Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra

TL;DR

This paper develops a lattice supersymmetry framework using Hopf algebraic structures, addressing previous inconsistencies by introducing a braided quantum field theory approach that ensures algebraic and perturbative consistency.

Contribution

It formulates lattice supersymmetry as a Hopf algebraic superalgebra, resolving ordering ambiguities via braided structures and defining a consistent path integral approach.

Findings

Hopf algebraic supersymmetry on the lattice is established.

The Leibniz rule breakdown is managed through coproduct operations.

A braided quantum field theory formulation ensures consistency.

Abstract

The lattice superalgebra of the link approach is shown to satisfy a Hopf algebraic supersymmetry where the difference operator is introduced as a momentum operator. The breakdown of the Leibniz rule for the lattice difference operator is accommodated as a coproduct operation of (quasi)triangular Hopf algebra and the associated field theory is consistently defined as a braided quantum field theory. Algebraic formulation of path integral is perturbatively defined and Ward-Takahashi identity can be derived on the lattice. The claimed inconsistency of the link approach leading to the ordering ambiguity for a product of fields is solved by introducing an almost trivial braiding structure corresponding to the triangular structure of the Hopf algebraic superalgebra. This could be seen as a generalization of spin and statistics relation on the lattice. From the consistency of this braiding structure of fields a grading nature for the momentum operator is required.