A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, TMF, 177:1 (2013), 3–67; Theoret. and Math. Phys., 177:1 (2013), 1281

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Teoreticheskaya i Matematicheskaya Fizika, 2013, Volume 177, Number 1, Pages 3–67
DOI: https://doi.org/10.4213/tmf8551 (Mi tmf8551)

This article is cited in 26 scientific papers (total in 26 papers)

Modifications of bundles, elliptic integrable systems, and related problems

A. V. Zotov^abc, A. V. Smirnov^ad

^a Institute for Theoretical and Experimental Physics, Moscow, Russia
^b Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, Russia
^c Steklov Mathematical Institute, RAS, Moscow, Russia
^d Department of Mathematics, Columbia University, New York, USA

Full-text PDF (1047 kB) Citations (26)

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DOI: https://doi.org/10.4213/tmf8551

Abstract: We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik–Zamolodchikov–Bernard equations, classical and quantum

R

$R$ -matrices, monopoles, spectral duality, Painlevé equations, and the classical–quantum correspondence. For an

$SL(N,\mathbb C)$ -bundle on an elliptic curve with nontrivial characteristic classes, we obtain equations of isomonodromy deformations.

Keywords: integrable system, Painlevé equation, Hitchin system, modification of bundles.

Funding agency	Grant number
Russian Foundation for Basic Research	12-01-00482 12-02-00594 12-01-33071_мол_а_вед
Dynasty Foundation

Received: 20.05.2013

English version:
Theoretical and Mathematical Physics, 2013, Volume 177, Issue 1, Pages 1281–1338
DOI: https://doi.org/10.1007/s11232-013-0106-1

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, TMF, 177:1 (2013), 3–67; Theoret. and Math. Phys., 177:1 (2013), 1281–1338

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This publication is cited in the following 26 articles:

K. R. Atalikov, A. V. Zotov, “Gauge equivalence of $1+1$ Calogero–Moser–Sutherland field theory and a higher-rank trigonometric Landau–Lifshitz model”, Theoret. and Math. Phys., 219:3 (2024), 1004–1017
V. A. Pavlenko, “Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of $H^{3+2}$ Hamiltonian systems”, Theoret. and Math. Phys., 212:3 (2022), 1181–1192
E. Trunina, A. Zotov, “Lax equations for relativistic $\mathrm{G}\mathrm{L}(NM,\mathbb{C})$ Gaudin models on elliptic curve”, J. Phys. A, 55:39 (2022), 395202–31
I. A. Sechin, A. V. Zotov, “Quadratic algebras based on $SL(NM)$ elliptic quantum $R$-matrices”, Theoret. and Math. Phys., 208:2 (2021), 1156–1164
E. S. Trunina, A. V. Zotov, “Multi-pole extension of the elliptic models of interacting integrable tops”, Theoret. and Math. Phys., 209:1 (2021), 1331–1356
Atalikov K. Zotov A., “Field Theory Generalizations of Two-Body Calogero-Moser Models in the Form of Landau-Lifshitz Equations”, J. Geom. Phys., 164 (2021), 104161
B. I. Suleimanov, “Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom”, St. Petersburg Math. J., 33:6 (2022), 995–1009
I. A. Sechin, A. V. Zotov, “Integrable system of generalized relativistic interacting tops”, Theoret. and Math. Phys., 205:1 (2020), 1291–1302
Vasilyev M. Zotov A., “On Factorized Lax Pairs For Classical Many-Body Integrable Systems”, Rev. Math. Phys., 31:6 (2019), 1930002
A. V. Zotov, “Relativistic interacting integrable elliptic tops”, Theoret. and Math. Phys., 201:2 (2019), 1565–1580
Grekov A. Sechin I. Zotov A., “Generalized Model of Interacting Integrable Tops”, J. High Energy Phys., 2019, no. 10, 081
A. Grekov, A. Zotov, “On $R$-matrix valued Lax pairs for Calogero–Moser models”, J. Phys. A-Math. Theor., 51:31 (2018), 315202
A. V. Zotov, “Calogero–Moser model and $R$-matrix identities”, Theoret. and Math. Phys., 197:3 (2018), 1755–1770
V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102
V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107
A. Zotov, “Relativistic elliptic matrix tops and finite Fourier transformations”, Mod. Phys. Lett. A, 32:32 (2017), 1750169
D. P. Novikov, B. I. Suleimanov, ““Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom”, Theoret. and Math. Phys., 187:1 (2016), 479–496
B. I. Suleimanov, “Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential”, Ufa Math. J., 8:3 (2016), 136–154
I. Sechin, A. Zotov, “Associative Yang–Baxter equation for quantum (semi-)dynamical $r$-matrices”, J. Math. Phys., 57:5 (2016), 053505
JETP Letters, 101:9 (2015), 648–655

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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