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On the Geometry of Some Braid Group Representations

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Knots, Low-Dimensional Topology and Applications (KNOTS16 2016)

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Abstract

In this note we report on recent differential geometric constructions aimed at devising representations of braid groups in various contexts, together with some applications in different domains of mathematical physics. First, the classical Kohno construction for the 3- and 4-strand pure braid groups \(P_3\) and \(P_4\) is explicitly implemented by resorting to the Chen-Hain-Tavares nilpotent connections and to hyperlogarithmic calculus, yielding unipotent representations able to detect Brunnian and nested Brunnian phenomena. Physically motivated unitary representations of Riemann surface braid groups are then described, relying on Bellingeri’s presentation and on the geometry of Hermitian–Einstein holomorphic vector bundles on Jacobians, via representations of Weyl-Heisenberg groups.

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Acknowledgements

The author is grateful to the Organizers of the Workshop “Knots in Hellas 2016”, held at International Olympic Academy, Ancient Olympia, Greece, 17th–23rd July 2016, and to the staff of IOA, for the opportunity given to him to present a talk therein and for the excellent scientific level, atmosphere and hospitality in a marvellous historical and natural landscape. He also acknowledges financial support from D1-funds (Catholic University) (ex 60% Italian MIUR funds). This work has been carried out within the activities of INDAM (GNSAGA).

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “Niccolò Tartaglia”, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121, Brescia, Italy

    Mauro Spera

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  1. Mauro Spera
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Correspondence to Mauro Spera .

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Editors and Affiliations

  1. Department of Mathematics, Williams College, Williamstown, MA, USA

    Colin C. Adams

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, USA

    Cameron McA. Gordon

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Vaughan F.R. Jones

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Louis H. Kauffman

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  6. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA

    Kenneth C. Millett

  7. Department of Mathematics, Columbian College of Arts & Sciences, George Washington University, Washington, DC, USA

    Jozef H. Przytycki

  8. Department of Mathematics and Applications, University of Milano-Bicocca, Milano, Italy

    Renzo Ricca

  9. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Radmila Sazdanovic

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Spera, M. (2019). On the Geometry of Some Braid Group Representations. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_14

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