A numerical solution to the local cohomology problem in U(1) chiral gauge theories

TL;DR

This paper introduces a numerical method for solving the local cohomology problem in U(1) chiral gauge theories, enabling better analysis of gauge anomalies through explicit differentiation and integration techniques.

Contribution

It presents a novel numerical approach combining rational approximation and Gaussian quadrature for cohomological analysis in lattice gauge theories.

Findings

Successfully computed the current associated with trivial chiral anomaly parts.

Demonstrated good convergence of the numerical integration method.

Verified the locality properties of the computed current.

Abstract

We consider a numerical method to solve the local cohomology problem related to the gauge anomaly cancellation in U(1) chiral gauge theories. In the cohomological analysis of the chiral anomaly, it is required to carry out the differentiation and the integration of the anomaly with respect to the continuous parameter for the interpolation of the admissible gauge fields. In our numerical approach, the differentiation is evaluated explicitly through the rational approximation of the overlap Dirac operator with Zolotarev optimization. The integration is performed with a Gaussian Quadrature formula, which turns out to show rather good convergence. The Poincare lemma is reformulated for the finite lattice and is implemented numerically. We compute the current associated with the cohomologically trivial part of the chiral anomaly in two-dimensions and check its locality properties.