Axial anomaly and topological charge in lattice gauge theory with Overlap Dirac operator

TL;DR

This work establishes, non-perturbatively, that the axial anomaly density derived from the overlap Dirac operator reproduces the continuum density $q^A(x)$ in the classical limit for $0<m_0<2$ across broad gauge fields, including topologically non-trivial backgrounds, by a novel power-series expansion around $H^2=L-V$ and an integral representation of $1/ oot\{H^2\}$. It also analyzes the lattice index in infinite volume, showing it is generally ill-defined, but proves that in finite-volume settings with suitable pure-gauge behavior at infinity, the index recovers the continuum topological charge $Q(A)$ via the $k=2$ contribution; the finite-volume analysis further yields $ ext{index} ext{D}^U=Q(A)$ when $0<m_0<2$. A key result is the explicit computation of the momentum-space integral $I(m)$, which equals 1 for $0<m<2$ and takes a piecewise constant sequence for other intervals, guiding when the lattice index matches the continuum topology. The paper thus clarifies the conditions under which the overlap formulation faithfully encodes topological and anomaly data, with implications for non-perturbative studies and potential generalizations to higher dimensions and torus geometries.

Abstract

An explicit, detailed evaluation of the classical continuum limit of the axial anomaly/index density of the overlap Dirac operator is carried out in the infinite volume setting, and in a certain finite volume setting where the continuum limit involves an infinite volume limit. Our approach is based on a novel power series expansion of the overlap Dirac operator. The correct continuum expression is reproduced when the parameter $m_0$ is in the physical region $0<m_0<2$. This is established for a broad range of continuum gauge fields. An analogous result for the fermionic topological charge, given by the index of the overlap Dirac operator, is then established for a class of topologically non-trivial fields in the aforementioned finite volume setting. Problematic issues concerning the index in the infinite volume setting are also discussed.