## Abstract

Topological invariants for the 4D gapped system are discussed with application to the quantum vacua of relativistic quantum fields. Expression N~3 for the 4D systems with mass gap defined in Volovik (2010) [13] is considered. It is demonstrated that 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 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 N~5. Namely, 2n _{F} is equal to the jump δN~5 across the transition. The jump δN~3 at the transition determines the number of only those massless fermions, which live near the hypersurface ω=0. The considered invariants are calculated for the lattice model with Wilson fermions.

Original language | English |
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Pages (from-to) | 295-309 |

Number of pages | 15 |

Journal | Nuclear Physics B |

Volume | 860 |

Issue number | 2 |

DOIs | |

State | Published - 11 Jul 2012 |

Externally published | Yes |

## Keywords

- Fermionic vacua
- Lattice gauge theory
- Momentum space topological invariants
- Semimentals
- Topological insulators
- Wilson fermions