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Coloured Alexander polynomials and KP hierarchy. (English) Zbl 1414.37030
The authors study the relation between the scaling 1-hook property of the coloured Alexander polynomial \(\mathcal {A}_R^{\mathcal{K}}\left( q \right)\) and the KP hierarchy. The Alexander polynomial arises as a special case of the HOMFLY polynomial \(\mathcal {H}_R^{\mathcal{K}}\left(q,a \right)\) of the knot \(\mathcal{K}\) coloured with representation \(R\) [E. Witten, Commun. Math. Phys. 121, No. 3, 351–399 (1989; Zbl 0667.57005); D. Bar-Natan, J. Knot Theory Ramifications 4, No. 4, 503–547 (1995; Zbl 0861.57009)]: \[{\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \frac{1} {Z}\int {DA{e^{ - \frac{i} {\hbar }{S_{CS}}\left[ A \right]}}{W_R}\left( {K,A} \right)}, \] where the Wilson loop is \[{W_R}\left( {K,A} \right) = {\text{t}}{{\text{r}}_R}P \exp \left( {\oint {A_\mu ^a\left( {\text{x}} \right){T^a}d{{\text{x}}^\mu }} } \right),\] and the Chern-Simons action is \[{S_{CS}}\left[ A \right] = \frac{\kappa } {{4\pi }}\int\limits_M {\text{Tr} \left( {A \wedge dA + \frac{2} {3}A \wedge A \wedge A} \right)}, \] with \(q = {e^\hbar },a = N\hbar \) and \(\hbar = \frac{{2\pi i}} {{\kappa + N}}.\) The limiting case \(\hbar \to 0,N \to \infty \) such that \(N\hbar\) remains fixed, i.e., \(q=1\), of the HOMFLY polynomials gives the special polynomials \({\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \sigma _R^{\mathcal {K}}\left( a \right)\), whose \(R\) dependence makes them expressible in the form \(\sigma _R^{\mathcal {K}}\left( a \right) = {\left( {\sigma _{\left[ 1 \right]}^{\mathcal {K}}\left( a \right)} \right)^{\left| R \right|}}\) for the Young diagram \(R = \left\{ {{R_i}} \right\},{R_1} \geqslant {R_2} \geqslant \ldots \geqslant {R_{l\left( R \right)}},\left| R \right|: = \sum\nolimits_i {{R_i}} \). This provides the construction of a KP \(\tau\)-function (see for example [P. Dunin-Barkowski et al., J. High Energy Phys. 2013, No. 3, Paper No. 021, 85 p. (2013; Zbl 1342.57004)]).

The dual limit as \(a \to 1\) of the HOMFLY polynomials, i.e., \({\mathcal {H}}_R^{\mathcal {K}}\left( {q,1} \right)\), for the fundamental representation \(R\) gives the Alexander polynomial, the coloured version of which exhibits a dual property with respect to \(R\), viz., \({\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)\), which holds only for the representations corresponding to 1-hook Young diagrams \({R = \left[ {r,{1^L}} \right]}.\)

In this paper the authors study this property perturbatively and claim that while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials induce the equations of the KP hierarchy.

The main result of the paper is stated in Section 5, where, by considering the generating function of the KP hierarchy, replacing the Hirota operators with the Casimir eigenvalues and symmetrizing the identity, the authors find that Hirota KP bilinear equations are satisfied if and only if \({\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)\). The authors give only the first half of the proof of this result.
The paper explores interesting connections between the KP hierarchy and the coloured Alexander polynomials.

MSC:
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
58J28 Eta-invariants, Chern-Simons invariants
81T55 Casimir effect in quantum field theory
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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References:
[1] Witten, E., Commun. Math. Phys., 121, 351, (1989)
[2] Chern, S.-S.; Simons, J., Ann. Math., 99, 48-69, (1974)
[3] E. Guadagnini, M. Martellini, M. Mintchev, Clausthal 1989, Procs., pp. 307-317.
[4] E. Guadagnini, M. Martellini, M. Mintchev, Clausthal 1989, Procs., pp. 307-317.
[5] Wu, F. Y., Knot theory and statistical mechanics, Rev. Mod. Phys., Rev. Mod. Phys., 65, 577, (1993), Erratum
[6] Kaul, R. K.; Govindarajan, T. R.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Zodinmawia; Ramadevi, P.; Zodinmawia; Ramadevi, P., Nucl. Phys. B, Nucl. Phys. B, Nucl. Phys. B, 422, 291-306, (1994)
[7] Ooguri, H.; Vafa, C.; Labastida, J. M.F.; Mariño, M.; Labastida, J. M.F.; Mariño, M.; Vafa, C.; Labastida, J. M.F.; Mariño, M.; Mariño, M.; Vafa, C., Nucl. Phys. B, Commun. Math. Phys., J. High Energy Phys., 0011, 423-449, (2000)
[8] Aravind, P. K.; Kauffman, L.; Lomonaco, S.; Kauffman, L.; Lomonaco, S.; Balasubramanian, V.; Fliss, J. R.; Leigh, R. G.; Parrikar, O.; Balasubramanian, V.; DeCross, M.; Fliss, J.; Kar, A.; Leigh, R. G.; Parrikar, O.; Melnikov, D.; Mironov, A.; Mironov, S.; Morozov, A.; Morozov, An., (Cohen, R. S.; etal., Potentiality, Entanglement and Passion-at-a-Distance, (1997), Kluwer), New J. Phys., New J. Phys., J. High Energy Phys., Nucl. Phys. B, 926, 6, 491-508, (2018)
[9] Bar-Natan, D.; Sleptsov, A., J. Knot Theory Ramif., Int. J. Mod. Phys. A, 29, 503-547, (2014)
[10] Smirnov, A.; Morozov, A.; Smirnov, A., (Proceedings of International School of Subnuclear Physics in Erice, Italy, (2012)), Nucl. Phys. B, 835, 284-313, (2010)
[11] Labastida, J. M.F.; Perez, E.; Dunin-Barkowski, P.; Sleptsov, A.; Smirnov, A., J. Math. Phys., Int. J. Mod. Phys. A, 28, 5183-5198, (2013)
[12] Dunin-Barkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., J. High Energy Phys., 03, (2013)
[13] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 2012, (2012)
[14] Zhu, Shengmao, J. High Energy Phys., 10, 1-24, (2013)
[15] Mironov, A. D.; Morozov, A. Yu.; Sleptsov, A. V.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., Theor. Math. Phys., Nucl. Phys. B, 889, 757, (2013)
[16] Mironov, A.; Morozov, A.; Sleptsov, A., Eur. Phys. J. C, 73, 2492, (2013)
[17] Akutsu, Y.; Deguchi, T.; Ohtsuki, T.; Murakami, J.; Cho, J.; Murakami, J., J. Knot Theory Ramif., Osaka J. Math., J. Knot Theory Ramif., 18, 09, 1271-1286, (2009)
[18] Mironov, A.; Morozov, A., Eur. Phys. J. C, 78, 284, (2018)
[19] Chmutov, S.; Duzhin, S., Acta Appl. Math., 66, 155, (2001)
[20] Dunin-Barkowski, P.; Sleptsov, A.; Smirnov, A., Int. J. Mod. Phys. A, 28, (2013)
[21] Alvarez, M.; Labastida, J. M.F., Nucl. Phys. B, 433, 555-596, (1995)
[22] Chmutov, S.; Duzhin, S.; Mostovoy, J., Introduction to Vassiliev knot invariants · Zbl 1245.57003
[23] Prasolov, V. V., Elements of homology theory, Graduate Studies in Mathematics, vol. 81, (2007) · Zbl 1120.55001
[24] Naculich, S. G.; Schnitzer, H. J., Nucl. Phys. B, 347, 687, (1990)
[25] Naculich, S. G.; Riggs, H. A.; Schnitzer, H. J.; Mlawer, E. J.; Naculich, S. G.; Riggs, H. A.; Schnitzer, H. J., Phys. Lett. B, Nucl. Phys. B, 352, 863, (1991)
[26] Liu, K.; Peng, P., Commun. Number Theory Phys., 5, (2011)
[27] Zhelobenko, D. P., Compact Lie groups and their representations, (1973), American Mathematical Society
[28] Mironov, A.; Morozov, A.; Natanzon, S.; Mironov, A.; Morozov, A.; Natanzon, S., Theor. Math. Phys., J. Geom. Phys., 62, 148-155, (2012)
[29] Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Orlov, A. Yu.; Scherbin, D. M.; Alexandrov, A.; Mironov, A.; Morozov, A.; Natanzon, S.; Alexandrov, A.; Mironov, A.; Morozov, A.; Natanzon, S., Int. J. Mod. Phys. A, Physica D, J. Phys. A, J. High Energy Phys., 11, 51-65, (2014)
[30] Okounkov, A.; Pandharipande, R., Ann. Math., 163, 517, (2006)
[31] Miwa, T.; Jimbo, M.; Date, E.; Jimbo, M.; Miwa, T., Solitons: differential equations, symmetries and infinite dimensional algebras, Publ. Res. Inst. Math. Sci., 19, 943-1001, (1983), Cambridge University Press · Zbl 0557.35091
[32] Kharchev, S.
[33] Kashiwara, M.; Miwa, T.; Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Proc. Jpn. Acad., Ser. A, Math. Sci., Physica D, 4, 343, (1982) · Zbl 0571.35100
[34] Dubrovin, B. A.; Natanzon, S. M., Real theta-function solutions of the Kadomtsev-Petviashvili equation, Math. USSR, Izv., 32, 2, (1989)
[35] Mironov, A.; Morozov, A.; Morozov, An., (Rebhan, A.; Katzarkov, L.; Knapp, J.; Rashkov, R.; Scheidegger, E., Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, (2013), World Scientific Publishing Co.Pte.Ltd.), 101-118
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