\section{Main theorem}
The main theorem states that the Witten class of a complex manifold $X$ arises when one considers the factorization homology of a certain sheaf (on $X$) of translation-invariant factorization algebras on $\C$.   Before I state this theorem, I need to recall the definition of the Witten class.

\numberedparagraph
Let $E$ be an elliptic curve, and let $\omega$ be a translation-invariant volume element on $E$.  The Witten class
$$
\op{Wit}(X,E,\omega) \in \R\Gamma(X, \Omega^{-\ast}(X))   = \oplus H^i( X, \Omega^i(X)) 
$$
is a cohomology class, defined as follows.

Let
$$
E_{2k}(E,\omega) = \sum_{\lambda \in \Lambda} \lambda^{-2k}
$$
be the Eisenstein series of the marked elliptic curve $(E,\omega)$. Here, as before, we are writing $E$ as the quotient of $\C$ by a lattice $\Lambda$, in such a way that $\omega$ corresponds to $\d z$.

The Witten class of $X$ is defined by
$$
\op{Wit}(X, E,\omega) = \exp \left\{\sum_{k \ge 2}  \frac{(2k-1)!}{ (2 \pi i )^{2k} }  E_{2k}(E,\omega) ch_{2k} ( T X) \right\}.
$$

If $\tau$ is in the upper half-plane, let $(E_\tau, \omega_\tau)$ denote the elliptic curve associated to the lattice generated by $(1,\tau)$, with volume form $\omega_\tau$ corresponding to $\d z$.  Then, the Witten class has the property that
$$
\lim_{\tau \to i \infty} \op{Wit}(X,E_\tau, \omega_\tau) = e^{-c_1(T_X) / 2} \op{Td}(T X) .
$$
This follows from the identities
\begin{align*}
\lim_{\tau \to i \infty} E_{2k} ( E_\tau, \omega_\tau ) &= 2 \zeta(2k ) \\
\sum_{k \ge 1} 2 \zeta(2k)  \frac{x^{2k}}{ 2k (2 \pi i)^{2k} }  &= \log \left( \frac{x}{1 - e^{-x} } \right) - \frac{x}{2}
\end{align*}
where $\zeta$ is the Riemann zeta function.

\numberedparagraph
Now we can state the theorem.
\begin{theorem}
Let $X$ be a complex manifold, equipped with a trivialization of the second Chern character $ch_2(T X)$.  Then, there is a sheaf $D^{\hbar}_{X,ch}$ of translation-invariant factorization algebras on $\C$, over the algebra $\C[\hbar]$, such that, for every elliptic curve $E$ with volume element $\omega$, there is a natural isomorphism of BD algebras
$$
FH( E, D^{\hbar}_{X,ch} ) \simeq  \left( \Omega^{-\ast} (T^\ast X)[\hbar], \hbar L_\pi + \hbar \{ \log \op{Wit}(X,E,\omega) , - \} \right).
$$

Alternatively, there is an isomorphism of cochain complexes
$$
FH( E, D^{\hbar}_{X,ch} ) \simeq  \left( \Omega^{-\ast} (T^\ast X)[\hbar], \hbar L_\pi\right).
$$
sending
$$
1 \to \op{Wit}(X,E,\omega).
$$
\end{theorem} 

\numberedparagraph

The factorization algebra I construct is  an analytic avatar of the chiral differential operators constructed by Gorbounov, Mailkov and Schechtman \cite{GorMalSch00}.  Note that in their work, the $q$-expansion of the Witten genus appears as the character of the algebra of chiral differential operators. The way the Witten genus appears in this paper is somewhat different, and has the advantage that we see the Witten genus directly as a function on the moduli space of elliptic curves, and not just as a $q$-expansion.   A further analysis of the relationship between the $q$-expansion of the Witten genus and the chiral differential operators has been undertaken by Cheung \cite{Che08}.
{"config": "arxiv", "file": "1006.5422/d_main_theorem.tex"}