Geometric Presentations of Braid Groups for Particles on a Graph

Byung Hee An, Tomasz Maciazek

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
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Abstract

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.
Original languageEnglish
Pages (from-to)1109-1140
Number of pages32
JournalCommunications in Mathematical Physics
Volume384
Issue number2
Early online date28 Apr 2021
DOIs
Publication statusPublished - Jun 2021

Bibliographical note

Funding Information:
The authors gratefully acknowledge the support of the American Institute of Mathematics (AIM) where this collaboration was initiated. We would like to thank Adam Sawicki and Jon Harrison for useful discussions during the workshop at AIM. TM would like to also thank Jonathan Robbins and Nick Jones for helpful discussions about graph configuration spaces and anyons. Byung Hee An was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) No. 2020R1A2C1A01003201. TM acknowledges the support of the Foundation for Polish Science (FNP), START programme.

Funding Information:
The authors gratefully acknowledge the support of the American Institute of Mathematics (AIM) where this collaboration was initiated. We would like to thank Adam Sawicki and Jon Harrison for useful discussions during the workshop at AIM. TM would like to also thank Jonathan Robbins and Nick Jones for helpful discussions about graph configuration spaces and anyons. Byung Hee An was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) No. 2020R1A2C1A01003201. TM acknowledges the support of the Foundation for Polish Science (FNP), START programme.

Publisher Copyright:
© 2021, The Author(s).

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