  We develop a systematic approach to obtain an effective Lagrangian for 2D
non-Abelian topological BF theory. A general expression is presented in a
diagrammatic representation containing solely scalar fields. Expressions for
the SU(2) and SU(3) effective actions are explicitly stated. In the case of
SU(2), we show that the effective action can be interpreted as a winding
number. By using the SU(2) effective action, the partition function on a sphere
for SU(2) Yang-Mills theory is calculated. Moreover, we generalise the theory
to include a source term for the gauge field as well as calculate the vacuum
expectation value of the Wilson loop based on the effective theory.
