Commit e4411c26 authored by Jannes Bantje's avatar Jannes Bantje

proper intro of elliptic genera

parent ecf6e352
Pipeline #59704 failed with stages
in 39 seconds
......@@ -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 \equiv 1 \mod 2\\
......@@ -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
\[
\varphi \enbrace*{\HP^k} = \begin{cases}
0 &\text{ if }k \equiv 1 \mod 2\\
\varepsilon^{k/2} &\text{ if } k\equiv 0 \mod 2
\end{cases}
\]
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:
\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} \\
......@@ -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}.
% section ellipctic_genera_markus (end)
\section{Modular Forms -- Jens} % (fold)
......
Markdown is supported
0%
or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment