  We give an interpretation of the holographic correspondence between
two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm
PSL}(2,\mathbb R)$ and Schwarzian quantum mechanics in terms of a
Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence
of certain edge states imposing a first class constraint on the model. The
constrained path integral localizes over exceptional Virasoro coadjoint orbits.
The reduced theory is governed by the Schwarzian action functional generating a
Hamiltonian $S^1$-action on the orbits. The partition function is given by a
sum over topological sectors (corresponding to the exceptional orbits), each of
which is computed by a formal Duistermaat-Heckman integral.
