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Higher Gauss sums of modular categories

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Abstract

The definitions of the nth Gauss sum and the associated nth central charge are introduced for premodular categories \(\mathcal {C}\) and \(n\in \mathbb {Z}\). We first derive an expression of the nth Gauss sum of a modular category \(\mathcal {C}\), for any integer n coprime to the order of the T-matrix of \(\mathcal {C}\), in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these n, the higher Gauss sums are d-numbers and the associated central charges are roots of unity. In particular, if \(\mathcal {C}\) is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.

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References

  1. Anderson, G., Moore, G.: Rationality in conformal field theory. Commun. Math. Phys. 117(3), 441–450 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Bantay, P.: The Frobenius–Schur indicator in conformal field theory. Phys. Lett. B 394(1–2), 87–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. Calegari, F., Morrison, S., Snyder, N.: Cyclotomic integers, fusion categories, and subfactors. Commun. Math. Phys. 303(3), 845–896 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coste, A., Gannon, T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323(3–4), 316–321 (1994)

    Article  MathSciNet  Google Scholar 

  7. Davydov, A., Müger, M., Nikshych, D., Ostrik, V.: The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math. 677, 135–177 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Davydov, A., Nikshych, D., Ostrik, V.: On the structure of the Witt group of braided fusion categories. Sel. Math. 19(1), 237–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. de Boer, J., Goeree, J.: Markov traces and \({\rm II}_1\) factors in conformal field theory. Commun. Math. Phys. 139(2), 267–304 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong, C., Lin, X., Ng, S.-H.: Congruence property in conformal field theory. Algebra Number Theory 9(9), 2121–2166 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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  MATH  Google Scholar 

  12. Eilenberg, S., MacLane, S.: Cohomology theory of Abelian groups and homotopy theory. I. Proc. Natl. Acad. Sci. USA 36, 443–447 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eilenberg, S., MacLane, S.: Cohomology theory of Abelian groups and homotopy theory. II. Proc. Natl. Acad. Sci. USA 36, 657–663 (1950)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. (2) 162(2), 581–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Etingof, P., Ostrik, V.: On semisimplification of tensor categories. arXiv:1801.04409

  17. Evans, D.E., Gannon, T.: The exoticness and realisability of twisted Haagerup–Izumi modular data. Commun. Math. Phys. 307(2), 463–512 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gauss, C.F.: Summatio serierum quarundam singularium. Comment. Soc. Regiae Sci. Gott. 1 (1811)

  19. Gauss, C.F.: Disquisitiones arithmeticae (Trans. into English by Arthur A. Clarke, S. J.). Yale University Press, New Haven, Conn.-London (1966)

  20. Greiter, G.: A simple proof for a theorem of Kronecker. Am. Math. Mon. 85(9), 756–757 (1978)

    Article  MathSciNet  Google Scholar 

  21. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Volume 84 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1990)

    Book  MATH  Google Scholar 

  22. Kashina, Y., Montgomery, S., Ng, S.-H.: On the trace of the antipode and higher indicators. Isr. J. Math. 188, 57–89 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kashina, Y., Sommerhäuser, Y., Zhu, Y.: On higher Frobenius–Schur indicators. Mem. Am. Math. Soc. 181(855), viii+65 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Kač, V.G., Peterson, D.H.: Infinite-dimensional lie algebras, theta functions and modular forms. Adv. Math. 53(2), 125–264 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kirby, R., Melvin, P.: The \(3\)-manifold invariants of Witten and Reshetikhin–Turaev for \({\rm sl}(2,{ C})\). Invent. Math. 105(3), 473–545 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kirillov, A., Ostrik, V.: On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_2\) conformal field theories. Adv. Math. 171(2), 183–227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lang, S.: Algebra. Volume 211 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2002)

    Book  MATH  Google Scholar 

  28. Lejeune Dirichlet, G.: Recherches sur diverses applications de l’Analyse infinitésimale à la Théorie des Nombres. Seconde Partie. J. Reine Angew. Math. 21, 134–155 (1840)

    MathSciNet  Google Scholar 

  29. Lickorish, W.B.R.: Invariants for \(3\)-manifolds from the combinatorics of the Jones polynomial. Pac. J. Math. 149(2), 337–347 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Linchenko, V., Montgomery, S.: A Frobenius–Schur theorem for Hopf algebras. Algebr. Represent. Theory 3(4), 347–355 (2000). (Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lusztig, G.: Introduction to Quantum Groups, Modern Birkhäuser Classics. Birkhäuser, Boston (2010)

    Book  Google Scholar 

  32. Müger, M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Murakami, H., Ohtsuki, T., Okada, M.: Invariants of three-manifolds derived from linking matrices of framed links. Osaka J. Math. 29(3), 545–572 (1992)

    MathSciNet  MATH  Google Scholar 

  34. Ng, S.-H., Schauenburg, P.: Frobenius–Schur indicators and exponents of spherical categories. Adv. Math. 211(1), 34–71 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ng, S.-H., Schauenburg, P.: Higher Frobenius–Schur indicators for pivotal categories. In: Hopf algebras and generalizations. Contemporary Mathematics, vol. 441, pp. 63–90. American Mathematical Society, Providence, RI (2007)

  36. Ng, S.-H., Schauenburg, P.: Central invariants and higher indicators for semisimple quasi-Hopf algebras. Trans. Am. Math. Soc. 360(4), 1839–1860 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ng, S.-H., Schauenburg, P.: Congruence subgroups and generalized Frobenius–Schur indicators. Commun. Math. Phys. 300(1), 1–46 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ostrik, V.: On formal codegrees of fusion categories. Math. Res. Lett. 16(5), 895–901 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ostrik, V.: Pivotal fusion categories of rank 3. Mosc. Math. J. 15(2), 373–396 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pareigis, B.: On braiding and dyslexia. J. Algebra 171, 413–425 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Penneys, D., Tener, J.E.: Subfactors of index less than 5, part 4: vines. Int. J. Math. 23(3), 18 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Reshetikhin, N., Turaev, V.G.: Invariants of \(3\)-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rowell, E.C.: From quantum groups to unitary modular tensor categories. Contemp. Math. 413, 215–230 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rowell, E.C., Wang, Z.: Mathematics of topological quantum computing. Bull. Am. Math. Soc. (N.S.) 55(2), 183–238 (2018)

    Article  MathSciNet  Google Scholar 

  45. Scharlau, W.: Quadratic and Hermitian Forms, Volume 270 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1985)

  46. Schopieray, A.: Classification of \(\mathfrak{sl}_3\) relations in the Witt group of nondegenerate braided fusion categories. Commun. Math. Phys. 353(3), 1103–1127 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Schopieray, A.: Level bounds for exceptional quantum subgroups in rank two. Int. J. Math. 29(5), 1850034 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Schopieray, A: Prime decomposition of modular tensor categories of local modules of Type D. arXiv:1810.09057

  49. Shimizu, K.: Frobenius–Schur indicators in Tambara–Yamagami categories. J. Algebra 332, 543–564 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Shimizu, K.: Some computations of Frobenius–Schur indicators of the regular representations of Hopf algebras. Algebr. Represent. Theory 15(2), 325–357 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tambara, D., Yamagami, S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209(2), 692–707 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tucker, H.: Frobenius–Schur indicators for near-group and Haagerup–Izumi fusion categories. Pac. J. Math. arXiv:1510.05696

  53. Turaev, V.: Reciprocity for Gauss sums on finite abelian groups. Math. Proc. Cambr. Philos. Soc. 124(2), 205–214 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Volume 18 of De Gruyter Studies in Mathematics, revised edn. Walter de Gruyter & Co., Berlin (2010)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  56. Vaughan, S.M., Jones, F.R., Snyder, N.: The classification of subfactors of index at most 5. Bull. Am. Math. Soc. 51(2), 277–327 (2014)

    MathSciNet  MATH  Google Scholar 

  57. Wan, Z., Wang, Y.: Classification of spherical fusion categories of Frobenius–Schur exponent 2. Algebra Colloq. arXiv:1811.02004

  58. Wang, Z.: Topological Quantum Computation, Volume 112 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence, RI (2010)

  59. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The second author would like to thank Victor Ostrik for his suggestion to explore the notion of higher Gauss sums. Both the second and third authors would like to thank MSRI (Summer Graduate School 791) for providing the opportunity to initiate this collaboration. The third author would like to thank Thomas Kerler and James Cogdell, and the first author would like to thank Ling Long for fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Siu-Hung Ng & Yilong Wang

  2. School of Mathematics and Statistics, UNSW, Sydney, NSW, 2052, Australia

    Andrew Schopieray

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  1. Siu-Hung Ng
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Correspondence to Yilong Wang.

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The first author was partially supported by NSF DMS1664418.

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Ng, SH., Schopieray, A. & Wang, Y. Higher Gauss sums of modular categories. Sel. Math. New Ser. 25, 53 (2019). https://doi.org/10.1007/s00029-019-0499-2

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