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On Gauging Symmetry of Modular Categories

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Abstract

Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group G, gauging is a 2-step process: first extend the UMC to a G-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the \({H^4}\)-obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Shawn X. Cui & Zhenghan Wang

  2. Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A -10, Bogotá, Colombia

    César Galindo

  3. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Julia Yael Plavnik

  4. Microsoft Research, Station Q, University of California, Santa Barbara, CA, 93106, USA

    Zhenghan Wang

Authors
  1. Shawn X. Cui
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  2. César Galindo
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  3. Julia Yael Plavnik
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  4. Zhenghan Wang
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Corresponding author

Correspondence to Zhenghan Wang.

Additional information

Communicated by Y. Kawahigashi

S.-X. C and Z. W are partially supported by NSF DMS 1108736 and J.P. by CONICET, ANPCyT and Secyt-UNC. C. G. was partially supported by the FAPA funds from vicerrectoria de investigaciones de la Universidad de los Andes. This project began while J.P. was at Universidad de Buenos Aires, and the support of that institution is gratefully acknowledged. Part of this work was done during visits of C.G. to Microsoft Research Station Q and J.P. to University of California, Santa Barbara.

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Cui, S.X., Galindo, C., Plavnik, J.Y. et al. On Gauging Symmetry of Modular Categories. Commun. Math. Phys. 348, 1043–1064 (2016). https://doi.org/10.1007/s00220-016-2633-8

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  • DOI: https://doi.org/10.1007/s00220-016-2633-8

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