@@ -168,7 +168,7 @@ The quaternionic projective spaces are important as well and \textcite[16]{hirze

This is sometimes called duplication formula.

Let us look at an example, what this means in practice:

\begin{example}

\begin{example}\label{ex:Ahat-genus}

Consider the power series

\[

Q(x)=\frac{x}{\sinh x}

...

...

@@ -185,8 +185,12 @@ Let us look at an example, what this means in practice:

In general the \Ahat-genus is not an integer!

\end{example}

\begin{example}

For the $L$-genus one can easily calculate, that

\begin{example}\label{ex:L-genus-duplication}

For the $L$-genus one can easily calculate, that

\[

h(y)=\frac{1}{1-y^4}=1+y^4+y^8+\ldots

\]

and therefore

\[

L \enbrace*{\HP^k}=\begin{cases}

0&\text{ if }k \equiv1\mod2\\

...

...

@@ -222,7 +226,7 @@ The main observation is here, that the Witten genus maps string manifolds to \em

By means of homotopy theory there already is a map of spectra $\sigma\colon\M\Strng\to\tmf$.

The million dollar question is: Is there a geometrical description as for the \Ahat-genus?

Some of the conjectures in this direction are:

Some of the \emph{conjectures} in this direction are:

\begin{itemize}

\item The Witten genus is given by the index of a Dirac like operator on a certain loop space (Witten)

\item The generalised Witten genus is an obstruction to positive Ricci curvature (Stolz)

...

...

@@ -235,10 +239,31 @@ Another interesting aspect ist, that $\tmf$ arises canonically from chromatic ho

\section{Ellipctic Genera -- Markus}% (fold)

\label{sec:ellipctic_genera_markus}

Most of the genera, that are of particular interest, belong to the following class:

We already made contact with the duplication formula

\[

\sum_{k=0}^{\infty}\varphi\enbrace*{\HP^k}y^{2k}= h(y)\qquad\text{ where }\quad h(f(x))=\frac{f(2x)}{2\,f(x)f'(x)}

\]

and have seen, that $h(y)$ has a rather simple form for the \Ahat- and $L$-genus, \cref{ex:Ahat-genus,ex:L-genus-duplication}.

Namely, they are of the form

\[

h(y)=\frac{1}{1-\varepsilon\cdot y^4}

\]

which yields

\[

\varphi\enbrace*{\HP^k}=\begin{cases}

0&\text{ if }k \equiv1\mod2\\

\varepsilon^{k/2}&\text{ if } k\equiv0\mod2

\end{cases}

\]

The solution $f(x)$ of this kind of duplication formula is not unique and still depends on a parameter $\delta$, which is the value of the genus on $\CP^2$, i.e. $g'(y)=1+\delta y^2+\ldots$.

\begin{example}

For the $L$-genus we have $\delta=\varepsilon=1$ and the \Ahat-genus is defined by $\delta=-\frac{1}{8}$ and $\varepsilon=0$.

\end{example}

We turn this phenomenon into a definition, which has several equivalent descriptions:

\begin{definition}

A genus $\varphi$ is called an \Index[genus!elliptic]{elliptic genus}, if the corresponding power seires $f(x)=x/Q(x)$ satisfies on of the three equivalent conditions:

A genus $\varphi$ is called an \Index[genus!elliptic]{elliptic genus}, if the corresponding power series $f(x)=x/Q(x)$ satisfies on of the three equivalent conditions:

@@ -246,9 +271,8 @@ Most of the genera, that are of particular interest, belong to the following cla

\end{align*}

\end{definition}

\begin{remark}

\todo[inline]{add motivation}

\end{remark}

There is a interesting procedure to construct such elliptic genera, which also explains the name: There is an intimate connection with elliptic functions from complex analysis.

This is what \citeauthor{hirzebruch_modularforms} explain in \cite[Chap.~2]{hirzebruch_modularforms}.