Categorical Time-Reversal Symmetries

[2603.28720] Categorical Time-Reversal Symmetries Skip to main content Search Submit Donate Log in Condensed Matter > Strongly Correlated Electrons arXiv:2603.28720 (cond-mat) [Submitted on 30 Mar 2026 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: Categorical Time-Reversal Symmetries Authors: Rui Wen , Sakura Schafer-Nameki View PDF HTML (experimental) Abstract: The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over $\mathbb{C}$. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry $\mathbb{Z}_2^T$, and show that the relevant physical structures are naturally described by fusion categories over $\mathbb{R}$. A class of real fusion categories, which we call Galois-real fusion categories, provides the correct categorical model for anti-unitary symmetries. A simple example is the time-reversal symmetry $\mathbb{Z}_2^T$ itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories $(G^T)^{\omega}$ and $\mathsf{Rep}(G^T)$ associated to anti-linear groups $G^T$, as well as non-invertible time-reversal symmetries described by a real analogue of Tambara--Yamagami fusion categories. We then classify gapped phases enriched with anti-linear symmetries in terms of module categories over Galois-real fusion categories. We furthermore apply the categorical formulation to prove dualities (i.e. gauge or Morita equivalences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry Topological Field Theory (SymTFT) framework, in which Galois-real fusion categories arise as boundary conditions of a $\mathbb{Z}_2^T$-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same $\mathbb{Z}_2^T$-enriched SymTFT. Comments: 43 pages. v2: typos fixed Subjects: Strongly Correlated Electrons (cond-mat.str-el) ; High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Category Theory (math.CT) Cite as: arXiv:2603.28720 [cond-mat.str-el] (or arXiv:2603.28720v3 [cond-mat.str-el] for this version) https://doi.org/10.48550/arXiv.2603.28720 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rui Wen [ view email ] [v1] Mon, 30 Mar 2026 17:36:31 UTC (78 KB) [v2] Thu, 2 Apr 2026 17:08:12 UTC (78 KB) [v3] Tue, 25 Aug 2026 07:26:39 UTC (79 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.str-el < prev | next > new | recent | 2026-03 Change to browse by: cond-mat hep-th math math-ph math.CT math.MP References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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