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On the monodromy of KZ-connections with irregular singularities

Xia Gu, Babak Haghighat, Pavel Putrov

Abstract

We study Knizhnik-Zamolodchikov (KZ) connection in the presence of irregular singularities, that is, poles of higher order. We consider both the case of a universal connection and the case when it is associated with a specific simple Lie algebra, such as $\mathfrak{su}(2)$. We give some general results about the monodromies of such flat connections in the configuration spaces of points, and provide explicit examples of topological invariants of links (more generally, tangles) realized by the monodromy.

On the monodromy of KZ-connections with irregular singularities

Abstract

We study Knizhnik-Zamolodchikov (KZ) connection in the presence of irregular singularities, that is, poles of higher order. We consider both the case of a universal connection and the case when it is associated with a specific simple Lie algebra, such as . We give some general results about the monodromies of such flat connections in the configuration spaces of points, and provide explicit examples of topological invariants of links (more generally, tangles) realized by the monodromy.
Paper Structure (22 sections, 143 equations, 7 figures)

This paper contains 22 sections, 143 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Braids, links and the flat connections
  3. The monodromy representation of the braid group
  4. From braids to links and tangles
  5. The KZ equation
  6. The space of chord diagrams and the associator
  7. Matrix forms and WZW realizations
  8. Irregular singularities
  9. The Stokes phenomenon
  10. The scaling transformation
  11. Examples of Invariants
  12. One regular and one irregular singularity
  13. Two regular singularities and one irregular
  14. Two irregular singularities and one irregular: universal perturbative invariant
  15. Two irregular operators
  16. ...and 7 more sections

Figures (7)

  • Figure 1: A path from $(z_1,z_2,z_3)$ to $(z_1',z_2',z_3')$ in $Conf_3$, the configuration space of 3 points in $\mathbf{C}$. The vertical direction corresponds to the interval $I$ parametrizing the path, while the horizontal directions correspond to $\mathbf{C}$.
  • Figure 2: Braids
  • Figure 3: A braid and its closure in a solid torus $D^2\times S^1$.
  • Figure 4: A tangle in $\mathbf{C}\times I$ and a closure of its limit in the solid torus $D^2\times S^1$.
  • Figure 5:
  • ...and 2 more figures