@@ -31,7 +31,7 @@ In studying such invariant, we will

\bigskip

We start by recalling some facts from bordism theory starting with an example of a bordism invariant:

\begin{example}

\begin{example}\label{ex:signature}

Let $M$ be a closed, $4n$-dimensional oriented smooth manifold.

The cup product gives rise to a symmetric bilinear form on $H^{2n}(M;\mathbb{Q})$.

The \Index{signature}

...

...

@@ -157,42 +157,98 @@ We therefore have

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

\]

The genus obtained by this power series is called \Index[L-genus@$L$-genus]{$L$-genus}.

It was proven by \textcite{hirzebruch_Lgenus}, that this genus coincides with the signature.

It was proven by \textcite{hirzebruch_Lgenus}, that this genus coincides with the signature, see \cref{ex:signature}.

\end{example}

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

\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:

We have already seen, that the values of a genus $\varphi$ on the complex projective spaces are central to understanding $\varphi$.

The quaternionic projective spaces are important as well and \textcite[16]{hirzebruch_modularforms} prove, that for a genus $\varphi$ and the corresponding power series $Q(x)=x/f(x)$ one has

\[

\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)}

\]

This is sometimes called duplication formula.

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

\begin{example}

The $\hat{A}$-genus is elliptic.\todo[inline]{extend}

Consider the power series

\[

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

\]

Then we have $f(x)= x/Q(x)=\sinh x$ and $f(2x)=2 f(x) f'(x)$ and therefore $h\equiv1$.\marginnote{in \cite[16]{hirzebruch_modularforms} mistakenly $\sin x$}

More generally for all power series with $f(x)=\sinh(\alpha x)/\alpha$ we have $h \equiv1$ and therefore their genera vanish on all projective spaces $\HP^k$ for $k>0$.

The genus corresponding to

\[

Q(x)=\frac{x/2}{\sinh(x/2)}

\]

is called the \Index[Ahat-genus@$\Ahat$-genus]{$\Ahat$-genus}.

The $\hat{A}$-genus can be generalised to the spin bordism ring, where a morphism

By the Atiyah--Singer Index Theorem the \Ahat-genus of a spin manifold equals the index of the associated Dirac operator, which is an integer.

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

\end{example}

\begin{example}

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

\[

\alpha\colon\Omega^{\Spin}_*\to\KO^{-*}(\pt)

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

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

1&\text{ if }k \equiv0\mod2

\end{cases}

\]

emerges.

This map can be represented by a map of spectra $\M\Spin\to\KO$.

\end{example}

These two examples are so-called \emph{elliptic} genera, which we will introduce in the next section.

For the remainder of this section we will however give a first overview on how the story is going to unfold from here.

The $\hat{A}$-genus can be generalised to the spin bordism ring, where a morphism

\[

\alpha\colon\Omega^{\Spin}_*\To{}\KO^{-*}(\pt)

\]

emerges.

In some degrees the group on the right is $\mathbb{Z}/2$ and in particular has torsion.

One could say, this $\alpha$-invariant remedies the need to tensor with $\mathbb{Q}$.

This map can be represented by a map of spectra $\M\Spin\to\KO$, which factors through the connective cover $\kO$ since $\M\Spin$ is connective.

This generalisation process is illustrated in the following diagram

In the end we are interested in the so-called Witten genus, which will be a genus $\varphi_W$, that can be \enquote{lifted} to string cobordism and maps into $\mathbb{Q}\llbracket p\rrbracket$, $\mathbb{Z}\llbracket p \rrbracket$ and a \enquote{spectrum of modular forms} respectively.

One now asks for a generalisation of the Witten genus in the same fashion as the generalisation of the \Ahat-genus.

The main observation is here, that the Witten genus maps string manifolds to \emph{topological modular forms}, \tmf.

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:

\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)

\item The kernel of $\sigma_*\colon\Omega^{\Strng}_*\to\pi_*\tmf$ is given by Cayley plane bundles

\end{itemize}

The Stolz--Teichner program tries to construct $\tmf$ geometrically using approaches of theoretical physics.

Another interesting aspect ist, that $\tmf$ arises canonically from chromatic homotopy theory.

% section introduction_by_julian (end)

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

\label{sec:ellipctic_genera_markus}

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

\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: