \mathbb{R}$
and the big question is: what is the image of this map? This is complicated, and the answer in general is not known.
But a theorem of Beilinson says that for
$R$ the cyclotomic field as above,
there exist $q \in \mathbb{Q}$
and $x \in K_{2n1}(R)$ such that
$r_\sigma(x) =
q Re\left[
\frac{1}{(2 \pi i)^n} Li_{2 n 1}(\mu)
\right]$
where
$Li_k(z) := \sum_{n \geq 1}^\infty \frac{z^n}{n^k}$
is a “higher logarithm”.
The proof that Beilinson gives is complicated.
The claim in the following is: this is also a consequence of a statement that naturally ought to be true in index theory for differental algebraic Ktheory.
Index theorem is about comparison of two pushforwards, one analytic, one homotopy theoretic.
Fix a bundle $\pi : W \to B$ with compact fibers and
being a proper submersion
(no need for any kind of orientation in the following).
In the running example we look at lens spaces
$S^{2n1}/{\mathbb{Z}_p} \simeq
L_p^{2n+1} \to \mathbb{C}P^n
\,.$
Consider local systems of $R$modules on $W$ and $B$. The claim is then that there is a diagram as follows, to be explained now:
$\array{
Loc(W)
&\stackrel{cycle}{\to}&
\hat K R^0(W)
\\
{}^{\mathllap{analytic transfer}}\downarrow
&\neArrow_{\simeq}^{higher analytic torsion}&
\downarrow^{\mathrlap{BecherGottlieb transfer}}
\\
Loc(B)
&\stackrel{cycle}{\to}&
\hat K R^0(B)
}$
Here $\hat K R$ is the differential Ktheory spectrum
$Loc(B)$ is isomorphism classes of geometrically constant sheaves of finitely generated projective $R$modules on $B$,
where the “geometric” is choices of hermitean metrics on the complexification of the sheaves (which are sheaves of sections of complexes of flat vector bundles)
In the example we have
$\pi_1(L_p^{2n+1}) = \mathbb{Z}_p$
take $V$ te rank 1, free, with holonomy $\xi$
Now definition of $\hat K R$ following HopkinsSinger.
[to be polished from here on, no time now]
We have the spectrum $K R$ and can form smash product
with the Moore spectrum, the result being
equivalent to an EilenbergMacLane spectrum
$K R \to K R \wedge M \mathbb{R}
\stackrel{\simeq}{\to}
H A
\,.$
where $A_\bullet = K\bullet(R) \otimes \mathbb{R}$
give a de Rham construction for this
$H(\Omega(B,A))
\stackrel{de Rham theorem}{\to}
H(A)^B$
form the homotopy pullback
$\array{
\hat K R^0(B)
&\to&
H(\sigma^{\geq 0} \Omega(B,A))
\\
{}^{\mathllap{I}}\downarrow && \downarrow
\\
K R^B &\to& H(\Omega(B,A))
}$
$K R^{1}
\stackrel{r}{\to}
\Omega^{1}(B,A)/im D
\stackrel{a}{\to}
\hat K R^0(B)$
the kernel of $a$ is the Borel regulator from before
task: give geometric models for elements in this hopullback
Theorem: there exists a cycle map
$cycl : Loc \to \hat K R$
$g^V$ is the geometry
$I(cycle(V, g^V)) = [V]$
$in K R^0(B) [M \stackrel{V}{\to} B GL(R) \to \Omega^\infty K R]$
$R(cycle(V, g^V)) = contains ch(\nabla^* , \nabla)$
now
$\hat B G$
transfer choose Riem structure on $\Pi$
pushforward induced from the following morphism of diagrams
$\array{
K R^W
&\to&
H(\Omega(W,A))
&\leftarrow&
H(\Sigma^{\geq 0} \Omega(B,A))
\\
\downarrow
&\neArrow_{canonical}&
\downarrow^{\int_{W/R} \wedge e^{\nabla}}
&&
\downarrow^{\int () \wedge ...}
\\
K R^B
&\to&
H(\Omega(W,A))
&\leftarrow&
H(\Sigma^{\geq 0} \Omega(B,A))
}$
analytic transfer
$(V, g^V) \mapsto
(R^i \pi_* V, g_{L^R}^V)$
Conjecture:
$cycle(transfer^{analytic}(V, g^V))

\hat tr(cycle(V, G^V))
=
a(\tau)$
$\tau$
is higher analytic torsion form.
Getzler, “Higher analytic stacks”
Ezra Getzler speaks about joint work with Kai Behrend.
Here a brief and incomplete note on what he said.
Let $A$ be a dgalgebra. Form the simplicial set
$N_\bullet A := MC(C^\bullet(\Delta[\bullet]), A)$
of MaurerCartan elements $\alpha$
$\delta \alpha + D \alpha + \alpha^2 = 0$
on the $A$valued cochains on a simplicial set, where $\delta$ is the simplicial differential and $D$ is the “internal” differential on $A$.
Claim: This simplicial set is a quasicategory.
If $A$ is an ordinary algebra, then this is the nerve of the corresponding multiplicative monoid.
(ed.: Hence it makes sense to write $B A := N(A)$.)
Consider now the maximal Kan complex inside this, we may think of it as the delooping of the general linear group with coefficient in $A$
$B GL(A) := core(N(A)) \subset N(A) \,.$
Theorem: This also works for differential graded Banach algebras.
Application: for $E \to X$ a homomorphic vector bundle, its algebra of forms
$A := \oplus_i \Omega^{0,i}(X, End(E))$
is a dgBanach algebra. Hence we have an application to Kuranishi deformation theory.