arXiv subjects

Andrew James Bruce

Publications and source records attributed to Andrew James Bruce.

At least 19 recordsLinked to original sources

arXiv2026

Derived Associative Algebras and Cyclic Cohomology of Q-manifolds

Loday's derived product is revisited in the setting of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that `squares to zero'. We interpret the Grassmann odd product as a standard product upon shifting the grading, which implies the existence of a derived (noncommutative) associative $\mathbb{Z}_2$-graded algebra associated with any Q-manifold. We apply Connes' cyclic cohomology to the derived associative algebra, giving a new cohomology on Q-manifolds distinct from the standard cohomology: we refer to this as the derived cyclic cohomology of a Q-manifold. The derived cyclic cocycles are interpreted as classically BRST-invariant functionals within the BV--BFV--BRST formalism or generalised Ruelle--Sullivan currents when applied to regular foliations via their foliation Lie algebroids.

math-ph↗

arXiv2026

Go Fishing with Alice: Compatible Carrollian and Poisson Structures

We introduce, motivated by noncommutative Carrollian geometry, the notion of a Carrollian--Poisson bundle understood as a Carrollian manifold with a compatible Poisson structure. Compatibility is expressed as the inclusion of the Carroll distribution in the image of the Poisson anchor. We explore the geometric consequences of this compatibility and present several simple examples of Carrollian--Poisson bundles, including the (outer) Kerr horizon, the BTZ horizon, and future null infinity. Furthermore, we establish that every Carrollian--Poisson bundle can be equipped with a Vaisman (contravariant) connection that is both metric-compatible and clock-compatible, and thus geometric kinematics can be defined and studied.

math.DG↗

arXiv2026

Frozen Motion: Why Single Carrollian Scalars Cannot Propagate

We investigate a class of first-order scalar field theories minimally coupled to a Carrollian connection that are defined intrinsically on the Carrollian plane, i.e., the theories are not defined via limits of Lorentzian theories. The theories built are invariant under the extended Carrollian transformations which include supertranslations. The symmetry allows for a large class of Lagrangians, independence of spacetime coordinates is all that is required. However, invariance under supertranslations (which include boosts as linear supertranslations) forces the energy density to be static and the momentum density to vanish -- this precludes on-shell propagation of fields. Thus, to have propagating theories, one must move beyond single field theories that are minimally coupled to the geometry.

gr-qc↗

arXiv2026

Modular Classes and Supersymmetric Berezin Volumes

We argue that modular classes of Q-manifolds provide an efficient method for addressing the existence of supersymmetric Berezin volumes in the supergeometric representation theory of the $\mathcal{N}=2$ $d=1$ supertranslation algebra. We establish a cohomological coherence criterion for the existence of a Berezin volume that is invariant under both of the supercharges.

hep-th↗

arXiv2026

Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond

We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \matthbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles, and a large class of Carrollian manifolds can be analysed in this formalism. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics.

math.DG↗

arXiv2026

Carrollian Lie Algebroids: Taming Singular Carrollian Geometries

Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map. The Carroll distribution is, in general, a singular Stefan--Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections.

math.DG↗

arXiv2026

Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $ρ$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $ρ$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

math-ph↗

arXiv2026

Lie Superheaps and their Groupification

We introduce the notion of a Lie superheaps as a generalisation of Lie supergroups. We show that the well-known `groupification' and `heapification' functors generalise to the ambience of supergeometry. In particular, we show that there is an isomorphism between the categories of pointed Lie superheaps and Lie supergroups. To do this we make extensive use of the functor of points.

math-ph↗

arXiv2026

Carrollian $\mathbb{R}^\times$-bundles II: Sigma Models on Event Horizons

Carrollian field theories are usually understood as limits of relativistic theories. In this note, we use Carrollian $\mathbb{R}^\times$-bundles equipped with a principal connection to construct Carrollian sigma models intrinsically. The resulting theories are neither ``electric'' nor ``magnetic'' in the usual sense. As a physically suggestive example, we derive a Carrollian wave equation governing the dynamics of a scalar field on the event horizon of a Schwarzschild black hole.

gr-qc↗

arXiv2026

Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians

Carrollian $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \mathbb{R}\setminus \{0\}$) offer a novel perspective on intrinsic Carrollian geometry using the powerful tools of principal bundles. Given a choice of principal connection, a canonical Lorentzian metric exists on the total space. This metric enables the development of Hodge theory on a Carrollian $\mathbb{R}^\times$-bundle; specifically, the Hodge star operator and Hodge--de Rham Laplacian are constructed. These constructions are obstructed on a Carrollian manifold due to the degenerate metric. The framework of Carrollian $\mathbb{R}^\times$-bundles bridges the gap between Carrollian geometry and (pseudo)-Riemannian geometry. As an example, the question of the Hodge--de Rham Laplacian on the event horizon of a Schwarzschild black hole is addressed. A Carrollian version of electromagnetism is also proposed.

math.DG↗

arXiv2026

On a Grassmann odd analogue of Carrollian Manifolds

We define a Grassmann odd analogue of a Carrollian manifold as a supermanifold of dimension $n|1$ with an even degenerate metric such that the kernel is generated by a non-singular odd vector field that is a supersymmetry generator. Alongside other results, we establish that the reduced manifold is a pseudo-Riemannian manifold, and show that compatible affine connections always exist, albeit they must carry torsion. As a physically relevant example, we examine an Inönü--Wigner contraction of the supertranslation algebra on standard superspace $\mathbb{R}^{4|4}$.

math.DG↗

arXiv2026

The Carrollian Superplane and Supersymmetry

This note provides an intrinsic construction of the Carrollian superplane $Π\mathbb{S}\simeq \mathbb{R}^{2|4}$ as a supermanifold generalisation of the Carrollian plane. Moving away from the $c\rightarrow 0$ limit of relativistic spinors, we define Carroll spinors as sections of a degenerate Clifford module. We show that the Carrollian superplane is a principal $\mathbb{R}^{1|2}$-bundle. Once clock forms and a complementary basic one-form are specified, there is a pair of odd vector fields that generate novel $N =2$ Carrollian supersymmetry transformations, not all of which come from an Inönü--Wigner contraction of a Poincaré superalgebra

hep-th↗

arXiv2026

Affine Supertrusses and Superbraces

Brzeziński's trusses are ``ring-like'' algebraic structures in which the addition is replaced with an abelian heap operation and the binary product satisfies a natural distributivity rule of the ternary product. The question of how to define ($\mathbb{Z}_2$-graded) super-versions of trusses is addressed in this note. Taking our cue from the theory of algebraic supergroups, we define an affine supertruss as a representable functor from the category of unital associative supercommutative superalgebras to the category of trusses. The representing superalgebras are equipped with a `cotruss' structure--a new concept in itself. We show that from an affine supertruss one can construct an affine superbrace, and so generalise Rump's braces to supermathematics. As an application of these constructions, we propose a generalisation of the set-theoretic Yang--Baxter equation to the setting of affine superschemes.

math-ph↗

arXiv2025

Para-associative Algebroids

We introduce para-associative algebroids as vector bundles whose sections form a ternary algebra with a generalised form of associativity. We show that a necessary and sufficient condition for local triviality is the existence of a differential connection, i.e., a connection that satisfies the Leibniz rule over the ternary product.

math.DG↗

arXiv2025

A First Look at Supersymmetry

These are expanded notes for a short series of lectures, presented at the University of Luxembourg in 2017, giving an introduction to some of the ideas of supersymmetry and supergeometry. In particular, we start from some motivating facts in physics, pass to the theory of supermanifolds, then to spinors, ending up at super-Minkowski space-times. We examine some salient mathematical issues with understanding supersymmetry in a classical setting and make no attempt to discuss phenomenologically interesting models. Moreover, the presentation is "light" in the sense that nothing is carefully proved. The audience of the seminars ranged from Ph.D. students to more experienced postdocs in mathematics. The audience was assumed to have a working knowledge of differential geometry, elementary category theory, and basic ideas from classical & quantum mechanics. No knowledge of supersymmetry and supergeometry is assumed.

math.DG↗

arXiv2024

Principal bundles in the category of $\mathbb{Z}_2^n$-manifolds

We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading replaces $\mathbb{Z}_2$-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of $\mathbb{Z}_2^n$-manifolds is essentially different than that of supermanifolds, as for $n>1$ we have formal variables of even parity, so local smooth functions are formal power series. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this `higher graded' geometry, with properly defined frame bundles of $\mathbb{Z}_2^n$-vector bundles as canonical examples. However, formulating these concepts and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to $\mathbb{Z}_2^n$-manifolds is therefore included together with basic examples.

math.DG↗

arXiv2024

A Novel Generalisation of Supersymmetry: Quantum $\mathbb{Z}_2^2$-Oscillators and their `superisation'

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.

math-ph↗

arXiv2024

The Ternary Structure of Lie Algebroid Connections

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.

math.DG↗