Commit e4411c26 by Jannes Bantje

### proper intro of elliptic genera

parent ecf6e352
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 ... ... @@ -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) ... ...
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