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Numerical Results for U(1) Gauge Theory on 2d and 4d Non-Commutative Spaces

We present non-perturbative results for U(1) gauge theory in spaces, which include a non-commutative plane. In contrast to the commutative space, such gauge theories involve a Yang-Mills term, and the Wilson loop is complex on the non-perturbative level. We first consider the 2d case: small Wilson loops follows an area law, whereas for large Wilson loops the complex phase rises linearly with the area. In four dimensions the behavior is qualitatively similar for loops in the non-commutative plane, whereas the loops in other planes follow closely the commutative pattern. In d=2 our results can be extrapolated safely to the continuum limit, and in d=4 we report on recent progress towards this goal.