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Communications in Mathematical Physics

, Volume 369, Issue 3, pp 1167–1185 | Cite as

Domain Walls in Topological Phases and the Brauer–Picard Ring for \({{\rm Vec} (\mathbb{Z}/p\mathbb{Z})}\)

  • Daniel Barter
  • Jacob C. BridgemanEmail author
  • Corey Jones
Article

Abstract

We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer–Picard ring for the fusion category \({{\bf Vec} (\mathbb{Z}/p\mathbb{Z})}\) . Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer–Picard ring from a physical perspective.

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Notes

Acknowledgements

This work is supported by the Australian Research Council (ARC) via the Centre of Excellence in Engineered Quantum Systems (EQuS) Project Number CE170100009, Discovery Project “Subfactors and symmetries“ DP140100732 and Discovery Project “Low dimensional categories“ DP16010347. We thank Andrew Doherty, Steve Flammia, Dominic Williamson, Cain Edie-Michell, Scott Morrison, and Christoph Schweigert. This work would not have been possible without their input and feedback. We thank Paul Wedrich and Christopher Chubb for feedback on the draft manuscript. We thank Noah Snyder for explaining how to prove Proposition 11.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Daniel Barter
    • 1
  • Jacob C. Bridgeman
    • 2
    Email author
  • Corey Jones
    • 1
  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.Centre for Engineered Quantum Systems, School of PhysicsThe University of SydneySydneyAustralia

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