No-Go Theorem of Leibniz Rule and Supersymmetry on the Lattice

TL;DR

This paper proves a no-go theorem showing the impossibility of maintaining Leibniz rule, translation invariance, and locality simultaneously in lattice supersymmetry, but suggests a workaround using infinite flavors to preserve supersymmetry.

Contribution

The paper establishes a no-go theorem for lattice supersymmetry and proposes a novel approach with infinite flavors to realize exact supersymmetry on the lattice.

Findings

Proof of no-go theorem for lattice supersymmetry

Introduction of infinite flavors to bypass the theorem

Construction of an N=2 supersymmetric lattice model

Abstract

An obstacle to realize supersymmetry on a lattice is the breakdown of Leibniz rule. We give a proof of a no-go theorem that it is impossible to construct a lattice field theory in an infinite lattice volume with any nontrivial field products and difference operators that satisfy the following three properties: (i) translation invariance, (ii) locality and (iii) Leibniz rule. We then propose a way to escape from the no-go theorem by introducing infinite flavors, and present a lattice model of N=2 supersymmetric quantum mechanics equipped with the full exact supersymmetry.