Analytic index and chiral fermions

TL;DR

This paper reviews how an index relation can generate chiral fermions in gauge theories and clarifies the generalized Pauli-Villars regularization based on this mechanism.

Contribution

It introduces a novel application of the index relation to generate chiral fermions and clarifies the regularization scheme for chiral gauge theories.

Findings

Index relation explains chiral fermion generation.

Regularization scheme based on index relation is clarified.

Chiral structure arises from a non-trivial index mass term.

Abstract

A recent application of an index relation of the form, $dim\ ker\ M - dim\ ker\ M^{\dagger} = \nu$, to the generation of chiral fermions in a vector-like gauge theory is reviewed. In this scheme the chiral structure arises from a mass term with a non-trivial index.The essence of the generalized Pauli-Villars regularization of chiral gauge theory, which is based on this mechanism, is also clarified.