  We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves
on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$
is smooth and of dimension four. Moreover, the Abel-Jacobi map to the
intermediate Jacobian of $X$ maps it birationally onto the theta divisor
$\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to the
singular point $0 \in \Theta$.
  We use this result to give a new proof of a categorical version of the
Torelli theorem for cubic threefolds, which says that $X$ can be recovered from
its Kuznetsov component $\operatorname{Ku}(X) \subset
\mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of the
description of the singularity of the theta divisor, and thus of the classical
Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its
intermediate Jacobian.
