Generalising the Ginsparg-Wilson relation: Lattice Supersymmetry from Blocking Transformations

TL;DR

This paper extends the Ginsparg-Wilson relation to lattice supersymmetry, proposing a framework for constructing local lattice actions that preserve supersymmetry through blocking transformations and specific derivative operators.

Contribution

It introduces a generalized Ginsparg-Wilson relation for interacting theories with linear symmetries, enabling the construction of local lattice supersymmetric actions.

Findings

Derived a non-local SLAC-type derivative satisfying the symmetry constraints.

Constructed supersymmetric lattice operators analogous to the overlap operator.

Identified conditions under which lattice actions remain polynomial in fields.

Abstract

The Ginsparg-Wilson relation is extended to interacting field theories with general linear symmetries. Our relation encodes the remnant of the original symmetry in terms of the blocked fields and guides the construction of invariant lattice actions. We apply this approach in the case of lattice supersymmetry. An additional constraint has to be satisfied because of the appearance of a derivative operator in the symmetry transformations. The solution of this constraint leads to non-local SLAC-type derivatives. We investigate the corresponding kinetic operators on the lattice within an exact solution of supersymmetric quantum mechanics. These solutions - analogues of the overlap operator for supersymmetry - can be made local through a specific choice of the blocking kernel. We show that the corresponding relation allows for local lattice symmetry operators as well as local lattice actions. We argue that for interacting theories the lattice action is polynomial in the fields only under special circumstances, which is exemplified within an exact solution.