arXiv subjects

Fiona Torzewska

Publications and source records attributed to Fiona Torzewska.

6 recordsLinked to original sources

arXiv2026

Paravortices: loop braid representations with both generators involutive

We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory. For reasons that will become clear, we call this quotient the mixed doubles category, $MD$. Then our main result is a theorem classifying all mixed doubles representations in rank-2. Each representation yields a mixed doubles group representation for every loop braid group $LB_n$, and we are able to analyse the unified linear representation theory of many of these sequences of representations, using a mixture of very classical, classical, and new techniques. In particular this is a motivating example for the `glue' generalisation of charge-conserving representation theory (a form of rigid higher non-semisimplicity) introduced recently.

math.QA↗

arXiv2025

Categorical 4-manifold invariants from trisection diagrams

We use Gay and Kirby's description of 4-manifolds in terms of trisections and trisection diagrams to define a new 4-manifold invariant. The algebraic data are an indecomposable finite semisimple bimodule category over a pair of spherical fusion categories, equipped with a bimodule trace, and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them. The 4-manifold invariant has a simple description in terms of a diagrammatic calculus for this data, in which the three spherical fusion categories correspond to the three colours of the trisection diagram. It includes the Hopf algebraic 4-manifold invariants by Chaidez, Cotler and Cui, which arise when the bimodule category is the category of finite-dimensional complex vector spaces. We also recover the 4-manifold invariants of Bärenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category.

math.QA↗

arXiv2023

Motion groupoids and mapping class groupoids

Here $\underline{M}$ denotes a pair $(M,A)$ of a manifold and a subset (e.g. $A=\partial M$ or $A=\emptyset$). We construct for each $\underline{M}$ its motion groupoid $\mathrm{Mot}_{\underline{M}}$, whose object set is the power set $ {\mathcal P} M$ of $M$, and whose morphisms are certain equivalence classes of continuous flows of the `ambient space' $M$, that fix $A$, acting on ${\mathcal P} M$. These groupoids generalise the classical definition of a motion group associated to a manifold $M$ and a submanifold $N$, which can be recovered by considering the automorphisms in $\mathrm{Mot}_{\underline{M}}$ of $N\in {\mathcal P} M$. We also construct the mapping class groupoid $\mathrm{MCG}_{\underline{M}}$ associated to a pair $\underline{M}$ with the same object class, whose morphisms are now equivalence classes of homeomorphisms of $M$, that fix $A$. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair $\underline{M}$ we explicitly construct a functor $\mathsf{F}\colon \mathrm{Mot}_{\underline{M}} \to \mathrm{MCG}_{\underline{M}}$, which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $π_0$ and $π_1$ of the appropriate space of self-homeomorphisms of $M$ are trivial. In particular, we have an isomorphism in the physically important case $\underline{M}=([0,1]^n, \partial [0,1]^n)$, for any $n\in \mathbb{N}$. We show that the congruence relation used in the construction $\mathrm{Mot}_{\underline{M}}$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). We examine several explicit examples of $\mathrm{Mot}_{\underline{M}}$ and $\mathrm{MCG}_{\underline{M}}$ demonstrating the utility of the constructions.

math-ph↗

arXiv2023

Classification of charge-conserving loop braid representations

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that representations fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree $N$.

math.QA↗

arXiv2022

Topological quantum field theories and homotopy cobordisms

We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class groupoid $\mathrm{MCG}_{M}^{A}$, and from the full subgroupoid of the motion groupoid $\mathrm{Mot}_{M}^{A}$, whose objects are homotopically 1-finitely generated. We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to \mathbf{Vect}$, one for each finite group $G$. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten. Given a space $X$, we prove that $\mathsf{Z}_G(X)$ can be expressed as the $\mathbb{C}$-vector space with basis natural transformation classes of maps $\{π(X,X_0)\to G\} $ for some finite representative set of points $X_0\subset X$, demonstrating that $\mathsf{Z}_G$ is explicitly calculable.

math-ph↗

arXiv2020

Non-ergodicity in open quantum systems through quantum feedback

It is well known that quantum feedback can alter the dynamics of open quantum systems dramatically. In this paper, we show that non-ergodicity may be induced through quantum feedback and resultantly create system dynamics that have lasting dependence on initial conditions. To demonstrate this, we consider an optical cavity inside an instantaneous quantum feedback loop, which can be implemented relatively easily in the laboratory. Non-ergodic quantum systems are of interest for applications in quantum information processing, quantum metrology and quantum sensing and could potentially aid the design of thermal machines whose efficiency is not limited by the laws of classical thermodynamics.

quant-ph↗