Is the $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon equation integrable?

We examine the question of the integrability of the recently defined $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-B\"{a}cklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.


Introduction
There has been some renewed interest in Z 2 × Z 2 -gradings (and higher) in physics, see for example [1,2,3,8,24] and references therein. It has long been appreciated that these "higher gradings" play a rôle in the theory of parastatistics, relatively recent works in this area include Tolstoy [23] (also see references therein). In particular, Toppan in [24] has shown that in multiparticle theories it is possible to distinguish between ordinary bosons/fermions, so Z 2 -graded particles, and the exotic Z 2 ×Z 2 -graded particles. It is thus conceivable that there is new physics behind these novel theories, especially in low-dimensional condensed matter physics where the spin-statistics theorem does not hold. In this paper, we continue the study of a 1+1-dimensional classical field theory that is Z 2 × Z 2 -graded.
Sigma models that exhibit a novel Z 2 × Z 2 -graded generalisation of supersymmetry (see [6]) were first presented by the author in [7]. As an example of one of these sigma models a Z 2 × Z 2graded version of the supersymmetric sine-Gordon theory was given and some initial study was made. In this short paper, we examine the natural question of the integrability of the Z 2 × Z 2graded sine-Gordon equation. There is no universal definition of integrability of systems with an infinite number of degrees of freedom. We will use the definition that a classical field theory is (formally) integrable if there is an infinite set of conserved currents (equivalently, conservation laws). Our approach to the question of integrability is to construct relevant auto-Bäcklund transformations and from there build the infinite set of conservation laws. The standard supersymmetric case was examined by Chaichian & Kulish [9] using a superspace auto-Bäcklund transformation following earlier calculations of the infinite set of conserved currents by Ferrara, Girardello & Sciuto [14]. There are some complications with the Z 2 × Z 2 -graded case due to the presence of a degree (1, 1) parameter in the Lagrangian and the subsequent need for spinor-valued parameters of degree (0, 1) and (1, 0). There is no general systematic method for constructing Bäcklund transformations and as such we take our cue from the classical transformations for the standard sine-Gordon equation making the necessary modifications.
Other approaches to the integrability of supersymmetric sine-Gordon exist, such as constructing Lax pairs via superconformally affine Toda theories (see [13,16]) and the construction of solitons (see for example [11,19]). For a study of the different aspects of the integrability of the supersymmetric sine-Gordon see Bertrand [5]. It would be interesting to see how these other approaches generalise to this "higher graded" setting.
We remark that the sine-Gordon model is found throughout mathematical physics and has a very wide range of applications such as coupled rigid pendula, Josephson junctions, crystal dislocations and 2-d gravitational physics. A review of some of these applications can be found in [4]. Furthermore, N = 2 supersymmetric sine-Gordon models can be found in string theory via reductions of superstring worldsheet theories on particular backgrounds, see for example [15]. More generally, two-dimensional field theories appear in condensed matter physics and statistical mechanics while often exhibiting the mathematical features of four-dimensional theories.

Z 2 × Z 2 -supersymmetry
The Z 2 × Z 2 -graded, d = 2, N = (1, 1) supertranslation algebra (in light-cone coordinates) is the Z 2 × Z 2 -Lie superalgebra (see [21,22]) with 5 generators of Z 2 × Z 2 -degree where they transform under 2-d Lorentz boosts (here β ∈ R is the rapidity) as Warning. Thinking of the following Z 2 × Z 2 -Lie superalgebra (2.1) as being embedded in an enveloping algebra, the first two brackets are, in the classical language, anticommutators, while the third is a commutator. The nature of the brackets is determined by the Z 2 × Z 2 -degree of the elements and the map Supposing we have two elements A and B from some enveloping algebra of Z 2 × Z 2 -degree a and b, respectively, the graded Lie bracket is . Note that we have the natural notion of the total degree of an element but this is not sufficient to determine the required sign factors. For example, given a pair of odd elements with Z 2 × Z 2 -degree (0, 1) and (1, 0), respectively, it is the commutator and not the anticommutator that we are interested in. This is, of course, quite different to standard supersymmetry. For an overview of supersymmetry and supermathematics the reader can consult [12].
We see that the generators consist of a pair of right-handed and left-handed vectors, a pair of right-handed and left-handed Majorana-Weyl spinors and a Lorentz scalar.
The d = 2, N = (1, 1) Z 2 × Z 2 -Poincaré algebra is the above Z 2 × Z 2 -Lie superalgebra with an additional degree (0, 0) generator M , understood as the generator of Lorentz boosts, together with the non-vanishing Z 2 × Z 2 -graded Lie brackets (all other brackets are zero) The geometric realisation of this novel supersymmetry is via d = 2, N = (1, 1) Z 2 ×Z 2 -Minkowski spacetime, which is the Z 2 × Z 2 -manifold (see [10]) that comes equipped with global coordinate systems of the form Under Lorentz boosts these coordinate transform as 2 β θ + , again, β is the rapidity. Thus, the underlying reduced manifold is two-dimensional Minkowski spacetime, here explicitly given in light-cone coordinates.
The following vector fields form a representation of the Z 2 × Z 2 -supertranslation algebra The reader can easily verify that these vector fields have the right transformation properties under a Lorentz boost.
In standard supersymmetry, the partial derivatives for the odd coordinates of superspace are not invariant under supertranslations and so we need covariant derivatives. The required Direct calculation gives the expected results We will take Z 2 × Z 2 -superfields to be scalar under the 2-d Lorentz transformations and of Z 2 × Z 2 -degree zero. Then to first order in z, In general, we have a formal power series in z, and so we have an infinite number of component fields. The first four types of component fields are For the Z 2 × Z 2 -superfield to be a Lorentz scalar we require that, under Lorentz boosts while X and F +− are Lorentz scalars. That is we have bosons, exotic boson that anticommute with the fermions while commuting with the bosons, and left-handed and right-handed fermions that mutually commute. This is quite different to the component fields on standard superspace.
For more details the reader should consult [6,7].
Remark. We will be mathematically lax in our definition of a Z 2 × Z 2 -superfield and simply allow the components thereof to depend on extra unspecified Z 2 × Z 2 -graded parameters as required. Furthermore, it is possible to consider Z 2 × Z 2 -superfield of non-zero Z 2 × Z 2 -degree and of a different nature with respect to Lorentz transformations.
3. The Z 2 × Z 2 -graded sine-Gordon equation is a scalar functional, we will comment on the Z 2 × Z 2 -degree shortly. Here W := W [Φ] is a scalar potential. The Z 2 × Z 2 -graded sine-Gordon Lagrangian is defined as where α is a Z 2 ×Z 2 -degree (1, 1) constant that satisfies α 2 −1 = 0. This constant was introduced because the Lagrangian should be homogeneous and of Z 2 × Z 2 -degree (1, 1), this implies that the action is Z 2 × Z 2 -degree (0, 0). The Clifford-relation is imposed so that this constant does not appear in the underlying classical equations of motion, in this case the classical sine-Gordon equation (see [7] for further details).
Remark. Coupling constants that are of Z 2 ×Z 2 -degree (1, 1) appear in Z 2 ×Z 2 -graded classical mechanics for exactly the same reason they appear in 2-dimensional field theories (see [2]). It is thus expected that non-trivial dynamics of Z n 2 -graded classical theories requires non-zero graded coupling constants. In the context of finding non-supersymmetric finite field theories, Clifford algebra-valued coupling constants were first conjectured by Kranner & Kummer (see [17]). Supergroups with Clifford algebra-valued parameters together with their relation with non-anticommuting supersymmetry were discussed by Kuznetsova, Rojas, & Toppan (see [18]).
An initial analysis of the Z 2 × Z 2 -graded sine-Gordon system was given in [7] including carefully showing how it is invariant under Z 2 × Z 2 -graded supersymmetry ( [6]) and the explicit presentation of the associated current.
Remark. For the integration theory to be sound there are constraints on the Z 2 ×Z 2 -superfield. It should be noted that the integration theory on Z n 2 -manifolds is not well-developed. As we will not directly use the integration theory we will not discuss this further. For details see [7,20].
The equations of motion for the lowest-order component fields (eliminating the auxiliary field using its equation of motion 2F +− = −α sin(X/2) ) are Setting ψ + = ψ − = 0 reduces the system to the classical sine-Gordon equation. Thus, the classical solutions, for example, the topological kinks and anti-kinks are solutions to the Z 2 × Z 2graded sine-Gordon equation once the non-zero degree component fields are set to zero.

Z 2 × Z 2 -graded auto-Bäcklund transformations
An auto-Bäcklund transformation for the Z 2 × Z 2 -supersymmetric sine-Gordon equation is defined as follows: where a ∈ R * , and λ + & λ − are constant spinor parameters of degree (0, 1) and (1, 0), respectively. More carefully, under Lorentz transformations we have Moreover, these spinor-valued parameters must satisfy We observe that (λ + ) 2 (λ − ) 2 = −1. We then set (there is a choice here with a sign) (λ Remark. There is another natural auto-Bäcklund upon changing D − and D + . We will use this other transformation later. We claim that if the field Φ is a solution to the Z 2 × Z 2 -supersymmetric sine-Gordon equation, then the field Φ is also a solution. We show this via direct computation. From (4.1a) & (4.1b) we have Adding and subtracting the two equations above produces Using (4.1c) and λ + λ − = α we obtain Thus, if the right-hand side vanishes then the left-hand side vanishes and so the claim is verified.
Writing the two fields out in components we have The auto-Bäcklund transformation (4.1c) tells us that both terms O(z) are equal. Thus, The auto-Bäcklund transformation (4.1a)-(4.1c) can then be written as and so we require the O(z) term to vanish, i.e., Z −+ Φ = Z −+ Φ = 0. 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 Z 2 × Z 2 -graded Berezin integral (see [7] for details). The equations (4.3a) and (4.3b) give a functional relationship between the Z 2 × Z 2 -superfields Φ and Φ. For small a ∈ R * we can consider the Maclaurin-Taylor series The auto-Bäcklund transformations then take the form ∞ n=0 a n D − Φ n − D − Φ = 2a λ + sin 1 4 ∞ n=0 a n Φ n + Φ , (4.5a) ∞ n=0 a n D + Φ n + D + Φ = 2a −1 λ − sin 1 4 ∞ n=0 a n Φ n − Φ , (4.5b) Expanding and equating terms in order of a allows us to deduce the relation between two Z 2 × Z 2 -superfields. From (4.5b) comparing the O(a −1 ) terms (the left-hand side is clearly zero) we deduce that Φ 0 = Φ. Comparing the O(a 0 ) terms we deduce from (4.5b) that 4D + Φ = λ − Φ 1 . Then using the properties of the spinor-valued parameters we see that Φ 1 = −4(λ + ) 2 λ − D + Φ.
Moving on to O(a) terms, from (4.5a) and (4.5b) we obtain the pair of equations The first equation is consistent and contains no new information. Indeed, we observe that By assumption we have no terms involving z and so D − and D + commute and the Z 2 × Z 2 -graded sine-Gordon equation simplifies and so we have D − Φ − = 2(λ + ) 2 λ − α sin(Φ/2) = 2λ + sin(Φ/2) using the algebraic condition on the spinorvalued parameters and their relation to α. The second equation Then, for small a ∈ R * we have Continuing, we make a few observations. We notice that, for n ≥ 1, Φ n ∝ D + Φ n−1 and thus ( Φ n ) k = 0 for all k ≥ 2. Then (4.5b) tells us that which can be solved iteratively following our earlier calculations. Specifically, we obtain Then we can write The system (4.5a) is redundant and automatically satisfied. To see this, using the Z 2 ×Z 2 -graded sine-Gordon equation together with the independence of Φ on z we obtain with p i > p i−1 · · · > p 1 and p i + · · · + p 1 = n − 1. We recognise the final line of (4.9) as being the O(a n ) term of (4.5a).

Currents and conservation laws
From the auto-Bäcklund transformation (4.3a)-(4.3b), we have the conserved (spinor) current Note that j + and j − are of Z 2 × Z 2 -degree (0, 1) and (1, 0), respectively. Under Lorentz transformations we have We show that we do indeed have a conserved current in this way via direct calculation: From the current (5.1) we have the conservation law We can then consider the expansion for small a (see (4.4)). In doing so we have a countable infinite set of conservation rules defined order-by-order in a. We observe that the right-hand-side of (5.2) vanishes as (D + Φ) k = 0 if k ≥ 2. Then using (4.7) we deduce that we obtain another countably infinite set of conservation laws (5.5) D + cos Φ/2 = 0 , D + sin Φ/2 (D − ) k Φ = 0 , k ≥ 1 .

Concluding remarks
Via construction of auto-Bäcklund transformations for the Z 2 × Z 2 -graded sine-Gordon equation we built a pair of infinite sets of conservation laws (see (5.3) and (5.5)). Thus, the Z 2 × Z 2 -graded sine-Gordon equation is integrable. The auto-Bäcklund transformations involve pairs of spinor-valued parameters that mutually anticommute that are related to the (1, 1)degree parameter. The presence of spinor-values parameters was expected as we are dealing with a generalisation of supersymmetry. The extra algebraic conditions are forced upon us as the Z 2 × Z 2 -graded sine-Gordon equation has a Clifford algebra-valued coupling constant of Z 2 × Z 2 -degree (1, 1). Such coupling constants are forced on us if we want an interacting theory of degree zero Z 2 × Z 2 -superfields, this properly discussed in [7].
It would be interesting to check if a similar analysis can be performed on Z 2 × Z 2 -graded versions of (supersymmetric) sinh-Gordon, Liouville or more generally Toda models. It is expected that appropriate auto-Bäcklund transformations can be found that involve spinorvalued parameters.
We stress that the analysis here is purely classical and it remains to be seen what survives quantisation and if any subtleties of quantisation with Z 2 × Z 2 -graded fields complicate the picture. That said, supersymmetry usually improves the quantum properties of a theory compared to the related pure bosonic theory. One conjectures the same is true of Z 2 × Z 2supersymmetry.