Wall Crossing Invariants: from quantum mechanics to knots

TL;DR

This paper reviews wall-crossing invariants, connecting quantum mechanics, knot theory, and algebraic structures, highlighting their universal forms and relations to conformal blocks, cluster algebras, and quantum dilogarithms.

Contribution

It provides a comprehensive overview of wall-crossing invariants, linking them to knot invariants, conformal blocks, and cluster algebra mutations, with insights into their universal structures.

Findings

Wall-crossing invariants relate to knot invariants via R- and Racah matrices.

In the quasiclassical limit, these matrices simplify to universal forms.

Knot invariants can be constructed using cluster coordinates and quantum dilogarithms.

Abstract

We offer a pedestrian level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In non-trivial situations, starting from spin chains and matrix models, the S-matrices are operator-valued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then, the Kontsevich-Soibelman invariants are nothing but the standard knot invariants made out of these data within the Reshetikhin-Turaev-Witten approach. The R- and Racah matrices acquire a relatively universal form in the quasiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this point of view are matrices provided by the modular transformations of conformal blocks (with the usual identification R=T and U=S), and in the simplest case of the first degenerate field (2,1), when the conformal blocks satisfy a second order Shroedinger-like equation, the invariants coincide with the Jones (N=2) invariants of the associated knots. Another possibility to construct knot invariants is to realize the cluster coordinates associated with reshufflings of the Stokes lines immediately in terms of check-operators acting on the solutions to the Knizhnik-Zamolodchikov equations. Then, the R-matrices are realized as products of successive mutations in the cluster algebra and are manifestly described in terms of quantum dilogarithms ultimately leading to the Hikami construction of knot invariants.