\chapter{Elliptic Genera -- Recapitulation from the Seminar}% (fold)

\chapter{Elliptic Genera -- Recapitulation of the Seminar}% (fold)

\label{cha:elliptic_genera_phd_seminar}

...

...

@@ -96,7 +96,9 @@ The general idea is to consider Chern resp. Pontryagin classes formally as eleme

A similar story can be told for the Pontryagin classes, see \cite[8]{hirzebruch_modularforms}.

\end{example}

\Textcite[Sec.~1.5]{hirzebruch_modularforms} continue to define the \Index{Chern character} and prove general formulas, that can be applied to $\Lambda^k E$ or $E^*$ for example.

\todo[inline]{mention splitting principle}

% \Textcite[Sec.~1.5]{hirzebruch_modularforms} continue to define the \Index{Chern character} and prove general formulas, that can be applied to $\Lambda^k E$ or $E^*$ for example.

Let us systematically construct genera:

We consider an even power series

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@@ -140,7 +142,7 @@ By resorting to the total Chern and Pontryagin classes of $\mathbb{C}\mathbb{P}^

\]

that is, its derivative essentially determined by the values of $\varphi_Q$ at the complex projective spaces, see \cite[15]{hirzebruch_modularforms}.

But this also canonically associates a genus $\varphi$ with an even power series by looking at its values on $\CP^n$.

\begin{proposition}

\begin{proposition}\label{prop:one-to-one_genera}

There is a one-to-one correspondence of genera $\varphi$ and multiplicative sequences $Q$.

\end{proposition}

...

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@@ -151,7 +153,7 @@ We therefore have

\]

\begin{example}

Letting $Q(x)=x/\tanh(x)$ yields an even power series (look at the Taylor expansion).

Letting $Q(x)=\frac{x}{\tanh(x)}$ yields an even power series (look at the Taylor expansion).

Following our earlier definitions, we get $f(x)=\tanh(x)$, $f'(x)=1-f(x)^2$ and hence

\[

g'(y)=\frac{1}{1-y^2}=1+ y^2+y^4+\ldots

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@@ -293,22 +295,33 @@ This seems to be mostly algebra and I'm unsure, whether it is actually relevant

\section{Index Theory -- Jannes}% (fold)

\label{sec:index_theory}

Before we can get started, we need yet another genus to get things going.

In order to define this properly we need to talk about \emph{complex} genera: in \cref{sec:introduction_julian} we used the real cobordism ring $\Omega^{\SO}_*$ as the domain for genera.

Defining an analogue for complex manifolds is a little more complicated, since a cobordism between complex manifolds cannot admit a complex structure for dimension reasons.

This can however be done for \Index[stably almost complex manifold]{stably almost complex manifolds}.

We start with a quick recap an genera as discussed in \cref{sec:introduction_julian}.

Following \textcite{hirzebruch_modularforms} a genus is a ring homomorphism

\[

\varphi\colon\Omega^{\SO}_*\otimes\mathbb{Q}\To{} R

\]

The main takeaway was, that even power series $Q(x)\in R\llbracket x\rrbracket$ (with constant term one) give rise to multiplicative sequences in which one can plug in the Pontryagin classes of an oriented, compact manifold to get a genus.

It even turned out, that there is a one-to-one correspondence of genera and such power series (see \cref{prop:one-to-one_genera}).

Before we can get started with index theory, we need to talk about the sibling of this correspondence.

Enter complex genera.

Cobordism between complex manifolds is somewhat more complicated\footnote{in particular a bordism between two complex manifolds cannot have a complex structure for dimension reasons}, but it works for \Index[stably almost complex manifold]{stably almost complex manifolds}.

\Textcite{novikov_complex_bordism,milnor_complex_bordism} proved results similar to the ones in \cref{sec:introduction_julian} for the complex bordism ring $\Omega^{\Unitary}$.\footnote{see \url{http://www.map.mpim-bonn.mpg.de/Complex_bordism} for a nice overview}

It turns out, that there is an analogous one-to-one correspondence between complex genera and power series $Q(x)$, but these need no longer be even!

Nonetheless, on can proceed in the same fashion: let $X$ be a compact, almost complex manifold of real dimension $2n$.

Then we write

By applying the splitting principle we write the total Chern class as

and get a genus out of $Q$ by setting $\varphi(X)=\enbrace[\big]{\prod_{i=1}^n Q(x_i)}[X]$.

From a computational point of view, one could say, that we now plug the Chern classes instead of the Pontryagin classes into a multiplicative sequence.

From a computational point of view, one could say, that we now just plug the Chern classes instead of the Pontryagin classes into a multiplicative sequence.

\textcite[Sec.~III.11]{lawson_spin} treat the real and complex case more side-by-side in comparison to \textcite{hirzebruch_modularforms}.

\todo[inline]{magic numbers $s_n$}

This digression into complex bordism arises from the need to define the following characteristic class, which features prominently in (classical) index theory.

\begin{example}

We consider the formal power series

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@@ -326,6 +339,23 @@ From a computational point of view, one could say, that we now plug the Chern cl

References: \cite[Ex.~III.11.11]{lawson_spin}

\end{example}

The Todd class can be viewed as the complex analogue of the \Ahat-class:

For any real oriented vector bundle $E$, the total Todd and \Ahat-classes satisfy

\[

\td(E \otimes\mathbb{C})=\Ahat(E)^2

\]

\end{proposition}

\begin{proof}

This can be seen on the level of power series:

By the splitting principle we consider the formal splitting $E \otimes\mathbb{C}=\ell_1\oplus\overline{\ell}_1\oplus\ldots\oplus\ell_n \oplus\overline{\ell}_n$.

For complex vector bundles $E,E'$ over $X$, we have

\begin{enumerate}[(i)]

\item$\ch(E \oplus E')=\ch(E)+\ch(E')$

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@@ -366,7 +396,15 @@ This gives a ring homomorphism

\]

\begin{remark}

This can be generalised \emph{drastically} to generalised cohomology theories! This is the arena of the Atiyah--Hirzebruch spectral sequence.

\begin{itemize}

\item For the purpose of the story told here, we mainly need to remember the Chern character as a construction, that takes us from \K-theory to cohomology.

This is arguably quite ad-hoc.

The Chern character can however be generalised \emph{drastically} to complex oriented cohomology theories!

There is also some connection with formal group laws.

\todo[inline]{look into this and maybe connect with Leon's talk, reference: \url{https://mathoverflow.net/questions/6144/explanation-for-the-chern-character}}

% This is the arena of the Atiyah--Hirzebruch spectral sequence.

\item One can also use the differential geometry approach from Chern--Weil theory to define the Chern character.

\end{itemize}

\end{remark}

\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.}