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TMF, 2013, Volume 177, Number 1, Pages 3–67 (Mi tmf8551)  

This article is cited in 15 scientific papers (total in 15 papers)

Modifications of bundles, elliptic integrable systems, and related problems

A. V. Zotovabc, A. V. Smirnovad

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

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 $R$-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


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

Full text: PDF file (1047 kB)
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English version:
Theoretical and Mathematical Physics, 2013, 177:1, 1281–1338

Bibliographic databases:

Document Type: Article
Received: 20.05.2013

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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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Classification of isomonodromy problems on elliptic curves”, Russian Math. Surveys, 69:1 (2014), 35–118  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. Levin, M. Olshanetsky, A. Zotov, “Planck constant as spectral parameter in integrable systems and KZB equations”, J. High Energy Phys., 2014, no. 10, 109  crossref  mathscinet  zmath  isi  scopus
    3. G. Aminov, S. Arthamonov, A. Smirnov, A. Zotov, “Rational top and its classical $r$-matrix”, J. Phys. A-Math. Theor., 47:30 (2014), 305207  crossref  mathscinet  zmath  isi  scopus
    4. A. Levin, M. Olshanetsky, A. Zotov, “Relativistic classical integrable tops and quantum $r$-matrices”, J. High Energy Phys., 2014, no. 7, 012  crossref  isi  scopus
    5. A. Levin, M. Olshanetsky, A. Zotov, “Classical integrable systems and soliton equations related to eleven-vertex $r$-matrix”, Nucl. Phys. B, 887 (2014), 400–422  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. JETP Letters, 101:9 (2015), 648–655  mathnet  crossref  crossref  isi  elib
    7. H. Rosengren, “Special polynomials related to the supersymmetric eight-vertex model: a summary”, Commun. Math. Phys., 340:3 (2015), 1143–1170  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. 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  mathnet  crossref  mathscinet  isi  elib
    10. I. Sechin, A. Zotov, “Associative Yang–Baxter equation for quantum (semi-)dynamical $r$-matrices”, J. Math. Phys., 57:5 (2016), 053505  crossref  mathscinet  zmath  isi  elib  scopus
    11. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107  mathnet  crossref  isi  elib
    12. A. Zotov, “Relativistic elliptic matrix tops and finite Fourier transformations”, Mod. Phys. Lett. A, 32:32 (2017), 1750169  crossref  mathscinet  zmath  isi  scopus
    13. A. Grekov, A. Zotov, “On $r$-matrix valued Lax pairs for Calogero–Moser models”, J. Phys. A-Math. Theor., 51:31 (2018), 315202  crossref  isi  scopus
    14. A. V. Zotov, “Calogero–Moser model and $R$-matrix identities”, Theoret. and Math. Phys., 197:3 (2018), 1755–1770  mathnet  crossref  crossref  adsnasa  isi  elib
    15. 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  mathnet  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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