Is the [formula omitted]-graded sine-Gordon equation integrable?

Andrew James Bruce

doi:10.1016/j.nuclphysb.2021.115514

Abstract

We examine the question of the integrability of the recently defined Z2×Z2-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-Bäcklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.

Highlights

There has been some renewed interest in Z2 × Z2-gradings in physics, see for example [1,2,3,8,24] and references therein
We examine the natural question of the integrability of the Z2 × Z2graded sine-Gordon equation
We show that we do have a conserved current in this way via direct calculation: D+ sin /2 (D−)j− + D− sin /2 (D+)j+

Summary

Introduction

There has been some renewed interest in Z2 × Z2-gradings (and higher) in physics, see for example [1,2,3,8,24] and references therein. Graded sine-Gordon equation once the non-zero degree component fields are set to zero. An auto-Bäcklund transformation for the Z2 × Z2-supersymmetric sine-Gordon equation is defined as follows: D− = D− + 2a λ+ sin. There is little loss in generality with this condition as, via mimicking standard superspace methods show that only the component fields independent of z remain after performing the Z2 × Z2-graded Berezin integral (see [7] for details). By assumption we have no terms involving z and so D− and D+ commute and the Z2 × Z2-graded sine-Gordon equation simplifies and so we have D− − = 2(λ+)2λ−α sin( /2) = 2λ+ sin( /2) using the algebraic condition on the spinorvalued parameters and their relation to α. We recognise the final line of (4.9) as being the O(an) term of (4.5a)

Currents and conservation laws

Concluding remarks