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Double-Graded Quantum Superplane Bruce, Andrew ; 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 ▲] Detailed reference viewed: 6 (1 UL)Linear Z2n-Manifolds and Linear Actions Bruce, Andrew ; Ibarguengoytia, Eduardo ; Poncin, Norbert E-print/Working paper (2020) Detailed reference viewed: 50 (2 UL)${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded supersymmetry: 2-d sigma models Bruce, Andrew 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 ▲] Detailed reference viewed: 11 (2 UL)Odd connections on supermanifolds: existence and relation with affine connections Bruce, Andrew ; in Journal of Physics. A, Mathematical and Theoretical (2020), 53(45), 455203 The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in ... [more ▼] The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type (1, 2), by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $\mathcal{N}=1$ super-Minkowski spacetime admits a natural odd connection. [less ▲] Detailed reference viewed: 27 (2 UL)Riemannian Structures on Z 2 n -Manifolds Bruce, Andrew ; 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 ▲] Detailed reference viewed: 23 (0 UL)The Schwarz-Voronov embedding of Z_2^n - manifolds Bruce, Andrew ; Ibarguengoytia, Eduardo ; Poncin, Norbert in Symmetry, Integrability and Geometry: Methods and Applications (2020), 16(002), 47 Detailed reference viewed: 191 (23 UL)Double-Graded Supersymmetric Quantum Mechanics Bruce, Andrew ; in Journal of Mathematical Physics (2020), 61 A quantum mechanical model that realizes the ℤ2×ℤ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and ... [more ▼] A quantum mechanical model that realizes the ℤ2×ℤ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics. [less ▲] Detailed reference viewed: 52 (1 UL)Functional analytic issues in Z_2 ^n Geometry Bruce, Andrew ; Poncin, Norbert in Revista de la Union Matematica Argentina (2020), 60(2), 611-636 Detailed reference viewed: 189 (19 UL)Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields Bruce, Andrew 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 ▲] Detailed reference viewed: 807 (0 UL)The super-Sasaki metric on the antitangent bundle Bruce, Andrew in International Journal of Geometric Methods in Modern Physics (2020) We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We ... [more ▼] We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalisation of Sasaki's construction of a Riemannian metric on the tangent bundle of a Riemannian manifold. [less ▲] Detailed reference viewed: 89 (0 UL)Almost Commutative Q-algebras and Derived brackets Bruce, Andrew 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 ▲] Detailed reference viewed: 91 (6 UL)Conference 'Supergeometry, Supersymmetry and Quantization' Bruce, Andrew ; Ibarguengoytia, Eduardo ; Poncin, Norbert Report (2019) Detailed reference viewed: 59 (11 UL)On a Z2n-Graded Version of Supersymmetry Bruce, Andrew 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 ▲] Detailed reference viewed: 94 (2 UL)Pre-Courant algebroids Bruce, Andrew ; in Journal of Geometry and Physics (2019), 142 Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some ... [more ▼] Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof. In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. We give the definition of (sub-)Dirac structures and study the naïve quasi-cochain complex within the setting of supergeometry. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids and our approach drastically simplifies working with them. [less ▲] Detailed reference viewed: 38 (1 UL)Products in the category of Z_2^n manifolds Bruce, Andrew ; Poncin, Norbert in Journal of Nonlinear Mathematical Physics (2019), 26(3), 420-453 Detailed reference viewed: 180 (21 UL)The graded differential geometry of mixed symmetry tensors Bruce, Andrew ; Ibarguengoytia, Eduardo in Archivum Mathematicum (2019), 55(2), 123-137 We show how the theory of $\mathbb{Z}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual ... [more ▼] We show how the theory of $\mathbb{Z}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed. [less ▲] Detailed reference viewed: 93 (8 UL)Representations up to Homotopy from Weighted Lie Algebroids Bruce, Andrew ; ; in Journal of Lie Theory (2018), 28(3), 715-737 Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB-algebroid. There ... [more ▼] Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB-algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB-algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects. [less ▲] Detailed reference viewed: 60 (1 UL)On the Concept of a Filtered Bundle Bruce, Andrew ; ; in International Journal of Geometric Methods in Modern Physics (2018), 15 We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of ... [more ▼] We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined. [less ▲] Detailed reference viewed: 57 (12 UL)Connections adapted to non-negatively graded structure Bruce, Andrew 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 ▲] Detailed reference viewed: 58 (14 UL)Workshop on Supergeometry and Applications Bruce, Andrew ; Poncin, Norbert Report (2017) Detailed reference viewed: 121 (13 UL) |
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