This article is cited in 6 papers

$SU(2)/SL(2)$ knot invariants and Kontsevich–Soibelman monodromies

D. M. Galakhov^ab, A. D. Mironov^cdea, A. Yu. Morozov<sup>dea

a</sup> New High Energy Theory Center, Department of Physics and Astronomy, Rutgers University, New Brunswick, NJ, USA
^b Institute for Information Transmission Problems, Moscow, Russia
^c Lebedev Physical Institute, RAS, Moscow, Russia
^d Institute for Theoretical and Experiment Physics, Moscow, Russia
^e National Research Nuclear University MEPhI, Moscow, Russia

Abstract: We review the Reshetikhin–Turaev approach for constructing noncompact knot invariants involving $R$-matrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich–Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials.

Keywords: Chern–Simons theory, Kontsevich–Soibelman monodromy, Wilson average, $R$-matrix, modular double, quantum A-polynomial.

PACS: 11.15.Yc, 02.10.Kn, 02.20.Uw

Received: 19.10.2015

DOI: 10.4213/tmf9068

 English version: Theoretical and Mathematical Physics, 2016, 187:2, 678–694

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