THE SPECTRAL ACTION PRINCIPLE IN NONCOMMUTATIVE GEOMETRY AND THE SUPERSTRING

A supersymmetric theory in two dimensions has enough data to define a noncommutative space thus making it possible to use all tools of nonco mmutative geometry. In particular, we apply this to the N=1 supersymme tric non-linear sigma model and derive an expression for the generaliz ed loop space Dirac operator, in presence of a general background, usi ng canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the strin g modes. (C) 1997 Published by Elsevier Science B.V.