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\section*{Oberwolfach Workshop, June 2009 -- Wednesday, June 10}

Here are notes by [[Urs Schreiber]] for Wednesday, June 10, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

\hypertarget{alexander_kahle_superconnections_and_index_theory}{}\subsection*{{Alexander Kahle: superconnections and index theory}}\label{alexander_kahle_superconnections_and_index_theory}

\begin{itemize}%
\item 1) superconnections


\item 2) index theory


\item 3) sketch some proofs



\end{itemize}
\hypertarget{1_superconnections}{}\subsubsection*{{1) superconnections}}\label{1_superconnections}

\textbf{definition} A \textbf{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 $\Omega^\bullet(M, End(V))^{odd}$

\begin{displaymath}
End(V)
\end{displaymath}
\begin{displaymath}
\nabla_s = \omega_0 + \nabla + \omega_2 + \omega_3
\end{displaymath}
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

\begin{displaymath}
\nabla_s
  =
  \left(
    \itexarray{
      & f^*
      \\
      f & 
    }
  \right)
  + \nabla
\end{displaymath}
\textbf{Chern character} by the usual formulas

\begin{displaymath}
ch(\nabla_s) := sTr e^{\nabla^2}
\end{displaymath}
\hypertarget{2_index_theory}{}\subsubsection*{{2) index theory}}\label{2_index_theory}

\textbf{definition} Let $M$ be smooth Riemannian and $Spin$, The Dirac operator associated to $(V \to M, \nabla_s)$ is defined by

\begin{itemize}%
\item \begin{itemize}%
\item \begin{displaymath}
D(\nabla_s) : \Gamma(S \otimes V) 
\stackrel{\nabla_s \otimes 1 \oplus 1 \otimes \nabla_s}{\to} \Omega^\bullet(M, S \otimes V) 
\stackrel{c(.)}{\to} \Gamma(S \otimes V)
\end{displaymath}


\end{itemize}


\end{itemize}
This is

\begin{itemize}%
\item an elliptic operator;


\item formally self adjoint


\item of the form

\begin{displaymath}
D(\nabla_s) = \left(
  \itexarray{
     & D'(\nabla_s)
     \\
     D'(\nabla_s)
  }
 \right)
\end{displaymath}

\item \textbf{theorem} (corollary of Atiyah-singer index theory)

\begin{displaymath}
index(D(\nabla_s))
  =
  index(D(\nabla))
  =
  \int_M \hat A(\Omega^m) ch(\nabla_s)
\end{displaymath}


\end{itemize}
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

\begin{displaymath}
Tr \exp(-t D(\nabla_s)^2 ) = index(D(\nabla_s))
\end{displaymath}
the heat semi-group is smoothing, therefore it is represented by a kernel

\begin{displaymath}
\exp(-t D(\nabla_s)^2) \psi(x)
  =
  \int_M p_t(x,y) \psi(y) d y
\end{displaymath}
\begin{displaymath}
Tr \exp(-t  D(\nabla_s)^2) =
  \int_M Tr p_t(x,x) d vol
\end{displaymath}
the following expected formula which holds for ordinary connections (due to Ezra Getzler) \emph{no longer} holds directly for superconnections

\begin{displaymath}
\lim_{t \to 0}
  Tr p_t(x,x) d vol
  \neq
  (2 \pi i)^{-n/2}
  [
    \hat A(\Omega^m) ch(\nabla_s)
  ]_n
\end{displaymath}
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

\begin{displaymath}
\nabla_s^t := |t|^{-1/2} \omega_0 + \nabla + |t|^{1/2} \omega_2 + \cdots
\end{displaymath}
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,

\begin{displaymath}
p : T(M) \to T(M/B)
\end{displaymath}
\begin{displaymath}
\itexarray{
    V, \nabla_s
    \\
    \downarrow
    \\
    M
    \\
    \downarrow^\pi
    \\
    B
  }
\end{displaymath}
$\pi_* (V)$ : 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 $V$

\begin{displaymath}
\pi_! \nabla_s = \pi_! \nabla + \pi_! \omega
\end{displaymath}
with $\nabla_s = \nabla + \omega$

\begin{displaymath}
[\pi_! \omega_!]_{\omega}(\xi_1, \cdots, \xi_i)
  =
  c^{M/B}(2 (\tilde \xi_1), 2(\tilde \xi_2) \cdots 2(\tilde \xi_k))
\end{displaymath}
\begin{displaymath}
\pi^r = (\pi, r g^{M/B}, P)
\end{displaymath}
\begin{displaymath}
\lim_{t \to 0}
  ch(\pi_!^t \nabla_s)
  =
  (2 \pi i)^{dim M/B}
  \pi_*
  [
    \hat A(\Omega^{M/B} ch(\nabla_s))
  ]
\end{displaymath}
the scalings are related by

\begin{displaymath}
\pi_!^t(\nabla_s) = [\pi_! \nabla_s^{1/t}]^t
\end{displaymath}
\hypertarget{determinant_line_bundles}{}\subsubsection*{{determinant line bundles}}\label{determinant_line_bundles}

(\ldots{}skipping a bunch of remarks\ldots{})

\hypertarget{3_sketch_of_some_proofs}{}\subsubsection*{{3) sketch of some proofs}}\label{3_sketch_of_some_proofs}

(no time, as expected)

\hypertarget{operads}{}\subsubsection*{{$\infty$-operads}}\label{operads}

Baronikov-Kontsevich passage

\hypertarget{gabriel_drummondcole_operads__and_}{}\subsection*{{Gabriel Drummond-Cole; $\infty$-operads, $BV_\infty$ and $HyperComm_\infty$}}\label{gabriel_drummondcole_operads__and_}

\begin{quote}%
(was hard to take typed notes of this otherwise pretty cool talk, does anyone have handwriitten notes?)


\end{quote}
\hypertarget{scott_wilson_categorical_algebra_mapping_spaces_and_applications}{}\subsection*{{Scott Wilson: Categorical algebra, mapping spaces and applications}}\label{scott_wilson_categorical_algebra_mapping_spaces_and_applications}

(for closely related blog entry see

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2008/04/higher_hochschild_cohomology_a.html}{Higher Hochschild Cohomology and Differential Forms on Mapping Spaces}

\end{itemize}
)

outline

\begin{itemize}%
\item language for some elementary algebraic topology


\item application to generalizatons of Hochschild complexes


\item Examples

\begin{itemize}%
\item invariants on mapping spaces


\item contributions related to def of Laplacian



\end{itemize}


\end{itemize}
\textbf{def/lema}

A commutative associative [[differential graded algebra]] is (equivalently given by) a strict monoidal functor

\begin{displaymath}
(FinSet, \coprod) \to (ChainComplexes, \otimes)
\end{displaymath}
generalize this

\textbf{def} a \textbf{partial DGA} is a monoidal functor with coherence map given by weak equivalence in the model structure

\begin{displaymath}
A :  (FinSet, \coprod) \to (ChainComplexes, \otimes)
\end{displaymath}
i.e. there exists a natural weak equivalence

\begin{displaymath}
A(j \sqcup k) \stackrel{T}{\to} A(j) \otimes A(k)
\end{displaymath}
that respects the obvious coherence properties

generalized

\begin{itemize}%
\item 1) co-algebras


\item 2) any operad


\item 3) note that $FinSet_*$ (pointed finite sets) is a module over $FinSet$, so generalize to modules, comodules, etc.



\end{itemize}
Then weak partial algebras can be functorially replaced by $E_\infty$-algebras

\textbf{example}

$X$ be a space $j \stackrel{f}{\to} k$

\begin{displaymath}
X^j = Map(j,X) \leftarrow Map(k,X) = X^k
\end{displaymath}
pass to the chains version of this

\begin{displaymath}
Ch_*(X^j) \leftarrow Ch_*(X^k)
\end{displaymath}
\begin{displaymath}
Ch^*(X^j) \to Ch^*(X^k)
\end{displaymath}
by Kuenneth formula we have a chain equivalence

\begin{displaymath}
C_*(X^j) \otimes C_*(X^k) \to C_*(X^{j+k})
\end{displaymath}
and similarly for cochains.

so this gives two things:

\begin{itemize}%
\item a partial coalgebra on $C_*(X)$


\item a partial algebra on $C^*(X)$



\end{itemize}
Let $Y$ be any finite simplicial space. A partial algebra

\begin{displaymath}
\Delta \stackrel{\gama}{\to} FinSet \stackrel{A}{\to} ChainCompl
\end{displaymath}
simplicial object in $ChainCompl$, so total complex

\begin{displaymath}
CH^\gamma(A)
\end{displaymath}
meaning generalization of Hochschild complex

\begin{itemize}%
\item this is joint work with Tradler and Zanelli (spelling? probably wrong)

\end{itemize}
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

\textbf{example}

let $A$ be a strict algebra, and $\gamma = Y = S^1$ then

\begin{displaymath}
CH^{S^1}(A) = \prod_{n \geq 0} A \otimes A^{\otimes n}
\end{displaymath}
is the Hochschild complex

there is also a shuffle product in the game, so this implies there is an exponential map

calculate:

\begin{displaymath}
\exp(| \otimes x)
  =
  | + | \otimes x + | \otimes  x \otimes x + | \otimes  x \otimes x \otimes x + \cdots +
\end{displaymath}
\begin{displaymath}
D \exp(1 \otimes x) = (1 \otimes d x + x \cdot x) \cdot e^{1 \otimes x}
\end{displaymath}
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 $M$

\begin{displaymath}
\itexarray{
    && CH^{S^1}(A) (\simeq \Omega(M^{S^1}))
    \\
    &\nearrow & \downarrow
    \\
    K(M)&\stackrel{ch}{\to}&\Omega(M)
  }
\end{displaymath}
commutes (due to some people)

\textbf{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

\begin{displaymath}
\Delta = [D, D^*]
\end{displaymath}
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\ldots{}


\vspace{.5em} \hrule \vspace{.5em}
[[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Next day]]



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