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We propose a very simple toy model of a $\mathbb{Z}_2^2$-supersymmetric quantum system and show, via Klein's construction, how to understand the system as being an $N=2$ supersymmetric system with an extra $\mathbb{Z}_2^2$-grading. That is, the commutation/anticommutation rules are defined via the standard boson/fermion rules, but the system still has an underlying $\mathbb{Z}_2^2$-grading that needs to be taken into account., Comment: 6 pages, accepted for publication in Acta Physica Polonica B

We examine the heap of linear connections on anchored vector bundles and Lie algebroids. Naturally, this covers the example of affine connections on a manifold. We present some new interpretations of classical results via this ternary structure of connections. Endomorphisms of linear connections are studied, and their ternary structure, in particular the endomorphism truss, is explicitly presented. We remark that the use of ternary structures in differential geometry is novel and that the endomorphism truss of linear connections provides a concrete geometric example of a truss., Comment: A rewritten and extended version that contains further results is published in Symmetry under the title "Heaps of Linear Connections and Their Endomorphism Truss"

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., Comment: 18 pages. Typos and editing artifacts were corrected, and the results of the paper are extended. The notion of left-invariant vector fields is now included. Comments are still welcomed

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., Comment: 8 pages. Comments are most welcomed

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., Comment: 7 pages, comments welcomed

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., Comment: Accepted for publication in Nuclear Physics B. 8 pages, comments still welcomed

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., Comment: 19 pages, minor typos corrected and clarifying comments added. The exposition has been improved. Comments welcomed

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.

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., Comment: The content is identical to the published version

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., Comment: 17 pages. Comments welcomed

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., Comment: 17 pages and includes two appendices. The analysis of the double-graded sine-Gordon theory has been extended and numerous sign mistakes have been corrected. Comments still welcomed

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., Comment: 17 pages including one Appendix. Parts of the exposition have been rewritte and further references added. Accepted for publication in the Journal of Physics A: Mathematical and Theoretical

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., Comment: 7 pages. This new version has been extensively rewritten. Clarifying points, examples and minor typos have been corrected. Comments welcomed

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., Comment: 12 pages. This preprint is a continuation of an earlier paper by the author published as Archivum Mathematicum, vol. 53 (2017), issue 4, pp. 203-219

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., Comment: 20 pages, amslatex, romp.sty, version 3: final journal version, to appear in "Reports on Mathematical Physics"

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 Fr\'echet 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.

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., Comment: 15 pages, amsart, in v.2: minor changes, bibliography added

We extend the notion of super-Minkowski space-time to include $\mathbb{Z}_2^n$-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 $\mathbb{Z}_2^n$-manifolds understood as locally ringed spaces. The formalism we present resembles $\mathcal{N}$-extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed., Comment: 15 pages. Comments welcomed. Part of this work was reported on at the miniworkshop "Supergeometry and Applications", University of Luxembourg, December 14-15, 2017

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation 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 generalises 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., Comment: 19 pages. Accepted for publication in the International Journal of Geometric Methods in Modern Physics

We prove that the category of $\mathbb{Z}_2 ^n$-manifolds has all finite products. Further, we show that a $\mathbb{Z}_2 ^n$-manifold (resp., a $\mathbb{Z}_2 ^n$-morphism) can be reconstructed from its algebra of global $\mathbb{Z}_2 ^n$-functions (resp., from its algebra morphism between global $\mathbb{Z}_2 ^n$-function algebras). These results are of importance in the study of $\mathbb{Z}_2 ^n$ Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two $\mathbb{Z}_2 ^n$-manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix., Comment: 38 pages. Comments welcomed

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit., Comment: 27 pages. Comments welcomed

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., Comment: 9 pages. Several changes as compared with earlier versions. In particular, we have added "supersymmetries" of mixed symmetry tensors and have included a section on mixed symmetry tensors with values in vector bundles

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\u{\i}ntrob'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., Comment: 15 pages. Accepted for publication in The Journal of Noncommutative Geometry

We present the notion of a filtered bundle as a generalisation 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 generalisations of double vector bundles and double affine bundles. Furthermore, we show that the linearisation of a filtered bundle - which can be seen as a partial polarisation of the admissible changes of local coordinates - is well defined., Comment: 23 pages

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds., Comment: 11 pages. Comments are welcomed

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., Comment: 23 pages (journal format)

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., Comment: Dedicated to the memory of James Alfred Bruce

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds., Comment: 19 pages. Comments welcomed. Minor typos corrected

We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear in the existing literature. We start with a discussion of graded spaces, passing through graded bundles and their linearisation to end with weighted (graded) Lie groupoids/algebroids., Comment: 11 pages, proceedings of the First International Conference of Differential Geometry, Fez (Morocco), April 11-15, 2016

We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising $N$-manifolds, and how one can use the full linearisation functor to "superise" a graded bundle.

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an $L_{\infty}$-algebra, which we refer to as a homotopy Kirillov algebra. We are then to higher Kirillov algebroids as higher generalisations of Jacobi algebroids. Furthermore, we show how to associate a higher Kirillov algebroid and a homotopy BV-algebra with every higher Kirillov manifold. In short, we construct homotopy versions of some of the well-known theorems related to Kirillov's local Lie algebras., Comment: Change of title, improved introduction section and examples. Minor typos etc corrected. No major revision of the key results. 15 pages. Further comments welcomed

We define and examine the notion of a Killing section of a Riemannian Lie algebroid as a natural generalisation of a Killing vector field. We show that the various expression for a vector field to be Killing naturally generalise to the setting of Lie algebroids. As an application we examine the internal symmetries of a class of sigma models for which the target space is a Riemannian Lie algebroid. Critical points of these sigma models are interpreted as generalised harmonic maps., Comment: 11 pages. Comments welcomed

Understanding how housing values evolve over time is important to policy makers, consumers and real estate professionals. Existing methods for constructing housing indices are computed at a coarse spatial granularity, such as metropolitan regions, which can mask or distort price dynamics apparent in local markets, such as neighborhoods and census tracts. A challenge in moving to estimates at, for example, the census tract level is the sparsity of spatiotemporally localized house sales observations. Our work aims at addressing this challenge by leveraging observations from multiple census tracts discovered to have correlated valuation dynamics. Our proposed Bayesian nonparametric approach builds on the framework of latent factor models to enable a flexible, data-driven method for inferring the clustering of correlated census tracts. We explore methods for scalability and parallelizability of computations, yielding a housing valuation index at the level of census tract rather than zip code, and on a monthly basis rather than quarterly. Our analysis is provided on a large Seattle metropolitan housing dataset.

We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson-Lie groupoids and weighted Lie bi-algebroids, as well as weighted Courant algebroids.