  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.

High Energy Physics - Theory;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons;Nuclear Theory