  We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that
it is a topological conformal field theory. This opens the possibility to
compute topological string amplitudes (Gromov-Witten invariants). We found that
the theory is exactly solvable in the sense that all correlators are given by
finite-dimensional convergent integrals. Surprisingly, this theory turns out to
be logarithmic in the sense that there are correlators given by polylogarithms
and powers of logarithms. Furthermore, we found fields with "logarithmic
conformal dimension" (elements of a Jordan cell for $L_0$). We also found
certain vertex operators with anomalous dimensions that depend on the
non-abelian coupling constant. The shift of dimension of composite fields may
be understood as arising from the dependence of subtracted singular terms on
local coordinates. This generalizes the well-known explanation of anomalous
dimensions of vertex operators in the free scalar field theory.
