Representations of the loop braid group and Aharonov–Bohm like effects in discrete $(3+1)$-dimensional higher gauge theory

Bullivant, Alex; Martins, João Faria; Martin, Paul; Martins, Jo o Faria

doi:10.4310/atmp.2019.v23.n7.a1

Cited by 11 publications

(11 citation statements)

References 71 publications

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“…Let us study the symmetry conditions satisfied by the block of tensors depicted in (41). As we mentioned earlier, the number of symmetries is now extensive and grows as we consider blocks of more and more tensors.…”

Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

“…For convenience, we highlighted sets of bonds that satisfy the same pattern of symmetry conditions, which reflects our choice of concatenation pattern. The choice of colours is in concordance with (41).…”

Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

“…More precisely, we can find an isomorphism of the form (37) for T X 3d T X 3d associated with the set of symmetry conditions described above. In order to show this, let us start by applying the isomorphism property to each tensor in (41), where the sets of independent loops are provided by seven of the eight triangles, respectively. More specifically, we choose to express the l.h.s and r.h.s tensors, respectively, as…”

Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

“…More precisely, we should interpret these topological sectors as a looplike flux excitation to which a charge is attached, while being threaded by another flux. Loop-like excitations can then only fuse and braid if they are threaded by the same flux, giving rise in particular to the so-called three-loop braiding statistics [36][37][38][39][40][41][42]. Furthermore, for more general models where the input group could be any finite group, the threading flux would have the effect of constraining the magnetic flux and electric charge quantum numbers of the loop-like excitation [37,[43][44][45].…”

Section: Transfer Operatormentioning

confidence: 99%

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On tensor network representations of the (3+1)d toric code

Delcamp

Schuch

2021

Quantum

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We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge Z2-symmetry.

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Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

Section: Renormalization Group Fixed Pointmentioning

confidence: 99%

Section: Transfer Operatormentioning

confidence: 99%

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On tensor network representations of the (3+1)d toric code

Delcamp

Schuch

2021

Quantum

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“…The technical engine of the construction notably reflects the combinatorial homotopy of Whitehead, Baues, et al [Bau91]. Relatively simple aspects of the construction such as loop-particle braiding [BFMM18] yield higher generalisations of classical results, for example in low-dimensional topology [FM09]. We discuss one such result, a lifting of Artin's representation of the braid group [Bir74] to the (extended) loop braid group [Lin08] (see [Dam17], for a survey).…”

Section: Introductionmentioning

confidence: 99%

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

Damiani¹,

Martins²,

Martin³

2019

Preprint

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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 Mn is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of Mn. 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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Motion Groupoids and Mapping Class Groupoids

Martins

Martin

Torzewska

2023

Commun. Math. Phys.

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Here $${\underline{M}}$$ M ̲ denotes a pair (M, A) of a manifold and a subset (e.g. $$A=\partial M$$ A = ∂ M or $$A=\varnothing $$ A = ∅ ). We construct for each $${\underline{M}}$$ M ̲ its motion groupoid$$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ , whose object set is the power set $$ {{\mathcal {P}}}M$$ P M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on $${{\mathcal {P}}}M$$ 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 $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ of $$N\in {{\mathcal {P}}}M$$ N ∈ P M . We also construct the mapping class groupoid$$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ associated to a pair $${\underline{M}}$$ 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 $${\underline{M}}$$ M ̲ we explicitly construct a functor $${\textsf{F}}:\textrm{Mot}_{{\underline{M}}} \rightarrow \textrm{MCG}_{{\underline{M}}}$$ F : Mot M ̲ → MCG M ̲ , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $$\pi _0$$ π 0 and $$\pi _1$$ π 1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case $${\underline{M}}=([0,1]^n, \partial [0,1]^n)$$ M ̲ = ( [ 0 , 1 ] n , ∂ [ 0 , 1 ] n ) , for any $$n\in {\mathbb {N}}$$ n ∈ N . We show that the congruence relation used in the construction $$\textrm{Mot}_{{\underline{M}}}$$ Mot 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 $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ and $$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ demonstrating the utility of the constructions.

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Representations of the loop braid group and Aharonov–Bohm like effects in discrete $(3+1)$-dimensional higher gauge theory

Abstract: This is a repository copy of Representations of the loop braid group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory.

Cited by 11 publications

References 71 publications

On tensor network representations of the (3+1)d toric code

On tensor network representations of the (3+1)d toric code

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

Motion Groupoids and Mapping Class Groupoids

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