Covariant Wigner--Weyl calculus for gauge theories: non-dissipative transport and chiral anomaly in inhomogeneous systems with nontrivial momentum-space topology

We consider fermions interacting with an external, in general non-Abelian, gauge field in an essentially inhomogeneous background: the Dirac operator Q(x,−iD) may be of arbitrary order in covariant derivatives and may depend explicitly on coordinates through background fields. For this setting we develop a covariant Wigner–Weyl calculus, in which the Wigner transform of the two-point Green function transforms as in the adjoint (two-index) representation of the gauge group. We derive the inverse transform, the star product in terms of a triangular Wilson loop, its differential representation and its expansion in powers of the field strength, and solve
the generalized Groenewold equation iteratively. The formalism is applied in two directions. (i) Non-dissipative transport: the responses of the vector and axial
currents to the non-Abelian field strength are expressed through topological invariants composed of Wigner-transformed Green functions; this gives the non-Abelian versions of the chiral separation effect and of the quantum Hall effect. (ii) The chiral anomaly: using point-splitting regularization and the Atiyah–Singer
theorem, we show that the anomaly factorizes into the instanton number times the momentum-space invariant N3 responsible for the stability of Fermi points, the same invariant that governs the chiral separation effect.

Following Phys. Rev. D 112, 056035 (2025) and Phys. Lett. B 871, 140021 (2025) [PLB]