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

TL;DR

This paper explores topological invariants in 4D relativistic quantum vacua, demonstrating their robustness under interactions, deriving new invariants via dimensional reduction, and establishing a theorem linking invariants to massless fermion counts.

Contribution

It introduces a new 4D topological invariant $ ilde{ }_5$, proves an index theorem relating invariants to massless fermions, and analyzes invariants' behavior during phase transitions.

Findings

$ ilde{ }_3$ remains invariant under interactions in the ultraviolet regime.

A new 4D invariant $ ilde{ }_5$ is proposed via dimensional reduction.

The index theorem relates the jump in invariants to the number of massless fermions.

Abstract

Topological invariants for the 4D gapped system are discussed with application to the quantum vacua of relativistic quantum fields. Expression $\tilde{\cal N}_3$ for the 4D systems with mass gap defined in \cite{Volovik2010} is considered. It is demonstrated that $\tilde{\cal N}_3$ 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 $\tilde{\cal N}_5$ is suggested. The index theorem is proved that defines the number of massless fermions $n_F$ in the intermediate vacuum, which exists at the transition line between the massive vacua with different values of $\tilde{\cal N}_5$. Namely, $ 2 n_F$ is equal to the jump $\Delta\tilde{\cal N}_5$ across the transition. The jump $\Delta\tilde{\cal N}_3$ at the transition determines the number of only those massless fermions, which live near the hypersurface $\omega=0$. The considered invariants are calculated for the lattice model with Wilson fermions.