Topology of generalized spinors and chiral anomaly

Weyl fermions with nonlinear dispersion have appeared in real-world systems, such as in the Weyl semimetals 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 literature, and also lead to a compact proof of the Nielsen-Ninomiya theorem.