Here are notes by Urs Schreiber for Wednesday, June 10, from Oberwolfach.
1) superconnections
2) index theory
3) sketch some proofs
definition A superconnection $\nabla_s$ on a $\mathbb{Z}_2$-graded vector bundle $V \to M$ is an odd derivation on $\Omega^\bullet(M,V)$
superconnections form an affine space modeled on $\Omega^\bullet(M, End(V))^{odd}$
class in $K$-theory given by a map $V \stackrel{f}{\to} W$
unitary superconnection on $\mathbb{Z}_2$-graded unitary bundles $V$ with map as a above look like
Chern character by the usual formulas
definition Let $M$ be smooth Riemannian and $Spin$, The Dirac operator associated to $(V \to M, \nabla_s)$ is defined by
This is
an elliptic operator;
formally self adjoint
of the form
theorem (corollary of Atiyah-singer index theory)
so superconnections don’t give new topological data: they are geometric objects with the same underlying topology as ordinary connections but refined “geometry”
recall that Atiyah-Singer says that
the heat semi-group is smoothing, therefore it is represented by a kernel
the following expected formula which holds for ordinary connections (due to Ezra Getzler) no longer holds directly for superconnections
here $n = dim X$ is the dimension of the manifold
problem is that components in a superconnections scale in a different
to make it true, we need to rescale
A Riemannian map is a triple $(\pi, g, P)$
$\pi : M \to B$
a family with fibers close Spin manifolds, $g^{M/B}$ a metric onm the fibers,
$\pi_* (V)$ : a fibre at $y \in B$ is
$\Gamma_y(S^{M/B} \otimes V)$
due to Bismut we get from a connection on the top a superconnecction on the bottom (which is one of the main original motivations to be interested in superconnection in the first place), which we tweak here a bit to get a superconnection on $B$ from a superconnection on $V$
with $\nabla_s = \nabla + \omega$
the scalings are related by
(…skipping a bunch of remarks…)
(no time, as expected)
Baronikov-Kontsevich passage
(was hard to take typed notes of this otherwise pretty cool talk, does anyone have handwriitten notes?)
(for closely related blog entry see
)
outline
language for some elementary algebraic topology
application to generalizatons of Hochschild complexes
Examples
invariants on mapping spaces
contributions related to def of Laplacian
def/lema
A commutative associative differential graded algebra is (equivalently given by) a strict monoidal functor
generalize this
def a partial DGA is a monoidal functor with coherence map given by weak equivalence in the model structure
i.e. there exists a natural weak equivalence
that respects the obvious coherence properties
generalized
1) co-algebras
2) any operad
3) note that $FinSet_*$ (pointed finite sets) is a module over $FinSet$, so generalize to modules, comodules, etc.
Then weak partial algebras can be functorially replaced by $E_\infty$-algebras
example
$X$ be a space $j \stackrel{f}{\to} k$
pass to the chains version of this
by Kuenneth formula we have a chain equivalence
and similarly for cochains.
so this gives two things:
a partial coalgebra on $C_*(X)$
a partial algebra on $C^*(X)$
Let $Y$ be any finite simplicial space. A partial algebra
simplicial object in $ChainCompl$, so total complex
meaning generalization of Hochschild complex
goes back to Pitashvili and more recently Gregory Ginot
For $A = \Omega(X)$, then $CH^\gamma(A)$ computes cohomology of $X^\gamma$, if $X$ is sufficiently connected
example
let $A$ be a strict algebra, and $\gamma = Y = S^1$ then
is the Hochschild complex
there is also a shuffle product in the game, so this implies there is an exponential map
calculate:
then: if $d x + x \cdot x = 0$ then $D e^{1 \otimes x} = 0$
this reminds us of curvature and connection
this can be taken further
let $A = \Omega^\bullet(M)$ be differential forms on $M$
commutes (due to some people)
example 2
$Y = I$ (the interval)
then $CH^I(A)$ is the 2-sided bar construction
more generally $CH(A, M, N) = \prod_{n \geq 0} M \otimes A^{\otimes n} \otimes N$
with $M$ and $N$ $A$-modules sitting on the end of the interval
consider the case $A = \Omega^\bullet(Riemannian manifold)$ and $M = A$ and $N = (\Omega^\bullet(...), d^* , (x\in A) \cdot (y\in N) = \star^{-1}(x \wedge \star y)))$
(the operatoin on $N$ here is the intersection product of forms)
Let $D$ be differential on $CH^I$
let $D$ be differential on $CH^I$ for normal structure, and and $D^*$ for $A, M, N$ as just described.
Set
then acting with this $\Delta$ on something produces interesting non-linear differential equations related to Witten’t Morse-theory deformation of susy quantum mechanics and to Navier-Stokes’ equations in fluid dynamics…