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These notes comprise the second part of two articles devoted to theconstruction of exact solutions of noncommutative gauge theory in two spacetimedimensions. Here we shall deal with the quantum field theory. Topics coveredinclude an investigation of the symmetries of quantum gauge theory on thenoncommutative torus within the path integral formalism, the derivation of theexact expression for the vacuum amplitude, and the classification of instantoncontributions. A section dealing with a new, exact combinatorial solution ofgauge theory on a two-dimensional fuzzy torus is also included.Comment: 33 pages, 3 figures, uses svmult.cls; Based on invited lectures given  by the second author at the ``International Workshop on Quantum Field Theory  and Noncommutative Geometry'', November 26-30 2002, Tohoku University,  Sendai, Japan; To be published in the proceedings by Springer-Verlag as  Lecture Notes in Physic

Lectures on Two-Dimensional Noncommutative Gauge Theory Quantization