# Wall-crossing invariants: from quantum mechanics to knots

## 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 nontrivial situations, starting from spin chains and matrix models, the S-matrices are operatorvalued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then the Kontsevich-Soibelman (KS) 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 semiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this standpoint 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 Shrödinger-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 solutions of the Knizhnik-Zamolodchikov equations. Then themore »

- Authors:

- ITEP (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22472362

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 120; Journal Issue: 3; Other Information: Copyright (c) 2015 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COORDINATES; MATHEMATICAL SOLUTIONS; QUANTUM MECHANICS; R MATRIX; S MATRIX; SCATTERING; SEMICLASSICAL APPROXIMATION; SPIN; TRANSFORMATIONS

### Citation Formats

```
Galakhov, D., E-mail: galakhov@itep.ru, E-mail: galakhov@physics.rutgers.edu, Mironov, A., E-mail: mironov@lpi.ru, and Morozov, A., E-mail: morozov@itep.ru.
```*Wall-crossing invariants: from quantum mechanics to knots*. United States: N. p., 2015.
Web. doi:10.1134/S1063776115030206.

```
Galakhov, D., E-mail: galakhov@itep.ru, E-mail: galakhov@physics.rutgers.edu, Mironov, A., E-mail: mironov@lpi.ru, & Morozov, A., E-mail: morozov@itep.ru.
```*Wall-crossing invariants: from quantum mechanics to knots*. United States. doi:10.1134/S1063776115030206.

```
Galakhov, D., E-mail: galakhov@itep.ru, E-mail: galakhov@physics.rutgers.edu, Mironov, A., E-mail: mironov@lpi.ru, and Morozov, A., E-mail: morozov@itep.ru. Sun .
"Wall-crossing invariants: from quantum mechanics to knots". United States.
doi:10.1134/S1063776115030206.
```

```
@article{osti_22472362,
```

title = {Wall-crossing invariants: from quantum mechanics to knots},

author = {Galakhov, D., E-mail: galakhov@itep.ru, E-mail: galakhov@physics.rutgers.edu and Mironov, A., E-mail: mironov@lpi.ru and Morozov, A., E-mail: morozov@itep.ru},

abstractNote = {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 nontrivial situations, starting from spin chains and matrix models, the S-matrices are operatorvalued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then the Kontsevich-Soibelman (KS) 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 semiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this standpoint 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 Shrödinger-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 solutions of 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.},

doi = {10.1134/S1063776115030206},

journal = {Journal of Experimental and Theoretical Physics},

number = 3,

volume = 120,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}