# Topological charge on the lattice: a field theoretical view of the geometrical approach

## TL;DR

This paper introduces a new perspective on lattice topological charge by constructing sequences of analytical operators that approach geometrical definitions, highlighting potential non-perturbative effects in the limits.

## Contribution

It proposes a novel interpretation of geometrical topological charges as limits of field theoretical operators, bridging analytical and geometrical approaches.

## Key findings

- Sequences of operators can approach geometrical definitions in models.
- Perturbative effects tend to vanish along these sequences.
- Non-perturbative renormalizations may persist in the limits.

## Abstract

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.