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Braided Zesting and Its Applications

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Abstract

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

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Notes

  1. It seems to be the case that the trivial representation appears in \(X_\mu ^{{\otimes }N}\) for any object \(X_\mu \in {\text {Rep}}(\mathfrak {sl}_N)\), but we could not find a proof in the literature.

References

  1. Bakalov, B., Kirillov Jr., A.: Lectures on Tensor Categories and Modular Functors, Volume 21 of University Lecture Series. American Mathematical Society, Providence, RI (2001)

    MATH  Google Scholar 

  2. Bourbaki, N.: Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer, Berlin (2002). Translated from the 1968 French original by Andrew Pressley

  3. Bruguières, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. Math. Ann. 316(2), 215–236 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bruillard, P., Galindo, C., Hagge, T., Ng, S.-H., Plavnik, J.Y., Rowell, E.C., Wang, Z.: Fermionic modular categories and the 16-fold way. J. Math. Phys. 58(4), 041704 (2017). 31

    Article  ADS  MathSciNet  Google Scholar 

  5. Bruillard, P., Galindo, C., Hong, S.-M., Kashina, Y., Naidu, D., Natale, S., Plavnik, J.Y., Rowell, E.C.: Classification of integral modular categories of Frobenius–Perron dimension \(pq^4\) and \(p^2q^2\). Can. Math. Bull. 57(4), 721–734 (2014)

    Article  Google Scholar 

  6. Bruillard, P., Gustafson, P., Plavnik, J.Y., Rowell, E.C.: Dimension as a quantum statistic and the classification of metaplectic categories. Topol. Phases Matter Quantum Comput. 747, 89 (2020)

    Article  MathSciNet  Google Scholar 

  7. Bruillard, P., Ng, S.-H., Rowell, E.C., Wang, Z.: Rank-finiteness for modular categories. J. Am. Math. Soc. 29(3), 857–881 (2016)

    Article  MathSciNet  Google Scholar 

  8. Cui, S.X., Galindo, C., Plavnik, J.Y., Wang, Z.: On gauging symmetry of modular categories. Commun. Math. Phys. 348(3), 1043–1064 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  9. Davydov, A., Nikshych, D.: Braided Picard groups and graded extensions of braided tensor categories. arXiv:2006.08022

  10. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II, volume 87 of Progr. Math., pp. 111–195. Birkhäuser Boston, Boston, MA (1990)

  11. Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories. I. Sel. Math. (N.S.) 16(1), 1–119 (2010)

    Article  MathSciNet  Google Scholar 

  12. Edie-Michell, C.: Auto-equivalences of the modular tensor categories of type \( {A} \), \( {B} \), \( {C} \) and \( {G}\). (2020) arXiv preprint arXiv:2002.03220

  13. Eilenberg, S., Lane, S.M.: On the groups of \(H(\Pi, n)\). I. Ann. Math. 2(58), 55–106 (1953)

    Article  Google Scholar 

  14. Eilenberg, S., Mac Lane, S.: On the groups \(H(\Pi, n)\). II. Methods of computation. Ann. Math. (2) 60, 49–139 (1954)

    Article  MathSciNet  Google Scholar 

  15. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence, RI (2015)

    Book  Google Scholar 

  16. Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quantum Topol. 1(3), 209–273 (2010). With an appendix by Ehud Meir

    Article  MathSciNet  Google Scholar 

  17. Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories. Adv. Math. 226(1), 176–205 (2011)

    Article  MathSciNet  Google Scholar 

  18. Gelaki, S., Naidu, D., Nikshych, D.: Centers of graded fusion categories. Algebra Number Theory 3(8), 959–990 (2009)

    Article  MathSciNet  Google Scholar 

  19. Gelaki, S., Nikshych, D.: Nilpotent fusion categories. Adv. Math. 217(3), 1053–1071 (2008)

    Article  MathSciNet  Google Scholar 

  20. Henriques, A., Penneys, D., Tener, J.: Categorified trace for module tensor categories over braided tensor categories. Doc. Math. 21(2016), 1089–1149 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hochschild, G., Serre, J.-P.: Cohomology of group extensions. Transactions of the American Mathematical Society, pp. 110–134 (1953)

  22. Kazhdan, D., Wenzl, H.: Reconstructing monoidal categories. In: I.M. Gelfand Seminar, volume 16 of Adv. Soviet Math., pp. 111–136. Am. Math. Soc., Providence, RI (1993)

  23. Mignard, M., Schauenburg, P.: Modular categories are not determined by their modular data (2017). arXiv preprint arXiv:1708.02796

  24. Naidu, D., Rowell, E.C.: A finiteness property for braided fusion categories. Algebr. Represent. Theory 14(5), 837–855 (2011)

    Article  MathSciNet  Google Scholar 

  25. Rowell, E.C.: From quantum groups to unitary modular tensor categories. In: Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, volume 413 of Contemp. Math., pp. 215–230. Amer. Math. Soc., Providence, RI (2006)

  26. Vafa, C.: Toward classification of conformal theories. Phys. Lett. B 206(3), 421–426 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  27. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, USA

    Colleen Delaney & Julia Plavnik

  2. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    César Galindo

  3. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Eric C. Rowell & Qing Zhang

Authors
  1. Colleen Delaney
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  2. César Galindo
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  3. Julia Plavnik
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  4. Eric C. Rowell
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  5. Qing Zhang
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Corresponding author

Correspondence to Eric C. Rowell.

Additional information

Communicated by Y. Kawahigashi

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The authors gratefully acknowledge the support of the American Institute of Mathematics, where this collaboration was initiated. C.G would like to thank the hospitality and excellent working conditions of the Department of Mathematics at the University of Hamburg, where he carried out this research as a Fellow of the Humboldt Foundation. C.G. is partially supported by Faculty of Science of Universidad de los Andes, Convocatoria para la Financiación de Programas de Investigación, proyecto “Condiciones de frontera para TQFT equivariantes”. J.P was partially supported by US NSF Grants DMS-1802503 and DMS-1917319. E.C.R. was partially supported by US NSF Grant DMS-1664359, a Simons Foundation Fellowship (Grant No. 614735), and a Texas A&M Presidential Impact Fellowship. Part of this research was carried out while CD, JP, ER and QZ participated in a semester-long program at MSRI, which is partially supported by NSF Grant DMS-1440140. We thank Richard Ng for useful discussions.

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Delaney, C., Galindo, C., Plavnik, J. et al. Braided Zesting and Its Applications. Commun. Math. Phys. 386, 1–55 (2021). https://doi.org/10.1007/s00220-021-04002-4

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