Commit ecf6e352 authored by Jannes Bantje's avatar Jannes Bantje

complete introduction

parent 95f9307f
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......@@ -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:
\begin{align*}
(f')^2 &= 1- 2 \delta \cdot f^2 + \varepsilon \cdot f^4 \\
f(u+v) &= \frac{f(u)f'(v) + f'(u)f(v)}{1- \varepsilon \cdot f(u)^2 f(v)^2} \\
f(2u) &= \frac{2f(u)f'(u)}{1-\varepsilon \cdot f(u)^4}
\end{align*}
\end{definition}
\begin{remark}
\todo[inline]{add motivation}
\end{remark}
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 $h\equiv 1$.\marginnote{in \cite[16]{hirzebruch_modularforms} mistakenly $\sin x$}
More generally for all power series with $f(x)= \sinh( \alpha x)/\alpha$ we have $h \equiv 1$ 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
\[
\alpha \colon \Omega^{\Spin}_* \to \KO^{-*}(\pt)
L \enbrace*{\HP^k} = \begin{cases}
0 &\text{ if }k \equiv 1 \mod 2\\
1 &\text{ if }k \equiv 0 \mod 2
\end{cases}
\]
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
\[
\begin{tikzcd}
\pi_* \M\Spin \dar \rar & \pi_* \kO \dar \\
\Omega^{\Spin}_* \rar["\alpha"] \dar & \KO_*(\pt) \dar \\
\Omega^{\Spin}_* \otimes \mathbb{Q} \rar["\Ahat"] \dar & \mathbb{Z} \dar \\
\Omega^{\SO}_* \otimes \mathbb{Q} \rar["\Ahat"] & \mathbb{Q}
\end{tikzcd}
\]
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:
\begin{align*}
(f')^2 &= 1- 2 \delta \cdot f^2 + \varepsilon \cdot f^4 \\
f(u+v) &= \frac{f(u)f'(v) + f'(u)f(v)}{1- \varepsilon \cdot f(u)^2 f(v)^2} \\
f(2u) &= \frac{2f(u)f'(u)}{1-\varepsilon \cdot f(u)^4}
\end{align*}
\end{definition}
\begin{remark}
\todo[inline]{add motivation}
\end{remark}
% section ellipctic_genera_markus (end)
\section{Modular Forms -- Jens} % (fold)
......
......@@ -228,8 +228,10 @@
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Strng}{String}
\newcommand{\CP}{\mathbb{C}\mathbb{P}}
\newcommand{\HP}{\mathbb{H}\mathbb{P}}
% \newcommand{\M}{\mathrm{M}\mkern-3mu}
\newcommand{\M}{M\mkern-3mu}
\newcommand{\tmf}{\ensuremath{\mathrm{tmf}}}
\DeclareMathOperator{\Map}{Map}
\newcommand{\sa}{\mathrm{sa}}
\DeclareMathOperator{\spectrum}{spec}
......@@ -240,6 +242,8 @@
\newcommand{\Bop}{\mathcal{B}}
\newcommand{\BH}{\Bop(\Hilb)}
\newcommand{\Ahat}{\ensuremath{\widehat{A}}}
\newcommand{\B}[1]{B{#1}} % classifying spaces
......
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