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Andrew James Bruce

Publications and source records attributed to Andrew James Bruce.

At least 37 records · Page 2Linked to original sources
arXiv2023

Homological sections of Lie algebroids

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that `self-commutes', we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably "superised" Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday-Leibniz bracket.

math-ph↗
arXiv2023

On Lie semiheaps and ternary principal bundles

We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known `heapification' functor to the ambience of Lie groups and principal bundles.

math.DG↗
arXiv2022

On the bundle of null cones

We examine the bundle structure of the field of nowhere vanishing null vector fields on a (time-oriented) Lorentzian manifold. Sections of what we refer to as the null tangent, are by definition nowhere vanishing null vector fields. It is shown that the set of nowhere vanishing null vector fields comes equipped with a para-associative ternary partial product. Moreover, the null tangent bundle is an example of a non-polynomial graded bundle.

math.DG↗
arXiv2021

Semiheaps and Ternary Algebras in Quantum Mechanics Revisited

We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.

math-ph↗
arXiv2021

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-Bäcklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.

math-ph↗
arXiv2021

Symplectic $\mathbb{Z}_2^n$-manifolds

Roughly speaking, $\mathbb{Z}_2^n$-manifolds are `manifolds' equipped with $\mathbb{Z}_2^n$-graded commutative coordinates with the sign rule being determined by the scalar product of their $\mathbb{Z}_2^n$-degrees. We examine the notion of a symplectic $\mathbb{Z}_2^n$-manifold, i.e., a $\mathbb{Z}_2^n$-manifold equipped with a symplectic two-form that may carry non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and results of symplectic geometry generalise to the `higher graded' setting, including a generalisation of Darboux's theorem.

math-ph↗
arXiv2021

Linear ${\mathbb Z}_2^n$-Manifolds and Linear Actions

We establish the representability of the general linear ${\mathbb Z}_2^n$-group and use the restricted functor of points - whose test category is the category of ${\mathbb Z}_2^n$-manifolds over a single topological point - to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.

math-ph↗
arXiv2020

Odd Connections on Supermanifolds: Existence and relation with Affine Connections

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 $N= 1$ super-Minkowski spacetime admits a natural odd connection.

math-ph↗
arXiv2020

Double-graded quantum superplane

A $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \emph{double-graded quantum superplane}. The commutation 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 $\mathbb{Z}_2^2$-algebra structure.

math.QA↗
arXiv2020

$\mathbb{Z}_2\times \mathbb{Z}_2$-graded supersymmetry: $2$-d sigma models

We propose a natural $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of $d=2$, $\mathcal{N}=(1,1)$ 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.

math-ph↗
arXiv2020

Riemannian structures on $\mathbb{Z}_2^n$-manifolds

Very loosely, $\mathbb{Z}_2^n$-manifolds are `manifolds' with $\mathbb{Z}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their $\mathbb{Z}_2^n$-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 $\mathbb{Z}_2^n$-manifold, i.e., a $\mathbb{Z}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of $\mathbb{Z}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

math-ph↗
arXiv2020

The super-Sasaki metric on the antitangent bundle

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.

math.DG↗
arXiv2020

The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ${\mathbb Z}_2^n$-manifolds into a subcategory of contravariant functors from the category of ${\mathbb Z}_2^n$-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ${\mathbb Z}_2^n$-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.

math-ph↗
arXiv2019

Pre-Courant Algebroids

Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive quasi-cochain complex. In particular we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. 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. We remark that examples of pre-Courant algebroids are plentiful - natural examples include the cotangent bundle of any almost Lie algebroid.

math-ph↗
arXiv2019

Double-Graded Supersymmetric Quantum Mechanics

A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-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.

math-ph↗
arXiv2018

Almost Commutative Q-algebras and Derived brackets

We introduce the notion of \emph{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 Va\uıntrob'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.

math.QA↗
arXiv2018

The Graded Differential Geometry of Mixed Symmetry Tensors

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.

math-ph↗