The topological renormalisation of the 0(3) sigma model

Abstract

Like other field theories of physical interest, the moduli-space integrals of the non-linear two-dimensional 0(3) sigma model diverge. We show that in the one-instanton sector the imposition of a cut-off in the moduli-space leads to an unacceptable dependence of the Green’s function on the way that the field is split into the quantum piece and the classical background. This dependence may be isolated in a term which may be interpreted as an anomaly to the Ward Identity of the theory. The moduli-space divergence is associated with degeneration of the field configurations to those of another topological sector. Hence it is possible that by modifying the Green’s function in, say, the zero-instanton sector will be able to cancel the divergence in the one- instanton sector. We show that the Ward Identity anomaly in the one-instanton sector may be written in the zero-instanton sector at next to leading order in powers of h, and hence we explicitly calculate the Green's function modification. We have called the process of applying this modification "Topological Renormalisation". A central piece of the modification term is the instanton contribution to the Green's function of the model. This is obtained by using two new methods of calculating the determinant of the fluctuation operator. The application of Topological Renormalisation to other theories is also investigated.