  It is shown that the BRS-formulated two-dimensional BF theory in the
light-cone gauge (coupled with chiral Dirac fields) is solved very easily in
the Heisenberg picture. The structure of the exact solution is very similar to
that of the BRS-formulated two-dimensional quantum gravity in the conformal
gauge. In particular, the BRS Noether charge has anomaly. Based on this fact, a
criticism is made on the reasoning of Kato and Ogawa, who derived the critical
dimension D=26 of string theory on the basis of the anomaly of the BRS Noether
charge. By adding the $\widetilde{B}^2$ term to the BF-theory Lagrangian
density, the exact solution to the two-dimensional Yang-Mills theory is also
obtained.
