Topology of Generalized Spinors and Chiral Anomaly

TL;DR

This paper explores the topological properties of generalized Dirac operators and their relation to chiral anomaly, providing new insights into topological invariants in various dimensions and clarifying debates on the chiral anomaly-Chern number connection.

Contribution

It introduces a comprehensive analysis of topological quantities associated with generalized Dirac operators, revealing their enhancement by a momentum-space winding number across dimensions.

Findings

Topological quantities are enhanced by a momentum-space winding number.

Chiral anomaly in 3+1 dimensions confirmed via Feynman diagrams.

Clarifies the relationship between chiral anomaly and Berry connection Chern number.

Abstract

Weyl fermions with nonlinear dispersion have appeared in real world systems, such as in the Weyl semi-metals and topological insulators. We consider the most general form of Dirac operators, and study its topological properties embedded in the chiral anomaly, in the index theorem, and in the odd-dimensional partition function, by employing the heat kernel. We find that all of these topological quantities are enhanced by a winding number defined by the Dirac operator in the momentum space, regardless of the spacetime dimensions. The chiral anomaly in $d=3+1$, in particular, is also confirmed via the conventional Feynman diagram. These interconnected results allow us to clarify the relationship between the chiral anomaly and the Chern number of the Berry connection, under dispute in some recent literatures, and also lead to a compact proof of the Nielsen-Ninomiya theorem.