Bott Periodicity and Realizations of Chiral Symmetry in Arbitrary Dimensions

TL;DR

This paper explores how chiral symmetries in vector-like gauge theories with massless fermions exhibit Bott periodicity across different dimensions, affecting the structure of fermion condensates and topological properties.

Contribution

It demonstrates that chiral symmetry patterns and residual symmetries follow Bott periodicity in arbitrary dimensions for various fermion representations.

Findings

Chiral symmetries follow Bott periodicity in all dimensions.

Quadratic fermion condensates are characterized for different representations.

Topological features of low-energy theories are influenced by these symmetry patterns.

Abstract

We compute the chiral symmetries of the Lagrangian for confining "vector-like" gauge theories with massless fermions in $d$-dimensional Minkowski space and, under a few reasonable assumptions, determine the form of the quadratic fermion condensates which arise through spontaneous breaking of these symmetries. We find that for each type (complex, real, or pseudoreal) of representation of the gauge group carried by the fermions, the chiral symmetries of the Lagrangian, as well as the residual symmetries after dynamical breaking, exactly follow the pattern of Bott periodicity as the dimension changes. The consequences of this for the topological features of the low-energy effective theory are considered.