Solving the local cohomology problem in U(1) chiral gauge theories within a finite lattice

TL;DR

This paper presents a method to solve the local cohomology problem in U(1) chiral gauge theories on finite lattices, enabling the construction of gauge-invariant measures for Weyl fermions.

Contribution

It reformulates the Poincaré lemma for finite lattices, allowing the local cohomology problem to be addressed with exponentially small corrections.

Findings

Reformulation of the Poincaré lemma on finite lattices

Solution to the local cohomology problem in abelian chiral gauge theories

Construction of Weyl fermion measures directly from finite lattice quantities

Abstract

In the gauge-invariant construction of abelian chiral gauge theories on the lattice based on the Ginsparg-Wilson relation, the gauge anomaly is topological and its cohomologically trivial part plays the role of the local counter term. We give a prescription to solve the local cohomology problem within a finite lattice by reformulating the Poincar\'e lemma so that it holds true on the finite lattice up to exponentially small corrections. We then argue that the path-integral measure of Weyl fermions can be constructed directly from the quantities defined on the finite lattice.