Critical comparison of different definitions of topological charge on the lattice

A detailed comparison is made between the field-theoretic and geometric definitions of topological charge density on the lattice. Their renormalizations with respect to the continuum are analyzed. The definition of the topological susceptibility chi, as used in chiral Ward identities, is reviewed. After performing the subtractions required by it, the different lattice methods yield results in agreement with each other. The methods based on cooling and on counting fermionic zero modes are also discussed.