Generalizing the Ginsparg-Wilson relation: Lattice supersymmetry from blocking transformations

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 nonlocal SLAC-type derivatives. We investigate the corresponding kinetic operators on the lattice within an exact solution of supersymmetric quantum mechanics. These solutions---analogs of the overlap operator for supersymmetry---can be made local through a specific choice of the blocking kernel. We show that the symmetry 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.