Topological charge on the lattice: a field theoretical view of the   geometrical approach
  We construct sequences of ``field theoretical'' (analytical) lattice topological charge density operators which formally approach geometrical definitions in 2-d $CP^{N-1}$ models and 4-d $SU(N)$ Yang Mills theories. The analysis of these sequences of operators suggests a new way of looking at the geometrical method, showing that geometrical charges can be interpreted as limits of sequences of field theoretical (analytical) operators. In perturbation theory renormalization effects formally tend to vanish along such sequences. But, since the perturbative expansion is asymptotic, this does not necessarily lead to well behaved geometrical limits. It indeed leaves open the possibility that non-perturbative renormalizations survive. 
Leonardo Rastelli, Paolo Rossi, and Ettore Vicari (University of Pisa)