Is the Z2xZ2-graded sine-Gordon equation integrable?

in Nuclear Physics B (2021), 971

We examine the question of the integrability of the recently defined Z2xZ2-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via ... [more ▼]

We examine the question of the integrability of the recently defined Z2xZ2-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. [less ▲]

Double-Graded Quantum Superplane

in Reports on Mathematical Physics (2020), 86(3), 383-400

A ℤ2 × ℤ2-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the double-graded quantum superplane. The ommutation ... [more ▼]

A ℤ2 × ℤ2-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the double-graded quantum superplane. The ommutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf -algebra structure. [less ▲]

${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded supersymmetry: 2-d sigma models

in Journal of Physics. A, Mathematical and Theoretical (2020), 53(45), 455201

We propose a natural ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded generalisation of d = 2, $\mathcal{N}=\left(1,1\right)$ supersymmetry and construct a ${\mathbb{Z}}_{2}^{2}$-space realisation ... [more ▼]

We propose a natural ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded generalisation of d = 2, $\mathcal{N}=\left(1,1\right)$ supersymmetry and construct a ${\mathbb{Z}}_{2}^{2}$-space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana–Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a 'double-graded' version of supersymmetric sine-Gordon theory. [less ▲]

Riemannian Structures on Z 2 n -Manifolds

in Mathematics (2020), 8(9), 1469

Very loosely, Zn2-manifolds are ‘manifolds’ with Zn2-graded coordinates and their sign rule is determined by the scalar product of their Zn2-degrees. A little more carefully, such objects can be ... [more ▼]

Very loosely, Zn2-manifolds are ‘manifolds’ with Zn2-graded coordinates and their sign rule is determined by the scalar product of their Zn2-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Zn2-manifold, i.e., a Zn2-manifold equipped with a Riemannian metric that may carry non-zero Zn2-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Zn2-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry [less ▲]

Functional analytic issues in Z_2 ^n Geometry

in Revista de la Union Matematica Argentina (2020), 60(2), 611-636

Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields

in Archivum Mathematicum (2020)

We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q ... [more ▼]

We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q-manifolds. We show that such Q-manifolds are unimodular, i.e., come equipped with a Q-invariant Berezin volume. [less ▲]

Almost Commutative Q-algebras and Derived brackets

in Journal of Noncommutative Geometry (2020)

We introduce the notion of almost commutative Q-algebras and demonstrate how the derived bracket formalism of Kosmann-Schwarzbach generalises to this setting. In particular, we construct ‘almost ... [more ▼]

We introduce the notion of almost commutative Q-algebras and demonstrate how the derived bracket formalism of Kosmann-Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaintrob’s Q-manifold understanding of classical Lie algebroids. We show that the basic tenets of the theory of Lie algebroids carry over verbatim to the almost commutative world. [less ▲]

On a Z2n-Graded Version of Supersymmetry

in Symmetry (2019), 11(1)(116),

We extend the notion of super-Minkowski space-time to include Zn2 -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst ... [more ▼]

We extend the notion of super-Minkowski space-time to include Zn2 -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of Zn2 -manifolds understood as locally ringed spaces. The formalism we present resembles N -extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed. [less ▲]

Products in the category of Z_2^n manifolds

in Journal of Nonlinear Mathematical Physics (2019), 26(3), 420-453

Connections adapted to non-negatively graded structure

in International Journal of Geometric Methods in Modern Physics (2018)

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we ... [more ▼]

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted A-connection on a graded bundle. In a natural sense weighted A-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear A-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted A-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles. [less ▲]