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\section*{Oberwolfach Workshop, June 2009 -- Friday, June 12}

Talk notes for the session of

\begin{itemize}%
\item Friday June 12

\end{itemize}
of

\begin{itemize}%
\item [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]].

\end{itemize}
\hypertarget{corbett_redden_string_structures_and_3forms}{}\subsection*{{Corbett Redden: String structures and 3-forms}}\label{corbett_redden_string_structures_and_3forms}

\begin{itemize}%
\item notational conventions

\begin{itemize}%
\item $M^n$ a smooth closed $n$-manifold


\item $g$ a Riem. metric


\item $Spin \to P \to M$ a principal $Spin(n)$-bundle


\item $A$ a connection on $P$


\item $S$ a string class



\end{itemize}


\end{itemize}
\hypertarget{outline}{}\subsubsection*{{Outline}}\label{outline}

\begin{itemize}%
\item I $\{String\ Structures\}/_{homotopy}$


\item II harmonic 4-forms on $P$


\item III geometry and tmf?



\end{itemize}
\hypertarget{i_string_structures}{}\subsubsection*{{I. String structures}}\label{i_string_structures}

\textbf{definition}

\begin{itemize}%
\item $B String$ is defined to be the homotopy fiber $B String(n) \to B Spin(n) \stackrel{p_1/2}{\to} K(\mathbb{Z},4)$


\item a \textbf{[[string structure|String structure]]} on $\pi : P \to M$ is a lift of the classifying map to $B Spin$ through $B String \to B Spin$

\begin{displaymath}
\itexarray{
      P && B String
      \\
      \downarrow &\nearrow& \downarrow
      \\
      M &\to& B Spin
  }
\end{displaymath}


\end{itemize}
\textbf{Prop}

\begin{itemize}%
\item String structure exists iff $\frac{1}{2}p_1(P) = 0 \in H^4(X,\mathbb{Z})$ (essentially by definition through homotopy fiber)


\item let $i^*_P$ the pullback to the fiber of $P$, then we have



\end{itemize}
\begin{displaymath}
\{ String\ structures \}/_{homotopy} \simeq_{canonical} \{ String\ classes \} := \{\rho \in H^3(P,\mathbb{Z})\ s.t.\ i_P^* \rho = 1 \in H^3(Spin, \mathbb{Z})\}
\end{displaymath}
\begin{itemize}%
\item $\{string\ classes\}$ is a torsor for $H^3(M, \mathbb{Z})$ under $\rho \mapsto \rho + \pi^ H^3(M)$

\end{itemize}
Proof. universal example

\begin{displaymath}
\itexarray{
    K(\mathbb{Z},3) &\simeq& \pi^* E Spin &\to & E Spin
    \\
    && B String &\to& B Spin &\to& K(\mathbb{Z}, 4)
  }
\end{displaymath}
\begin{quote}%
hm, here is my ([[Urs Schreiber|Urs]]) description of the situation:


\end{quote}
consider the following pasting diagram of [[homotopy limit|homtopy pullbacks]] ($P$ is the given $Spin$-bundle, $\hat P$ its String-lift)

\begin{displaymath}
\itexarray{
    String &\to& \hat P &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Spin &\to& P &\to&  B^2 U(1) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& X &\to& B String(n) &\to& B Spin(n)
  }
\end{displaymath}
\textbf{Why String structures?}

String structure on $P$ $\stackrel{transgresses}{\mapsto}$ Spin structure on loop space $L Spin \to L P \to L M$

but all the reps of this loop group are projective, so there is actually a central $S^1$ extension of $L Spin$ in the game

\begin{displaymath}
\hat {L Spin} \to L Spin
\end{displaymath}
Need:

\begin{displaymath}
\itexarray{
    \hat {L Spin} &\to & \hat {L P}
    \\
    \downarrow && \downarrow
    \\
    L Spin &\to& L P
  }
\end{displaymath}
\begin{displaymath}
\rho \in H^3(P, \mathbb{Z}) \mapsto
\end{displaymath}
\begin{displaymath}
\itexarray{
    (\pi_! ev^*) \rho &\in& H^2(L P, \mathbb{Z})
    \\
    \downarrow && \downarrow
    \\
    iniv. ext. && H^2(L G, \mathbb{Z})
  }
\end{displaymath}
(on the right: $S^1$-bundles)

\textbf{String orientation} of tmf = topological modular forms

\begin{displaymath}
M O\langle 8\rangle^{-n} = M String \stackrel{\sigma}{\to} tmf^{-n}(pt)
\end{displaymath}
so give a String manifold with a String class on $Spin(T M)$

\begin{displaymath}
M , \rho \mapsto [M, \rho] \mapsto \sigma(M,\rho)
\end{displaymath}
\begin{displaymath}
\itexarray{
    && tmf
    \\
    & {}^{\sigma}\nearrow & \downarrow
    \\
    M String &\stackrel{Witten\ genus}{\to}& Mod Forms
  }
\end{displaymath}
$Witten\ genus(M) =$ ``$index^{S^1} D_{L M}$''

(by the way, $\sigma$ is surjective on homotopy classes)

\textbf{warning}: I think above my $\rho$ should really be an $S$

\hypertarget{ii_harmonic_representative_of_}{}\subsubsection*{{II Harmonic representative of $S$}}\label{ii_harmonic_representative_of_}

\emph{reminder}

$(M,g) \mapsto \Delta = d d^* + d^* d$

\begin{displaymath}
H^k(M,\mathbb{R}) \simeq_{Hodge} \Delta^*_g \subset \Omega^k(M)
\end{displaymath}
\textbf{construction}

start with $(\pi : P  \to M, g_m, A)$

choose a bininvariant metric

\begin{displaymath}
g_\rho := \pi^* g_m \oplus g_{spin}
\end{displaymath}
where the direct sum comes from the splitting of tangent spaces $T_p P$ using the connection that we have

Introduce scaling factor $\delta \gt 0$

\begin{displaymath}
g_\rho := \pi^* g_m \oplus \delta^2 g_{spin}
\end{displaymath}
take the ``adiabatic limit'' $\lim_{\delta \to 0}$

so now there is a 1-parameter family of metric on the bundle, and for each one can look at its Laplacian, so as $\delta$ tends to 0 something becomes singular and one has to be careful, but fortunately others already did that for us\ldots{}

\textbf{theorem} (Mazzeo-Melrose, Dai, Forman)

the kernel $Ker \Delta_{g_\rho}$

\begin{itemize}%
\item extends smoothly to $\delta = 0$ \emph{(there is a smooth path in some Grassmannian space)}


\item and comes from a filtration isomorphic to Serre SS for $(Spin \to P \to M)$



\end{itemize}
this means that

\begin{displaymath}
\Rightarrow H^k(P, \mathbb{Z})
  \stackrel{\simeq}{\to}
   \lim_{\delta \to 0}
   Ker \Delta_{g_\rho}
   =: H^k(P)
\end{displaymath}
\textbf{Theorem} (Redden)

Given $P \stackrel{\pi}{\to} g, A$ and $\frac{1}{2}p_1(P) = 0$

then

\begin{displaymath}
\itexarray{
    H^3(P, \mathbb{Z}) &\to& H^3(P,\mathbb{R}) &\to& H^3(P)
    \\
    S &&\mapsto& CS_3(A)&& CS_3(A) - \pi^* H 
  }
\end{displaymath}
here $CS_3(A)$ is the Chern-Simons 3-form of the spin-connection

and recall $H^3(P)$ here denotes \emph{harmonic forms} on $P$ (should really be script font

\textbf{remark}

in genral

\begin{displaymath}
[\rho]_{g_\delta} = 
  CS_3(A) - \pi^* H + O(\delta) \notin \pi^* \Omega^3(M)
\end{displaymath}
if we have a product of two groups we accordingly would get CS of one connection minus CS of the other.

\textbf{What is $H$?}

(first digression)

\textbf{theorem} (Chern-Simons,\ldots{})

given $(P \to M , A)  \mapsto \widehat{\frac{1}{2}p_1}(A) \hat H^4(M)$

\begin{displaymath}
\itexarray{
    \Omega^3_{\mathbb{Z}} &\to& 
    (M)\Omega^3(M) &\stackrel{a}{to}& \hat H^4(M) &\to& H^4(M,\mathbb{Z}) &\to& 0
    \\
    && 
    H &\mapsto& \hat{\frac{1}{2}p_1(A)} 
    &\mapsto& 
    \frac{1}{2}p_1(P) = 0
    && 
  }
\end{displaymath}
in particular

\begin{displaymath}
\itexarray{
    && \mathbb{R}
    \\
    & {}^{\int H} \nearrow & \downarrow
    \\
    \mathbb{Z}_3(M)
    &\stackrel{\hat {\frac{1}{2}p_1(A)}}{\to}&
    \mathbb{R}/\mathbb{Z}
  }
\end{displaymath}
and secondly $d^* H = 0$

these two properties determine $H$ uniquely up to harmonic forms $\mathcal{H}^3_{\mathbb{Z}}(M) = ker \Delta_g$

\textbf{Equivariance}

\begin{displaymath}
H_{\rho + \pi^* \psi} = H_\rho + \pi^* H4
\end{displaymath}
where $\psi \in H^3(M, \mathbb{Z})$ $\mapsto H_4 \in \mathcal{H}^3(M)$

over all what this says is that if we go from

\begin{displaymath}
\itexarray{
    Metr(M)\times A(P) \times \{String\ Classes\} & (g,A,S)
    \\
    \downarrow & \downarrow
    \\
    \Omega^3(P) & CS_3(A) - \pi^* H_{q,A,S}
    \\
    \downarrow & \downarrow
    \\
    \Omega^3(M) & H_{g, A \rho}
 }
\end{displaymath}
\begin{displaymath}
\itexarray{
     && tmf^{-n}
     \\
     & \nearrow & \downarrow
    \\
    M String &\stackrel{}{\to}&
    MF
  }
\end{displaymath}
\textbf{conjecture} (S. Stolz) if $M$ is String and admits a positive Ricci curvature metric, then $Witten(M) = 0$

question: also $\sigma(M,\rho) = 0$? no, no way!

\textbf{hypothesis}

If $M$ is a Spin manifold that admits a metric and String structure $(g, S)$ and $A$ is the Levi-Civita connection

such that

\begin{itemize}%
\item $Ric(g) \gt 0$


\item $H_{g,s} = 0 \in \Omega^3(M)$



\end{itemize}
$\Rightarrow \sigma(M,S) = 0 \in tmf^{-n}(pt)$

\textbf{example} $M = S^3 = SU(2)$

\begin{displaymath}
p_1 \in H^4(S^3) = 0
\end{displaymath}
\begin{displaymath}
H^3(S^3, \mathbb{Z}) = \mathbb{Z} = number\ of\ string\ classes
\end{displaymath}
\begin{displaymath}
d H = d^* H = 0  \Rightarrow H \in H^3(S^3, \mathbb{R}) \simeq \mathbb{R}
\end{displaymath}
\begin{displaymath}
\itexarray{
    String\ Classes
    \\
    \downarrow
    \\
    M String^{-3} = \pi_3^S = tmf^{-3} = \mathbb{Z}/{24}
  }
\end{displaymath}
\begin{quote}%
(can't type the full diagram\ldots{})


\end{quote}
consider a 1-parameter family of ``berger metrics'' on $S^3$

rescaling the fiber in the Hopf fibration $S^1 \to S^3 \to S^2$

\hypertarget{konrad_waldorf}{}\subsection*{{Konrad Waldorf}}\label{konrad_waldorf}

\begin{quote}%
[[Urs Schreiber|Urs]]: I had to miss that and the following two talks, hopefully somebody else has notes. Konrad's talk is based on his new article


\end{quote}
\begin{itemize}%
\item [[Konrad Waldorf]], \emph{String connections and Chern-Simons theory} (\href{http://arxiv.org/abs/0906.0117}{arXiv})

\end{itemize}
\hypertarget{one_more}{}\subsection*{{one more}}\label{one_more}

\ldots{}

\hypertarget{mike_hopkins__kervaire_invariant_one}{}\subsection*{{Mike Hopkins : Kervaire invariant one}}\label{mike_hopkins__kervaire_invariant_one}

\ldots{}


\vspace{.5em} \hrule \vspace{.5em}
[[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- No next day



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