Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 081, 25 pages      arXiv:2103.09681

Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

Fatane Mobasheramini a and Marco Bertola ab
a) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montreal, QC H3G 1M8, Canada
b) SISSA, Area of Mathematics, via Bonomea 265, Trieste, Italy

Received March 19, 2021, in final form August 31, 2021; Published online September 07, 2021

We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.

Key words: quantization of Painlevé; Calogero-Painlevé; Harish-Chandra isomorphism.

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  1. Bergère M., Eynard B., Marchal O., Prats-Ferrer A., Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model, J. High Energy Phys. 2012 (2012), no. 3, 098, 78 pages, arXiv:1106.0332.
  2. Bertola M., Cafasso M., Rubtsov V., Noncommutative Painlevé equations and systems of Calogero type, Comm. Math. Phys. 363 (2018), 503-530, arXiv:1710.00736.
  3. Borot G., Eynard B., Majumdar S.N., Nadal C., Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. Theory Exp. 2011 (2011), P11024, 56 pages, arXiv:1009.1945.
  4. Brezin E., Kazakov V., Exactly solvable field theories of closed strings, in The Large $N$ Expansion in Quantum Field Theory and Statistical Physics: from Spin Systems to 2-Dimensional Gravity, Editors R. Brezin, S.R. Wadia, World Scientific, Singapore, 1993, 711-717.
  5. Douglas M.R., Shenker S.H., Strings in less than one dimension, Nuclear Phys. B 335 (1990), 635-654.
  6. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups, Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, arXiv:hep-th/9407018.
  7. Dubrovin B., Mazzocco M., Canonical structure and symmetries of the Schlesinger equations, Comm. Math. Phys. 271 (2007), 289-373, arXiv:math.DG/0311261.
  8. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, arXiv:math.AG/0011114.
  9. Eynard B., Garcia-Failde E., From topological recursion to wave functions and PDEs quantizing hyperelliptic curves, arXiv:1911.07795.
  10. Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65-116.
  11. Forrester P.J., Witte N.S., Random matrix theory and the sixth Painlevé equation, J. Phys. A: Math. Gen. 39 (2006), 12211-12233.
  12. Fuchs R., Sur quelques équations différentielles linéaires du second ordre, Gauthier-Villars, Paris, 1905.
  13. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1-55.
  14. Gross D.J., Migdal A.A., Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127-130.
  15. Inozemtsev V.I., Lax representation with spectral parameter on a torus for integrable particle systems, Lett. Math. Phys. 17 (1989), 11-17.
  16. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  17. Jimbo M., Nagoya H., Sun J., Remarks on the confluent KZ equation for $\mathfrak{sl}_2$ and quantum Painlevé equations, J. Phys. A: Math. Theor. 41 (2008), 175205, 14 pages.
  18. Kazhdan D., Kostant B., Sternberg S., Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-507.
  19. Lee S.-Y., Teodorescu R., Wiegmann P., Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé I transcendent, Phys. D 240 (2011), 1080-1091, arXiv:1005.0369.
  20. Malmquist J., Sur les équations différentielles du second ordre, dont l'intégrale générale a ses points critiques fixes, Almqvist & Wiksells, 1922.
  21. Mehta M.L., Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), Vol. 142, Elsevier/Academic Press, Amsterdam, 2004.
  22. Nagoya H., Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations, J. Math. Phys. 52 (2011), 083509, 16 pages, arXiv:1109.1645.
  23. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{{\rm VI}}$, Ann. Mat. Pura Appl. (4) 146 (1987), 337-381.
  24. Painlevé P., Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France 28 (1900), 201-261.
  25. Takasaki K., Painlevé-Calogero correspondence revisited, J. Math. Phys. 42 (2001), 1443-1473, arXiv:math.QA/0004118.
  26. Tracy C.A., Widom H., Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), 33-72, arXiv:hep-th/9306042.
  27. Zabrodin A., Zotov A., Quantum Painlevé-Calogero correspondence, J. Math. Phys. 53 (2012), 073507, 19 pages, arXiv:1107.5672.

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