 ### complete introduction

parent 95f9307f
Pipeline #59692 failed with stages
in 39 seconds
 ... ... @@ -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{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{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}{B{#1}} % classifying spaces ... ...
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!