  An effective theory for 2D non-Abelian topological BF theory is investigated.
We develop a systematic approach by integrating out the non-Abelian gauge
fields to obtain an effective theory 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. The
result obeys the area law agreeing with known results.
