Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 092, 20 pages      arXiv:2403.03211      https://doi.org/10.3842/SIGMA.2024.092

Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)

Thibault Didier Décoppet
Mathematics Department, Harvard University, Cambridge, Massachusetts, USA

Received March 11, 2024, in final form October 09, 2024; Published online October 17, 2024

Abstract
We study group graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.

Key words: extension theory; fusion 2-category; supercohomology.

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References

  1. Crans S.E., Generalized centers of braided and sylleptic monoidal $2$-categories, Adv. Math. 136 (1998), 183-223.
  2. Cui S.X., Four dimensional topological quantum field theories from $G$-crossed braided categories, Quantum Topol. 10 (2019), 593-676, arXiv:1610.07628.
  3. Davydov A., Nikshych D., Braided Picard groups and graded extensions of braided tensor categories, Selecta Math. (N.S.) 27 (2021), 65, 87 pages, arXiv:2006.08022.
  4. Décoppet T.D., Multifusion categories and finite semisimple 2-categories, J. Pure Appl. Algebra 226 (2022), 107029, 16 pages, arXiv:2012.15774.
  5. Décoppet T.D., Weak fusion 2-categories, Cah. Topol. Géom. Différ. Catég. 63 (2022), 3-24, arXiv:2103.15150.
  6. Décoppet T.D., The Morita theory of fusion 2-categories, High. Struct. 7 (2023), 234-292, arXiv:2208.08722.
  7. Décoppet T.D., Drinfeld centers and Morita equivalence classes of fusion 2-categories, Compos. Math., to appear, arXiv:2211.04917.
  8. Décoppet T.D., Finite semisimple module 2-categories, Selecta Math. (N.S.), to appear, arXiv:2107.11037.
  9. Décoppet T.D., On the dualizability of fusion 2-categoriess, Quantum Topol., to appear, arXiv:2311.16827.
  10. Décoppet T.D., Yu M., Fiber 2-functors and Tambara-Yamagami fusion 2-categories, arXiv:2306.08117.
  11. Douglas C.L., Reutter D.J., Fusion 2-categories and a state-sum invariant for 4-manifolds, arXiv:1812.11933.
  12. Elgueta J., Cohomology and deformation theory of monoidal 2-categories. I, Adv. Math. 182 (2004), 204-277, arXiv:math.QA/0204099.
  13. Elgueta J., Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces, Adv. Math. 213 (2007), 53-92, arXiv:math.CT/0408120.
  14. Etingof P., Gelaki S., Descent and forms of tensor categories, Int. Math. Res. Not. 2012 (2012), 3040-3063, arXiv:1102.0657.
  15. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr, Vol. 205, American Mathematical Society, Providence, RI, 2015.
  16. Etingof P., Nikshych D., Ostrik V., Fusion categories and homotopy theory, Quantum Topol. 1 (2010), 209-273, arXiv:0909.3140.
  17. Gaiotto D., Johnson-Freyd T., Symmetry protected topological phases and generalized cohomology, J. High Energy Phys. 2019 (2019), 007, 34 pages, arXiv:1712.07950.
  18. Gaiotto D., Johnson-Freyd T., Condensations in higher categories, arXiv:1905.09566.
  19. Gu Z.-C., Wen X.-G., Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory, Phys. Rev. B 90 (2014), 115141, 59 pages, arXiv:1201.2648.
  20. Johnson-Freyd T., On the classification of topological orders, Comm. Math. Phys. 393 (2022), 989-1033, arXiv:2003.06663.
  21. Johnson-Freyd T., (3+1)D topological orders with only a $\mathbb{Z}/2$-charged particle, Comm. Math. Phys., to appear, arXiv:2011.11165.
  22. Johnson-Freyd T., Reutter D.J., Super-duper vector spaces I, 2023, available at https://homepages.uni-regensburg.de/ lum63364/ConferenceFFT/Reutter.pdf.
  23. Johnson-Freyd T., Reutter D., Minimal nondegenerate extensions, J. Amer. Math. Soc. 37 (2024), 81-150, arXiv:2105.15167.
  24. Johnson-Freyd T., Yu M., Fusion 2-categories with no line operators are grouplike, Bull. Aust. Math. Soc. 104 (2021), 434-442, arXiv:2010.07950.
  25. Jones C., Penneys D., Reutter D., A 3-categorical perspective on $G$-crossed braided categories, J. Lond. Math. Soc. 107 (2023), 333-406, arXiv:2009.00405.
  26. Kapustin A., Thorngren R., Fermionic SPT phases in higher dimensions and bosonization, J. High Energy Phys. 2017 (2017), no. 10, 080, 48 pages, arXiv:1701.08264.
  27. Kong L., Tian Y., Zhou S., The center of monoidal 2-categories in (3+1)D Dijkgraaf-Witten theory, Adv. Math. 360 (2020), 106928, 25 pages, arXiv:1905.04644.
  28. Lan T., Kong L., Wen X.-G., Classification of (3+1)D bosonic topological orders (I): The case when pointlike excitations are all bosons, Phys. Rev. X 8 (2018), 021074, 24 pages, arXiv:1704.04221.
  29. Lan T., Wen X.-G., Classification of (3+1)D bosonic topological orders (II): The case when some pointlike excitations are fermions, Phys. Rev. X 9 (2019), 021005, 37 pages, arXiv:1801.08530.
  30. Lurie J., Higher algebra, 2017, available at https://www.math.ias.edu/ lurie/papers/HA.pdf.
  31. Nikolaus T., Schreiber U., Stevenson D., Principal $\infty$-bundles: general theory, J. Homotopy Relat. Struct. 10 (2015), 749-801, arXiv:1207.0248.
  32. Pstrągowski P., On dualizable objects in monoidal bicategories, Theory Appl. Categ. 38 (2022), 257-310, arXiv:1411.6691.
  33. Sanford S., Fusion categories over non-algebraically closed fields, Ph.D. Thesis, Indiana University, 2022, available at https://www.proquest.com/docview/2715420938/ACDC80FE6AEA477EPQ/1.
  34. Schommer-Pries C.J., The classification of two-dimensional extended topological field theories, Ph.D. Thesis, University of California, Berkeley, 2009, arXiv:1112.1000.
  35. Wang Q.-R., Gu Z.-C., Towards a complete classification of symmetry-protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory, Phys. Rev. X 8 (2018), 011055, 29 pages, arXiv:1703.10937.

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