Gauge Theory on the Fuzzy Sphere and Random Matrices

Gauge theories provide the best known description of the fundamental forces in nature. At very short distances however, physics is not known, and it seems unlikely that spacetime is a perfect continuum down to arbitrarily small scales. Indeed, physicists have started to learn in recent years how to formulate field theory on quantized, or noncommutative spaces. However, most attempts to (second-) quantize these field theories using modifications of the conventional, perturbative methods have failed up to now. It seems therefore worthwhile to try to develop new techniques for their quantization, trying to take advantage of the peculiarities of the non-commutative case. There is indeed one striking feature of some non-commutative gauge theories: they can be formulated as (multi-) matrix models. I will explain here the main ideas of [1], where such a matrix formulation of (pure) U(n) gauge theory on the fuzzy sphere has been used to calculate its partition function in the commutative limit. This is done using matrix techniques which cannot be applied in the commutative case.