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Homological and Monodromy Representations of Framed Braid Groups

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Abstract

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equation to have irregular singularities. We also give a conjectural equivalence between these two classes of representations.

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References

  1. Bigelow S.: Braid groups are linear. J. Am. Math. Soc. 14(2), 471–486 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bigelow, S.: The Lawrence–Krammer representation. In: Topology and Geometry of Manifolds (Athens, GA, 2001), volume 71 of Proceedings of Symposia in Pure Mathematics, pp. 51–68. American Mathematical Society,Providence, RI (2003)

  3. Babujian H.M., Kitaev A.V.: Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell–Bloch system with pumping. J. Math. Phys. 39(5), 2499–2506 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Burau, W.: über Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Sem. Univ. Hamburg 11(1), 179–186 (1936)

  5. Date, E., Jimbo, M., Matsuo, A., Miwa, T.: Hypergeometric-type integrals and the \({\mathfrak{sl}_2}\) Knizhnik–Zamolodchikov equation. In: Proceedings of the Conference on Yang–Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, vol. 4, pp. 1049–1057 (1990)

  6. Drinfeld V.G.: Quasi-Hopf algebras. Algeb Anal 1(6), 114–148 (1989)

    MathSciNet  Google Scholar 

  7. Feigin B., Frenkel E., Laredo V.T.: Gaudin models with irregular singularities. Adv. Math. 223(3), 873–948 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Felder G., Markov Y., Tarasov V., Varchenko A.: Differential equations compatible with KZ equations. Math. Phys. Anal. Geom. 3(2), 139–177 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gaiotto, D., Lamy-Poirier, J.: Irregular singularities in the \({H_3^+}\) WZW model. arXiv:1301.5342

  10. Gaiotto, D., Teschner, J.: Irregular singularities in Liouville theory and Argyres–Douglas type gauge theories. J. High Energy Phys. 12:050 (front matter + 76 (2012)

  11. Haraoka Y.: Confluence of cycles for hypergeometric functions on \({Z_{2,n+1}}\) . Trans. Am. Math. Soc. 349(2), 675–712 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  13. Haiden, F., Katzarkov, L., Kontsevich, M.: Flat surfaces and stability structures. arXiv:1409.8611

  14. Ikeda, A.: Framed Alexander polynomials and framed Burau representations (in preparation)

  15. Ito T.: Reading the dual Garside length of braids from homological and quantum representations invariants. Commun. Math. Phys. 335(1), 345–367 (2015)

    Article  ADS  MATH  Google Scholar 

  16. Ito T.: A homological representation formula of colored Alexander invariants. Adv. Math. 289, 142–160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iversen B.: Cohomology of sheaves, Universitext. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  18. Jimbo M., Nagoya H., Sun J.: Remarks on the confluent KZ equation for \({\mathfrak{sl}_2}\) and quantum Painlevé equations. J. Phys. A 41(17), 175205, 14 (2008)

    Article  MATH  Google Scholar 

  19. Kohno T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier (Grenoble) 37(4), 139–160 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kohno, T.: Quantum and homological representations of braid groups. In: Configuration Spaces, volume 14 of CRM Series, pp. 355–372. Ed. Norm., Pisa (2012)

  21. Krammer D.: The braid group \({B_4}\) is linear. Invent. Math. 142(3), 451–486 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Krammer D.: Braid groups are linear. Ann. Math. (2) 155(1), 131–156 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Ko K.H., Smolinsky L.: The framed braid group and 3-manifolds. Proc. Am. Math. Soc. 115(2), 541–551 (1992)

    MathSciNet  MATH  Google Scholar 

  24. Kassel, C., Turaev, V.: Braid groups, volume 247 of Graduate Texts in Mathematics. Springer, New York (2008). (With the graphical assistance of Olivier Dodane)

  25. Knizhnik V.G., Zamolodchikov A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247(1), 83–103 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Lawrence R.J.: Homological representations of the Hecke algebra. Commun. Math. Phys. 135(1), 141–191 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Nagoya H., Sun J.: Confluent primary fields in the conformal field theory. J. Phys. A 43(46), 465203, 13 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Paoluzzi L., Paris L.: A note on the Lawrence–Krammer–Bigelow representation. Algeb. Geom. Topol. 2, 499–518 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schechtman V.V., Varchenko A.N.: Hypergeometric solutions of Knizhnik–Zamolodchikov equations. Lett. Math. Phys. 20(4), 279–283 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Sysoeva I.: Dimension n representations of the braid group on n strings. J. Algebr. 243(2), 518–538 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on P 1 and monodromy representations of braid group. In: Conformal Field Theory and Solvable Lattice Models (Kyoto, 1986), volume 16 of Advanced Studies in Pure Mathematics, pp. 297–372. Academic Press, Boston, MA (1988)

  32. Tong D.M., Yang S.D., Ma Z.Q.: A new class of representations of braid groups. Commun. Theor. Phys. 26(4), 483–486 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Varchenko A.N.: A determinant formula for Selberg integrals. Funkt. Anal. Priloz. 25(4), 88–89 (1991)

    MathSciNet  MATH  Google Scholar 

  34. Wilson B.J.: Highest-weight theory for truncated current Lie algebras. J. Algeb. 336, 1–27 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zheng, H.: Faithfulness of the Lawrence representation of braid groups. arXiv:math/0509074

  36. Zheng H.: A reflexive representation of braid groups. J. Knot Theory Ramif 14(4), 467–477 (2005)

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Authors and Affiliations

  1. Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Akishi Ikeda

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  1. Akishi Ikeda
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Correspondence to Akishi Ikeda.

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Communicated by C. Schweigert

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Ikeda, A. Homological and Monodromy Representations of Framed Braid Groups. Commun. Math. Phys. 359, 1091–1121 (2018). https://doi.org/10.1007/s00220-017-3036-1

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  • DOI: https://doi.org/10.1007/s00220-017-3036-1

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