Generalisations of the Ginsparg-Wilson relation and a remnant of supersymmetry on the lattice

TL;DR

This paper introduces a generalized lattice symmetry relation extending the Ginsparg-Wilson relation, enabling the construction of lattice actions with remnant supersymmetry, including non-local derivative operators, and discusses solutions for both free and interacting theories.

Contribution

It generalizes the Ginsparg-Wilson relation to include supersymmetry on the lattice, providing a framework for invariant actions with non-local derivatives.

Findings

Remnant supersymmetry can be realized on the lattice using the new relation.

Local solutions exist for quadratic actions despite non-local derivatives.

Interacting theories generally lead to non-polynomial actions, reducible under certain conditions.

Abstract

We introduce a lattice symmetry relation for field theories with general linear symmetries. For chiral symmetry the well-known Ginsparg-Wilson relation is reproduced. The new relation encodes the remnant of the original symmetry on the lattice and guides the construction of invariant lattice actions. We apply this approach to lattice supersymmetry. There, an additional constraint has to be satisfied which originates in the derivative operator in the symmetry transformations. As a consequence the non-local SLAC derivative operator appears in the lattice transformation. Despite this non-local form we show how local solutions for quadratic actions can be found. For interacting theories the relation in general leads to a non-polynomial action that can be reduced to a finite polynomial order only under certain conditions.