The fuzzy disc

We introduce a finite dimensional matrix model approximation to the algebraof functions on a disc based on noncommutative geometry. The algebra is asubalgebra of the one characterizing the noncommutative plane with a * productand depends on two parameters N and theta. It is composed of functions whichdecay exponentially outside a disc. In the limit in which the size of thematrices goes to infinity and the noncommutativity parameter goes to zero thedisc becomes sharper. We introduce a Laplacian defined on the whole algebra andcalculate its eigenvalues. We also calculate the two--points correlationfunction for a free massless theory (Green's function). In both cases theagreement with the exact result on the disc is very good already for relativelysmall matrices. This opens up the possibility for the study of field theorieson the disc with nonperturbative methods. The model contains edge states, afact studied in a similar matrix model independently introduced byBalachandran, Gupta and Kurkcuoglu.Comment: 17 pages, 8 figures, references added and correcte