General solution of cyclic Leibniz rule

TL;DR

This paper derives the complete set of solutions to the cyclic Leibniz rule for lattice supersymmetry, revealing their structure and extending to general difference operators, thus advancing theoretical understanding in this area.

Contribution

It provides the first comprehensive characterization of solutions to the cyclic Leibniz rule, including explicit forms and extensions to general difference operators.

Findings

Solutions are uniquely expressed as combinations of fundamental and minimal solutions.

The general solution structure is characterized by cyclic invariant coefficients.

Extension to arbitrary difference operators is established.

Abstract

We study the general solution of the cyclic Leibniz rule (CLR) which was recently proposed as a new approach to the lattice supersymmetry. Introducing some mathematical preliminaries related to the cyclic symmetry, we find the general solution of the 2-body CLR for the naive symmetric difference operator. The main theorems of this paper state that the general solution can be uniquely expressed as (A) a linear combination of the two fundamental solutions with cyclic invariant coefficients, and (B) a linear combination of the minimal solutions with complex coefficients. Moreover, an extension to the general difference operators is also discussed.