The index of the overlap Dirac operator on a discretized 2d non-commutative torus

The index, which is given in terms of the number of zero modes of the Diracoperator with definite chirality, plays a central role in various topologicalaspects of gauge theories. We investigate its properties in non-commutativegeometry. As a simple example, we consider the U(1) gauge theory on adiscretized 2d non-commutative torus, in which general classical solutions areknown. For such backgrounds we calculate the index of the overlap Diracoperator satisfying the Ginsparg-Wilson relation. When the action is small, thetopological charge defined by a naive discretization takes approximatelyinteger values, and it agrees with the index as suggested by the index theorem.Under the same condition, the value of the index turns out to be a multiple ofN, the size of the 2d lattice. By interpolating the classical solutions, weconstruct explicit configurations, for which the index is of order 1, but theaction becomes of order N. Our results suggest that the probability ofobtaining a non-zero index vanishes in the continuum limit, unlike thecorresponding results in the commutative space.Comment: 22 pages, 8 figures, LaTeX, JHEP3.cls. v3:figures 1 and 2 improved (all the solutions included),version published in JHE