Onsager Algebras and Lattice Fermions

Abstract

We study the exact Onsager algebra of free lattice fermions in staggered, Wilson, and overlap incarnations, working in both Hamiltonian and Euclidean formulations in four spacetime dimensions. In each case, we show that the exact lattice algebra emanates the full chiral symmetry of the low-energy theory: if there are $N$ massless Dirac fermions in the continuum, the Onsager algebra reduces to a $U(2N)$ chiral symmetry. The details of which generators are quantised or ultra-local differ between formulations. There is also a trade-off between the locality of generators and the maximum compact subgroup of the Onsager algebra.