arXiv subjects

Paul Purdon Martin

Publications and source records attributed to Paul Purdon Martin.

3 recordsLinked to original sources

arXiv2024

Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopf-algebraic discrete higher gauge theory. We construct a family of crossed modules of semisimple Hopf algebras, $(\mathscr{F}_{\mathbb{C}}(X) \otimes \mathbb{C}E \xrightarrow{\partial} \mathscr{F}_{\mathbb{C}}(Y) \rtimes \mathbb{C} G, \triangleright)$, that depends on four finite groups, $E,G,X$ and $Y$. We calculate the ground-state spaces of the resulting model on a triangulated surface when $G=E=\{1\}$ and when $Y=\{1\}$, prove that those ground-state spaces are canonically independent of the triangulations, and so depend only on the underlying surface; and moreover we find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn's finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids. We leave it as an open problem whether the ground-state space of the Hopf-algebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from $E,G,X,Y$ can always be given a homotopical explanation.

math-ph↗

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.

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arXiv2019

On a canonical lift of Artin's representation to loop braid groups

Each pointed topological space has an associated $π$-module, obtained from action of its first homotopy group on its second homotopy group. For the $3$-ball with a trivial link with $n$-components removed from its interior, its $π$-module $\mathcal{M}_n$ is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of $\mathcal{M}_n$. We give a topological interpretation of this injection, showing that it is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups.

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