Probability distribution of the index in gauge theory on 2d non-commutative geometry

We investigate the effects of non-commutative geometry on the topologicalaspects of gauge theory using a non-perturbative formulation based on thetwisted reduced model. The configuration space is decomposed into topologicalsectors labeled by the index nu of the overlap Dirac operator satisfying theGinsparg-Wilson relation. We study the probability distribution of nu by MonteCarlo simulation of the U(1) gauge theory on 2d non-commutative space withperiodic boundary conditions. In general the distribution is asymmetric undernu -> -nu, reflecting the parity violation due to non-commutative geometry. Inthe continuum and infinite-volume limits, however, the distribution turns outto be dominated by the topologically trivial sector. This conclusion isconsistent with the instanton calculus in the continuum theory. However, it isin striking contrast to the known results in the commutative case obtained fromlattice simulation, where the distribution is Gaussian in a finite volume, butthe width diverges in the infinite-volume limit. We also calculate the averageaction in each topological sector, and provide deeper understanding of theobserved phenomenon.Comment: 16 pages,10 figures, version appeared in JHE