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This paper discusses the canonical quantization of 1+1-dimensional Yang-Millstheory on a spacetime cylinder, from the point of view of coherent states, orequivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,the coherent states are simply ordinary coherent states labeled by points in aninfinite-dimensional linear phase space. Gauge symmetry is imposed byprojecting the original coherent states onto the gauge-invariant subspace,using a suitable regularization procedure. We obtain in this way a new familyof "reduced" coherent states labeled by points in the reduced phase space,which in this case is simply the cotangent bundle of the structure group K.  The main result explained here, obtained originally in a joint work of theauthor with B. Driver, is this: The reduced coherent states are precisely thoseassociated to the generalized Segal-Bargmann transform for K, as introduced bythe author from a different point of view. This result agrees with that of K.Wren, who uses a different method of implementing the gauge symmetry. Thecoherent states also provide a rigorous way of making sense out of the quantumHamiltonian for the unreduced system.  Various related issues are discussed, including the complex structure on thereduced phase space and the question of whether quantization commutes withreduction

COHERENT STATES AND THE QUANTIZATION OF (1+1)-DIMENSIONAL YANG–MILLS THEORY