  Every conformal field theory (CFT) above two dimensions contains an infinite
set of Regge trajectories of local operators which, at large spin, asymptote to
"double-twist" composites with vanishing anomalous dimension. In two
dimensions, due to the existence of local conformal symmetry, this and other
central results of the conformal bootstrap do not apply. We incorporate exact
stress tensor dynamics into the CFT$_2$ analytic bootstrap, and extract several
implications for AdS$_3$ quantum gravity. Our main tool is the Virasoro fusion
kernel, which we newly analyze and interpret in the bootstrap context. The
contribution to double-twist data from the Virasoro vacuum module defines a
"Virasoro Mean Field Theory" (VMFT), its spectrum includes a finite number of
discrete Regge trajectories, whose dimensions obey a simple formula exact in
the central charge $c$ and external operator dimensions. We then show that VMFT
provides a baseline for large spin universality in two dimensions: in every
unitary compact CFT$_2$ with $c > 1$ and a twist gap above the vacuum, the
double-twist data approaches that of VMFT at large spin $\ell$. Corrections to
the large spin spectrum from individual non-vacuum primaries are exponentially
small in $\sqrt{\ell}$ for fixed $c$. We analyze our results in various large
$c$ limits. Further applications include a derivation of the late-time behavior
of Virasoro blocks at generic $c$, a refined understanding and new derivation
of heavy-light blocks, and the determination of the cross-channel limit of
generic Virasoro blocks. We deduce non-perturbative results about the bound
state spectrum and dynamics of quantum gravity in AdS$_3$.
