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\centerline{\bf Facets of the Witten
genus}
\medskip
\centerline{
\bf Topics in Topology Course
Fall 2002}
\medskip
\centerline{Stephan Stolz}
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Witten genus $\varphi_W(M)$ of an oriented
manifold $M$ is an element of $\Q[[q]]$ (i.e., a
power series with rational coefficients). It
should be thought of as a generalization of the
more familiar $\wh A$\nb-genus $\wh A(M)\in \Q$.
The following table shows the various aspects of
the
$\wh A$\nb-genus and the $\varphi_W$\nb-genus that
we will cover in the course:

\begin{center} 
\begin{tabular}{|p{.8in}|p{2.5in}|p{2.5in}|}
\hline
&$\hat A$-genus
&Witten genus $\varphi_W$\\
\hline
\hline
 assumption on $M$
& $M$ admits a spin structure 
&$M$ admits a string structure\\
\hline
formal properties
& $\wh A(M)$  is an integer
& Coefficients of $\phi_W(M)$ are integral;
$\phi_W(M)$ is the
`$q$\nb-expansion' of a modular form of weight $n/2$.\\
\hline
Topology 
& $M\mapsto \wh A(M)$ induces a ring homomorphism
$\wh A\colon \Omega_*^{SO}\to \Q$;
\newline
$\Omega_*^{SO}=$ bordism ring of oriented
manifolds&
$M\mapsto \varphi_W(M)$ induces a ring
homomorphism $\varphi_W\colon
\Omega_*^{SO}\to
\Q[[q]]$\\
\hline
Analysis&
$\wh A(M)$ is the index of the Dirac operator on a
manifold $M$&
heuristically: $\varphi_W(M)$ is the 
$S^1$\nb-equivariant index of the `Dirac operator'
on the free loop space $LM=\{\gamma\colon S^1\to
M\}$\\
\hline
Geometry
&Theorem (Lichnerowicz): If $M$
admits a Riemannian metric of positive scalar curvature,
then $\wh A(M)=0$. &Conjecture (H\"ohn, Stolz): If $M$
admits a Riemannian metric of positive Ricci curvature, then
$\phi_W(M)=0$.\\
\hline
Physics
& $\wh A(M)=$ partition function of
a $0+1$\nb-dimensional conformal field theory
& heuristically: $\varphi_W(M)=$ partition
function of a
$1+1$\nb-dimensional conformal field theory\\
\hline
\end{tabular}
\end{center}

{\bf Literature.}
The course will draw from many
sources: the book by Hirzebruch-Berger-Jung
\cite{HBJ} covers the topological aspects; for the
analytical and geometrical aspects of the Dirac
operator I recommend the book by
Lawson-Michelsohn \cite{LM}. The interpretation of
the Witten genus as
$S^1$\nb-equivariant index of the Dirac operator
on loop spaces and its interpretation as a
partition function is explained in the Witten
papers \cite{Wi1},\cite{Wi2}. The conjectural
relationship of the Witten genus and positive
Ricci curvature is explained in my paper
\cite{St}. Later parts of the course will be based
on recent joint work with Peter Teichner,
presented in a series of fourteen
lectures at the workshop `topics on conformal
field theory' in M\"unster, Germany in March of
this year.

{\bf Prerequisites.} The prerequisites for the
course are the first year graduate courses in
topology, algebra and analysis as well as a
willingness to absorb a lot of
definitions/ constructions/ theorems. To get
something out out this course, participants should
be willing to do some homework problems and to go
over notes of previous lectures. The emphasis will
be on developing concepts, doing examples and
stating results rather than on presenting detailed
proofs. The pace of the course will be somewhat
rapid, but largely determined by the audience and
suggestions from the audience. 

{\bf Related courses/seminars.} The course will
continue in the spring semester,  with focus on
possible generalizations of the Witten genus for
families of string manifolds parameterized by a
space $X$; this `families Witten genus' is an
element of the elliptic cohomology of $X$.
So far, there is only a {\it stable
homotopy theoretic} construction of
elliptic cohomology by Hopkins and Miller; we plan
a seminar on that subject this fall with talks by
the participants. Both, the
course and the seminar have the same goal: 
gaining an understanding of the Witten genus;
still, due to the differences of approach
(geometric in the course, homotopy theoretic in
the seminar),  the direct points of contact will
be somewhat spurious. Closely related to  the
conformal field theory point of view of the Witten
genus is Katrina Barron's course this fall.

\begin{thebibliography}{blah}
\bibitem[HBJ]{HBJ}
Hirzebruch, Friedrich;
Berger, Thomas; Jung, Rainer, 
\textit{Manifolds and modular forms},  Aspects of
Mathematics, E20.  Friedr. Vieweg \& Sohn,
Braunschweig, 1992. xii+211 pp.

\bibitem[LM]{LM}
Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise,
\textit{Spin geometry}. Princeton Mathematical
Series, 38. Princeton University Press, Princeton,
NJ, 1989. xii+427 pp.

\bibitem[St]{St} Stolz, Stephan, \textit{A
conjecture concerning positive Ricci curvature and
the Witten genus},
 Math. Ann. 304 (1996), no. 4, 785--800

\bibitem[Wi1]{Wi1}
Witten, Edward, \textit{The index of the Dirac
operator in loop space}, in `Elliptic curves and
modular forms in algebraic topology' (Princeton,
NJ, 1986), 161--181, Lecture Notes in Math., 1326,
Springer, Berlin, 1988

\bibitem[Wi2]{Wi2}
Witten, Edward,
\textit{Index of Dirac operators},
in `Quantum
fields and strings: a course for mathematicians',
Vol. 1, 2 (Princeton, NJ, 1996/1997), 475--511,
Amer. Math. Soc., Providence, RI, 1999
\end{thebibliography}
\end{document}
