  We consider the $ U(1) $ sigma model in the two dimensional space-time which
is a field-theoretical model possessing a nontrivial topology. It is pointed
out that its topological structure is characterized by the zero-mode and the
winding number. A new type of commutation relations is proposed to quantize the
model respecting the topological nature. Hilbert spaces are constructed to be
representation spaces of quantum operators. It is shown that there are an
infinite number of inequivalent representations as a consequence of the
nontrivial topology. The algebra generated by quantum operators is deformed by
the central extension. When the central extension is introduced, it is shown
that the zero-mode variables and the winding variables obey a new commutation
relation, which we call twist relation. In addition, it is shown that the
central extension makes momenta operators obey anomalous commutators. We
demonstrate that topology enriches the structure of quantum field theories.
