Commit 40b31a08 authored by Jannes Bantje's avatar Jannes Bantje

further work on talk summary

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......@@ -341,7 +341,7 @@ This digression into complex bordism arises from the need to define the followin
The Todd class can be viewed as the complex analogue of the \Ahat-class:
\begin{proposition}{\cite[Prop.~III.11.14]{lawson_spin}}
\begin{proposition}{\cite[Prop.~III.11.14]{lawson_spin}}\label{prop:todd_vs_Ahat}
For any real oriented vector bundle $E$, the total Todd and \Ahat-classes satisfy
\[
\td(E \otimes \mathbb{C}) = \Ahat(E)^2
......@@ -408,7 +408,7 @@ This gives a ring homomorphism
\end{remark}
\subsection*{Elliptic Operators and Elliptic Complexes}
In the following everything is considered to be smooth, this applies in particular to manifolds, vector bundles\footnote{meaning, that they are smooth manifolds} over them and their sections.
In the following everything is considered to be smooth, this applies in particular to manifolds, vector bundles\footnote{meaning, that they are smooth manifolds with smooth bundle projection} over them and their sections.
We now start define the main players of index theory, namely differential operators:
Let $X$ be a closed manifold and $E,F$ complex smooth vector bundles over $X$.
......@@ -435,6 +435,189 @@ Let $X$ be a closed manifold and $E,F$ complex smooth vector bundles over $X$.
\end{itemize}
\end{remark}
We will be interested in the \emph{symbol} of a differential operator:
\begin{definition}
The (principal) \Index{symbol} of an order $p$ differential operator is the bundle morphism
\[
\sigma_p(D) \colon \Tang^* X \To{} \Hom(E,F)
\]
defined as follows:
\[
\sigma_p(D)(x, \xi)(e) \coloneqq \frac{i^p}{p!} D(f^p s)(x) \in F_x
\]
where $f \in C^\infty(X,\mathbb{R})$ with $f(x)=0$, $\mathd f = \xi$ and $s \in \Gamma(E)$ with $s(x)=e$.
\end{definition}
\begin{lemma}
The symbol is well-defined.
In particular the order of $D$ is also well-defined.
\end{lemma}
See \cref{lem:symbol_welldefined} or \cite[167]{lawson_spin} for a detailed proof.
In proving this it turns out, that with respect to the local description of $D$, one has for $\xi = \sum_k \xi_k \mathd x_k $
\[
\sigma(D)(x,\xi) = i^p \sum_{\abs{\alpha}=p} A^\alpha(x) \xi^\alpha
\]
This justifies the description of the symbol as the \enquote{highest order term} of $D$, that one often hears.
\begin{remark}
Generally speaking the \emph{total} symbol of a differential operator arises -- roughly speaking -- by replacing the partial derivatives with a new variable.
The symbol naturally arises in the formula for the inverse Fourier transform.
Fourier analysis is also the backbone of \emph{elliptic regularity}, which we will get in a moment.
\end{remark}
\begin{definition}
$D$ is called \Index[differential operator!elliptic]{elliptic}, if $\sigma(D)(x,\xi)$ is invertible for every non-zero cotangent vector $0 \neq \xi \in \Tang_x^*X$.
\end{definition}
\begin{example}
The Laplace operator, the Dirac operator, the Hodge Laplacian.
\end{example}
We now use the symbol to produce elements in \K-theory out of $D$.
Let $\pi \colon \Tang^* X \to X$ be the projection.
Then we may view $\sigma(D)$ as a bundle map
\[
\sigma(D) \colon \pi^* E \To{} \pi^* F
\]
Ellipticity of $D$ is then equivalent to this map being an isomorphism away from zero.
We define an element in the \K-theory of the disk bundle associated to $\Tang^* X$:
\[
\sigma(D) \coloneqq \benbrace[\big]{\pi^* E,\pi^* F, \sigma(D)} \in \K \enbrace[\big]{D(\Tang^* X), S(\Tang^* X)} \cong \Kred \enbrace[\big]{\Thom(\Tang^* X)} = \K_\cpt(\Tang^* X)
\]
This uses the \enquote{clutching construction} due to \citeauthor{clifford_modules}.
Note, that we assume $X$ to be closed.
This is called the \Index{symbol class}.
\begin{remark}
As an advertisement for the topological importance of the symbol class, we should mention, that for an even-dimensional spin manifold the symbol of the Atiyah--Singer operator (which is the Dirac operator of the complex spinor bundle) gives a \K-theory orientation on the cotangent bundle.
In particular this makes this symbol class a generator of the Thom isomorphism.
An analogous remark holds in the real case for $8k$-dimensional spin manifolds.
\end{remark}
\subsection*{Generalisation to Complexes}
Instead of a single operator, we can also consider complex vector bundles $E_0, \ldots , E_m$ over $X$ and differential operators $D_i$
\[
\begin{tikzcd}[sep=large]
\Gamma(E_0) \rar["D_0"] & \Gamma(E_1) \rar["D_1"] & \ldots \rar["D_{m-1}"] & \Gamma(E_m)
\end{tikzcd}
\]
such that $D_{i+1} D_i =0$.
Such a complex is called \bet{elliptic}, if the corresponding complex of symbols
\[
\begin{tikzcd}[sep=large]
0 \rar & \pi^* E_0 \rar["\sigma_0"] & \pi^* E_1 \rar["\sigma_1"] & \ldots \rar["\sigma_{m-1}"] & \pi^* E_m \rar & 0
\end{tikzcd}
\]
is exact.
For a complex of length one, this clearly reduces to the earlier definition.
The particular interest in elliptic operator (and complexes) comes from the fact, that elliptic operators behave much more nicely from an operator theoretic perspective, a phenomenon sometimes called \Index{elliptic regularity}.
To cut a \emph{long} story short\footnote{for starters: differential operators are not even bounded operators, which makes the application of functional analytic methods a lot harder} on gets -- besides many other things like certain Fredholm properties -- the following theorem:
\begin{theorem}
The kernel (and therefore the cokernel!) of an elliptic differential operator on a closed manifold has finite dimension.
\end{theorem}
Finally, this allows us to define the index:
\begin{definition}
Let $D$ be an elliptic operator.
\[
\indeX(D) \coloneqq \dim \ker(D) - \dim \coker(D) \in \mathbb{Z}
\]
For an elliptic complex $(D_i)$ let $H^i = \ker D_i/ \im D_{i-1}$.
$H^i$ is finite dimensional and we set
\[
\indeX((D_i)) \coloneqq \sum_{i=0}^m (-1)^i \dim H^i \in \mathbb{Z}
\]
This is called the \Index{analytical index} of $D$ (or $(D_i)$).
\end{definition}
One should note, that the analytic index is an integer by design.
However computing it, might be really hard, since this amounts to solving a PDE.
Let us define another index for $D$ using the symbol class:
Pick a proper embedding $\iota \colon X \hookrightarrow \mathbb{R}^n$ and let $N$ be the normal bundle of $\Tang^* X$.
We also find a diffeomorphism of $N$ with a tubular neighbourhood $U$ of $\Tang^* X$ in $\Tang^* \mathbb{R}^n$.
We claim, that there is a shriek map
\[
\iota_! \colon \K_\cpt(\Tang^* X) \To{} \K_\cpt(\Tang^* \mathbb{R}^n)
\]
We want to invoke the Thom isomorphism at this point, but we need a complex structure to do so, see for example \cite[Thm.~C.8]{lawson_spin}.
More specifically we need a complex structure on $N$, the normal bundle of $\Tang^* X$.
But if $N'$ is the normal bundle of $X$ in $\mathbb{R}^n$, $N$ is just the pullback of $N' \oplus N'$ along $\Tang^* X \to X$ (the first factor should be though of as lying in the \enquote{manifold direction}, the second in the \enquote{fibre direction}).
We therefore have a canonical complex structure given by
\[
\begin{pmatrix}
0 & -\id \\ \id & 0
\end{pmatrix}
\]
We now get $\iota_!$ as the composition
\[
\begin{tikzcd}
\K_\cpt(\Tang^* X) \rar["cong"] & \K_\cpt(N) \rar["\cong"] \K_\cpt(U) \rar & \K_\cpt(\Tang^* \mathbb{R}^n)
\end{tikzcd}
\]
where the first map is the Thom isomorphism, the second induces by a diffeomorphism and the third by the inclusion of the \emph{open} subset $U$.
\begin{definition}
For $D$ elliptic the \Index{topological index} is defined as
\[
\topind(D) \coloneqq \iota_! \enbrace*{\sigma(D)} \in \K_\cpt(\Tang^* \mathbb{R}^n) \cong \K(\pt) = \mathbb{Z}
\]
\end{definition}
One obviously has to check, that this construction does not depend on the choices made.
\textquote[{\cite[244]{lawson_spin}}]{One of the basic results in mathematics is the following:}
\begin{theorem}{Atiyah--Singer}\label{thm:ind_eq_topind}
For an elliptic operator $D$ over a compact manifold $X$ of dimension $n$ we have
\[
\indeX(D) = \topind(D)
\]
\end{theorem}
\begin{remark}
The index theorem is a statement about \K-theory and not just the cohomological formula, that we will see in a moment!
This also fits well with modern index theory, which has indices of differential operators being elements of some \K-theory groups (of \Cstar-algebras).
It is not a coincidence, that the operator \K-theory of the \Cstar-algebra of Fredholm operators is $\mathbb{Z}$!
\end{remark}
\begin{theorem}
In the situation of \cref{thm:ind_eq_topind} we have
\[
\indeX D = (-1)^n \enbrace[\big]{\ch \sigma(D) \cdot \td(\Tang^*X \otimes \mathbb{C})}[\Tang^* X]
\]
\end{theorem}
\begin{proof}{Proof Idea}
Apply the Chern character.
It turns out, that the Chern character behaves well with respect to the composition of maps used to define the topological index with one exception: it does not commute with the Thom isomorphisms!
The Todd class is the error correction term for this defect, see \cite[Sec.~III.12]{lawson_spin}.
\end{proof}
There are now various variants of this formula: For example applying \cref{prop:todd_vs_Ahat} yields
\begin{theorem}
In the situation of \cref{thm:ind_eq_topind} we have
\[
\indeX D = (-1)^n \enbrace*{\ch \sigma(D) \cdot \Ahat(X)^2}[\Tang^* X]
\]
where $\Ahat(X)$ denotes the total \Ahat-class of $X$ pulled back to $\Tang^*X$.
\end{theorem}
Integrating over the fibre (if $X$ is oriented) can rid us of the cotangent bundle:
\[
\indeX D = (-1)^{\frac{n(n+1)}{2}} \enbrace*{\pi_! \ch(\sigma(D)) \cdot \Ahat(X)^2}[X]
\]
These formulas become even more intriguing in the special case, that the structure group of $X^{2n}$ is a connected subgroup of $\SO(2n)$, as the operator does not show up on the right hand side anymore!
\newpage
\todo[inline]{\textcite[Thm.~III.13.13]{lawson_spin} is the closest thing to the Atiyah--Singer formula in \cite{hirzebruch_modularforms} I found so far.}
% section index_theory (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
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