From a528c73f18741758941a455855f34222e201f161 Mon Sep 17 00:00:00 2001 From: Jannes Bantje Date: Sun, 24 May 2020 15:44:37 +0200 Subject: [PATCH] add definition of elliptic genus --- contents/elliptic-genera.tex | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/contents/elliptic-genera.tex b/contents/elliptic-genera.tex index 18ed3ae..9266dd7 100644 --- a/contents/elliptic-genera.tex +++ b/contents/elliptic-genera.tex @@ -159,6 +159,16 @@ We therefore have 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. \end{example} + +\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} + % section introduction_by_julian (end) \section{Ellipctic Genera -- Markus} % (fold) -- GitLab