Communications in Mathematical Physics

, Volume 359, Issue 3, pp 1091–1121 | Cite as

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. Big01.
    Bigelow S.: Braid groups are linear. J. Am. Math. Soc. 14(2), 471–486 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. Big03.
    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)Google Scholar
  3. BK98.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. Bur36.
    Burau, W.: über Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Sem. Univ. Hamburg 11(1), 179–186 (1936) Google Scholar
  5. DJMM90.
    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)Google Scholar
  6. Dri89.
    Drinfeld V.G.: Quasi-Hopf algebras. Algeb Anal 1(6), 114–148 (1989)MathSciNetGoogle Scholar
  7. FFTL10.
    Feigin B., Frenkel E., Laredo V.T.: Gaudin models with irregular singularities. Adv. Math. 223(3), 873–948 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. FMTV00.
    Felder G., Markov Y., Tarasov V., Varchenko A.: Differential equations compatible with KZ equations. Math. Phys. Anal. Geom. 3(2), 139–177 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. GLP.
    Gaiotto, D., Lamy-Poirier, J.: Irregular singularities in the \({H_3^+}\) WZW model. arXiv:1301.5342
  10. GT12.
    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)Google Scholar
  11. Har97.
    Haraoka Y.: Confluence of cycles for hypergeometric functions on \({Z_{2,n+1}}\) . Trans. Am. Math. Soc. 349(2), 675–712 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. Hat02.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  13. HKK.
    Haiden, F., Katzarkov, L., Kontsevich, M.: Flat surfaces and stability structures. arXiv:1409.8611
  14. Ike.
    Ikeda, A.: Framed Alexander polynomials and framed Burau representations (in preparation)Google Scholar
  15. Ito15.
    Ito T.: Reading the dual Garside length of braids from homological and quantum representations invariants. Commun. Math. Phys. 335(1), 345–367 (2015)ADSCrossRefMATHGoogle Scholar
  16. Ito16.
    Ito T.: A homological representation formula of colored Alexander invariants. Adv. Math. 289, 142–160 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. Ive86.
    Iversen B.: Cohomology of sheaves, Universitext. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  18. JNS08.
    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)CrossRefMATHGoogle Scholar
  19. Koh87.
    Kohno T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier (Grenoble) 37(4), 139–160 (1987)MathSciNetCrossRefMATHGoogle Scholar
  20. Koh12.
    Kohno, T.: Quantum and homological representations of braid groups. In: Configuration Spaces, volume 14 of CRM Series, pp. 355–372. Ed. Norm., Pisa (2012)Google Scholar
  21. Kra00.
    Krammer D.: The braid group \({B_4}\) is linear. Invent. Math. 142(3), 451–486 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Kra02.
    Krammer D.: Braid groups are linear. Ann. Math. (2) 155(1), 131–156 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. KS92.
    Ko K.H., Smolinsky L.: The framed braid group and 3-manifolds. Proc. Am. Math. Soc. 115(2), 541–551 (1992)MathSciNetMATHGoogle Scholar
  24. KT08.
    Kassel, C., Turaev, V.: Braid groups, volume 247 of Graduate Texts in Mathematics. Springer, New York (2008). (With the graphical assistance of Olivier Dodane)Google Scholar
  25. KZ84.
    Knizhnik V.G., Zamolodchikov A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247(1), 83–103 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. Law90.
    Lawrence R.J.: Homological representations of the Hecke algebra. Commun. Math. Phys. 135(1), 141–191 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. NS10.
    Nagoya H., Sun J.: Confluent primary fields in the conformal field theory. J. Phys. A 43(46), 465203, 13 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. PP02.
    Paoluzzi L., Paris L.: A note on the Lawrence–Krammer–Bigelow representation. Algeb. Geom. Topol. 2, 499–518 (2002)MathSciNetCrossRefMATHGoogle Scholar
  29. SV90.
    Schechtman V.V., Varchenko A.N.: Hypergeometric solutions of Knizhnik–Zamolodchikov equations. Lett. Math. Phys. 20(4), 279–283 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. Sys01.
    Sysoeva I.: Dimension n representations of the braid group on n strings. J. Algebr. 243(2), 518–538 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. TK88.
    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)Google Scholar
  32. TYM96.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. Var91.
    Varchenko A.N.: A determinant formula for Selberg integrals. Funkt. Anal. Priloz. 25(4), 88–89 (1991)MathSciNetMATHGoogle Scholar
  34. Wil11.
    Wilson B.J.: Highest-weight theory for truncated current Lie algebras. J. Algeb. 336, 1–27 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. Zhe.
    Zheng, H.: Faithfulness of the Lawrence representation of braid groups. arXiv:math/0509074
  36. Zhe05.
    Zheng H.: A reflexive representation of braid groups. J. Knot Theory Ramif 14(4), 467–477 (2005)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIASThe University of TokyoKashiwa, ChibaJapan

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