Chiral fermions on the lattice and index relations

TL;DR

This paper investigates the relationship between lattice and continuum formulations of chiral fermions, clarifying the index theorem and anomaly limits, and comparing with the Atiyah-Singer theorem to resolve longstanding puzzles.

Contribution

It provides a detailed analysis of the continuum limit of lattice chiral fermions and establishes the index relation, connecting lattice results with classical index theorems.

Findings

Established the continuum limit of the anomaly term.

Proved the index relation for lattice chiral fermions.

Compared lattice results with the Atiyah-Singer theorem.

Abstract

Comparing recent lattice results on chiral fermions and old continuum results for the index puzzling questions arise. To clarify this issue we start with a critical reconsideration of the results on finite lattices. We then work out various aspects of the continuum limit. After determining bounds and norm convergences we obtain the limit of the anomaly term. Collecting our results the index relation of the quantized theory gets established. We then compare in detail with the Atiyah-Singer theorem. Finally we analyze conventional continuum approaches.