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The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters

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Abstract

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are ‘categorified’ in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerbes. We also show how the 2-character of a 2-representation can be made functorial with respect to morphisms of 2-representations. Under the geometric correspondence, the 2-character of a 2-representation corresponds to the geometric character of its associated equivariant gerbe. This enables us to show that the complexified 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary conjugation equivariant vector bundles over the group.

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References

  1. Ambrose, W.: Structure theorems for a special class of Banach algebras. Trans. Amer. Math. Soc. 57, 364–386 (1945)

    MATH  MathSciNet  Google Scholar 

  2. Baez, J.C., Dolan, J.: Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36, 6073–6105 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baez, J.C.: Higher-dimensional algebra II: 2-Hilbert spaces. Adv. Math. 127, 125–189 (1997). arXiv:q-alg/9609018

    Article  MATH  MathSciNet  Google Scholar 

  4. Baez, J.C., Lauda, A.D.: Higher-dimensional algebra V: 2-groups. Theory Appl. Categ. 12(14), 423–491 (2004). arXiv:math.QA/0307200

    MATH  MathSciNet  Google Scholar 

  5. Baez, J.C.: This week’s finds in mathematical physics week 223. http://math.ucr.edu/home/baez/week223.html (2009)

  6. Baez, J.C., Baratin, A., Freidel, L., Wise, D.: Representations of 2-groups on higher Hilbert spaces. arXiv:0812.4969 (2009)

  7. Barrett, J., Mackaay, M.: Categorical representations of categorical groups. Theory Appl. Categ. 16(20), 529–557 (2006). arXiv:math.CT/math.CT/0407463

    MATH  MathSciNet  Google Scholar 

  8. Behrend, K., Xu, P.: Differentiable Stacks and Gerbes. arXiv:math/0605694 (2009)

  9. Bartlett, B.: On unitary 2-representations of finite groups and topological quantum field theory. Ph.D. thesis, University of Sheffield. http://math.sun.ac.za/~bbartlett/thesis/thesis.pdf (2008)

  10. Boyarchenko, M.: Introduction to modular categories, Lecture series at University of Chicago. Geometric Langlands Seminar. Available at www.math.uchicago.edu/~mitya/langlands.html

  11. Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics vol. 107. Birkhauser, Boston (1993)

    Google Scholar 

  12. Brylinski, J.-.L., McLaughlin, D.: The geometry of degree-four characteristic classes and of line bundles on loop spaces I. Duke Math. J. 75(3), 603–638 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caldararu, A., Willerton, S.: The Mukai pairing, I: a categorical approach. arXiv:math/0707.2052 (2009)

  14. Segal, G.: Lie groups. In: Carter, R., Segal, G., Macdonald, I. (eds.) Lectures on Lie Groups and Lie Algebras, vol. 32. London Mathematical Society, Student Texts (1995)

  15. Cegarra, A.M., Garzón, A.R., Grandjean, A.R.: Graded extensions of categories. J. Pure Appl. Algebra 154, 117–141 (2000). http://www.ugr.es/ anillos/Preprints/total.ps

    Article  MATH  MathSciNet  Google Scholar 

  16. Costello, K.: Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210(1), 165–214 (2007). arXiv:math/0412149

    MATH  MathSciNet  Google Scholar 

  17. Crane, L., Yetter, D.N.: Measurable categories and 2-groups. Appl. Categ. Structures 13(5–6), 501–516 (2005). arXiv:math.QA/0305176

    Article  MATH  MathSciNet  Google Scholar 

  18. Deligne, P.: Action du groupe des tresses sur une catégorie. Invent. Math. 128, 159–175 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Comm. Math. Phys. 129, 60–72 (1990)

    Article  MathSciNet  Google Scholar 

  20. Elgeueta, J.: Representation theory of 2-groups on Kapranov and Voevodsky’s 2-category 2Vect. Adv. Math. 213(1), 53–92 (2007) (arXiv:math/0408120v2)

    Article  MathSciNet  Google Scholar 

  21. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. of Math. 162(2), 581–642 (2005). arXiv:math/0203060v9

    Article  MATH  MathSciNet  Google Scholar 

  22. Fantechi, B., et al.: Fundamental Algebraic Geometry: Grothendieck’s FGA Explained, vol. 123. AMS Mathematical Surveys and Monographs. AMS, Providence (2006)

    Google Scholar 

  23. Freed, D.: Higher algebraic structures and quantization. Comm. Math. Phys. 159(2), 343–398 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Freed, D.: Quantum groups from path integrals. arXiv:q-alg/9501025 (2009)

  25. Freed, D., Hopkins, M., Teleman, C.: Loop groups and twisted k-theory II. arXiv:math/0511232 (2009)

  26. Freyd, P.J., Yetter, D.N.: Coherence theorems via knot theory. J. Pure Appl. Algebra 78, 49–76 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ganter, N., Kapranov, M.: Representation and character theory in 2-categories. Adv. Math. 217(5), 2268–2300 (2008). arXiv:math.KT/0602510

    Article  MATH  MathSciNet  Google Scholar 

  28. Grothendieck, A.: Catégories Fibrées et Descent (SGAI) Exposé VI. Lecture Notes in Mathematics, vol. 224, pp. 145–194. Springer, Berlin (1971)

    Google Scholar 

  29. Gurski, N.: An algebraic theory of tricategories. Ph.D. thesis, University of Chicago. http://gauss.math.yale.edu/~mg622/tricats.pdf (2006)

  30. Kapranov, M.M., Voevodsky, V.A.: 2-categories and the Zamolodchikov tetradedra equations. In: Proceedings of Symposia in Pure Mathematics, vol. 56, pp. 177–259. American Mathematical Society, Providence (1994)

    Google Scholar 

  31. Kirwin, W.D.: Coherent states in geometric quantization. J. Geom. Phys. 57(2), 531–548 (2007). arXiv:math/0502026

    Article  MATH  MathSciNet  Google Scholar 

  32. Lauda, A.: Frobenius algebras and planar open string topological field theories. arXiv:math/0508349 (2009)

  33. Leinster, T.: Basic bicategories. arXiv:math.CT/9810017 (2009)

  34. Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88, 55–112 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Joyal, A., Street, R.: Braided monoidal categories. Preprint, revised and expanded version of Macquarie Mathematics Reports No. 860081, Preprint (1986)

  36. Lawvere, F.W.: Metric spaces, generalized logic and closed categories. Rend. Sem. Mat. Fis. Milano XLIII, 135–166 (1973); republished in Reprints in Theory Appl. Categ. (1), 1–37 (2002)

    Article  MathSciNet  Google Scholar 

  37. Moerdijk, I.: Introduction to the language of Stacks and Gerbes. arXiv:math.AT/0212266 (2009)

  38. Morton, J.: Extended TQFT’s and quantum gravity. Ph.D. thesis, University of California Riverside. arXiv:0710.0032 (2007)

  39. Müger, M.: From subfactors to categories and topology Frobenius, I., algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180, 81–157 (2003). arXiv:math/0111204

    Article  MATH  MathSciNet  Google Scholar 

  40. Murray, M., Singer, M.: Gerbes, Clifford modules and the index theorem. Ann. Global Anal. Geom. 26(4), 355–367 (2004). arXiv:math.DG/0302096

    Article  MATH  MathSciNet  Google Scholar 

  41. Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8(2), 177–206 (2003). arXiv:math/0111139

    Article  MATH  MathSciNet  Google Scholar 

  42. Robinson, P.L., Rawnsley, J.H.: The metaplectic representation, Mp ℂ structures and geometric quantization. Mem. Amer. Math. Soc. 81(410) (1989)

  43. Street, R.: Categorical structures. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 1, pp. 529–577. Elsevier, Amsterdam (1995)

    Google Scholar 

  44. Tu, J.-L., Xu, P., Laurent-Gengoux, C.: Twisted K-theory of differentiable Stacks. Ann. Sci. École Norm. Sup. (4) 37(6), 841–910 (2004). arXiv:math/0306138v2

    MATH  MathSciNet  Google Scholar 

  45. Willerton, S.: The twisted Drinfeld double of a finite group via gerbes and finite groupoids. arXiv:math.QA/0503266 (2009)

  46. Spera, M.: On Kählerian coherent states. In: Mladenov, I., Naber, G. (eds.) Proceedings of the International Conference on Geometry, Integrability and Quantization, Varna Bulgaria 1999, pp. 241–256. Coral, St Albans. www.bio21.bas.bg/proceedings/Proceedings_files/vol1content.htm (2000)

  47. Woodhouse, N.M.J.: Geometric Quantization. Oxford Mathematical Monographs. Oxford University Press, Oxford (1992)

    Google Scholar 

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  1. Bruce Bartlett

    Present address: Stellenbosch University, Stellenbosch, South Africa

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  1. University of Sheffield, Sheffield, UK

    Bruce Bartlett

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Bartlett, B. The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters. Appl Categor Struct 19, 175–232 (2011). https://doi.org/10.1007/s10485-009-9189-0

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