Remark: The complex first-jet distinction

The symmetric real part suffices for the stress tensor but not for the current. If $H$ is replaced by $\mathrm{Re}H$ before taking the antisymmetric first derivative, the azimuthal current is lost.

In the periodic gauge,

The covariant derivatives in (eq:current-prescription) cancel this phase. The prescription is therefore gauge invariant. The parametrix must be transformed with the same parallel factor.