Momentum space topological invariants for the 4D relativistic vacua with mass gap Original Research Article

Topological invariants for the 4D gapped system are discussed with application to the quantum vacua of relativistic quantum fields. Expression image for the 4D systems with mass gap defined in Volovik (2010) is considered. It is demonstrated that image remains the topological invariant when the interacting theory in deep ultraviolet is effectively massless. We also consider the 5D systems and demonstrate how 4D invariants emerge as a result of the dimensional reduction. In particular, the new 4D invariant image is suggested. The index theorem is proved that defines the number of massless fermions image in the intermediate vacuum, which exists at the transition line between the massive vacua with different values of image. Namely, image is equal to the jump image across the transition. The jump image at the transition determines the number of only those massless fermions, which live near the hypersurface image. The considered invariants are calculated for the lattice model with Wilson fermions.