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A note on the eigenfunction in a relativistic closed Toda chain

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2018.05.001
  • Received Date: 01 March 2018
  • Accepted Date: 19 April 2018
  • Rev Recd Date: 19 April 2018
  • Publish Date: 31 May 2018
    Abstract

  • Abstract

    Recently, the eigenfunction for an N-particle relativistic closed Toda chain has been calculated by association with gauge theory. The results involve only the behavior of the eigenfunction of the energy level but do not include nonphysical energy. A conjecture was thus proposed to determine the eigenfunction of the nonphysical energy and a numerical calculation was conducted to prove that it is impossible to get an entire eigenfunction of the nonphysical energy. Because of the pole cancellation condition, the relation between the open and closed Bogomol’nyi-Prasad-Sommerfield (BPS) invariants could be derived.

    Abstract

    Recently, the eigenfunction for an N-particle relativistic closed Toda chain has been calculated by association with gauge theory. The results involve only the behavior of the eigenfunction of the energy level but do not include nonphysical energy. A conjecture was thus proposed to determine the eigenfunction of the nonphysical energy and a numerical calculation was conducted to prove that it is impossible to get an entire eigenfunction of the nonphysical energy. Because of the pole cancellation condition, the relation between the open and closed Bogomol’nyi-Prasad-Sommerfield (BPS) invariants could be derived.
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    • References(15)
  • [1]
    KHARCHEV S, LEBEDEV D. Integral representation for the eigenfunctions of a quantum periodic Toda chain[J]. Letters in Mathematical Physics, 1999, 50(1): 53-77.
    [2]
    FRANCO S, HATSUDA Y, MARIO M. Exact quantization conditions for cluster integrable systems[J]. Journal of Statistical Mechanics: Theory and Experiment, 2016, 2016(6): 063107.
    [3]
    HATSUDA Y, MARINO M. Exact quantization conditions for the relativistic Toda lattice[J]. Journal of High Energy Physics, 2016, 2016(5): 133.
    [4]
    SCIARAPPA A. Exact relativistic Toda chain eigenfunctions from separation of variables and gauge theory[J]. Journal of High Energy Physics, 2017, 2017(10): 116.
    [5]
    KASHANI-POOR A K. Quantization condition from exact WKB for difference equations[J]. Journal of High Energy Physics, 2016, 2016(6): 180.
    [6]
    MARINO M, ZAKANY S. Exact eigenfunctions and the open topological string[J]. Journal of Physics A: Mathematical and Theoretical, 2017, 50(32): 325401.
    [7]
    MARINO M, ZAKANY S. Wavefunctions, integrability, and open strings[DB/OL]. arXiv.org: arXiv 1706.07402, 2017.
    [8]
    FADDEEV L D. Modular double of the quantum group SLq (2, R)[C]// Lie Theory and Its Applications in Physics. Berlin: Springer, 2014: 21-31.
    [9]
    GRASSiA, HATSUDA Y, MARINO M. Topological strings from quantum mechanics[DB/OL]. arXiv.org: arXiv 1410.3382, 2014.
    [10]
    WANG Xin, ZHANG Guojun, HUANG Minxin. New exact quantization condition for toric Calabi-Yau geometries[J]. Physical Review Letters, 2015, 115(12): 121601.
    [11]
    KHARCHEV S, LEBEDEV D, SEMENOV-TIAN-SHANSKY M. Unitary representations of Uq(sl(2; R)), the modular double, and the multiparticle q-deformed Toda chains[J]. Communications in Mathematical Physics, 2002, 225(3): 573-609.
    [12]
    BULLIMORE M, KIM H C. The superconformal index of the (2, 0) theory with defects[J]. Journal of High Energy Physics, 2015, 2015(5): 48.
    [13]
    FADDEEV L D, KASHAEV R M. Quantum dilogarithm[C]// Fifty Years of Mathematical Physics: Selected Works of Ludwig Faddeev. Singapore: World Scientific, 2016: 502-509.
    [14]
    BARNES E W. The theory of the double gamma function[J]. Philosophical Transactions of the Royal Society of London, Series A, 1901, 196: 265-387.
    [15]
    BARNES E W. On the theory of multiple gamma function[J].Trans Cambridge Phil Soc, 1904, 19: 374-425.
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    [1]
    KHARCHEV S, LEBEDEV D. Integral representation for the eigenfunctions of a quantum periodic Toda chain[J]. Letters in Mathematical Physics, 1999, 50(1): 53-77.
    [2]
    FRANCO S, HATSUDA Y, MARIO M. Exact quantization conditions for cluster integrable systems[J]. Journal of Statistical Mechanics: Theory and Experiment, 2016, 2016(6): 063107.
    [3]
    HATSUDA Y, MARINO M. Exact quantization conditions for the relativistic Toda lattice[J]. Journal of High Energy Physics, 2016, 2016(5): 133.
    [4]
    SCIARAPPA A. Exact relativistic Toda chain eigenfunctions from separation of variables and gauge theory[J]. Journal of High Energy Physics, 2017, 2017(10): 116.
    [5]
    KASHANI-POOR A K. Quantization condition from exact WKB for difference equations[J]. Journal of High Energy Physics, 2016, 2016(6): 180.
    [6]
    MARINO M, ZAKANY S. Exact eigenfunctions and the open topological string[J]. Journal of Physics A: Mathematical and Theoretical, 2017, 50(32): 325401.
    [7]
    MARINO M, ZAKANY S. Wavefunctions, integrability, and open strings[DB/OL]. arXiv.org: arXiv 1706.07402, 2017.
    [8]
    FADDEEV L D. Modular double of the quantum group SLq (2, R)[C]// Lie Theory and Its Applications in Physics. Berlin: Springer, 2014: 21-31.
    [9]
    GRASSiA, HATSUDA Y, MARINO M. Topological strings from quantum mechanics[DB/OL]. arXiv.org: arXiv 1410.3382, 2014.
    [10]
    WANG Xin, ZHANG Guojun, HUANG Minxin. New exact quantization condition for toric Calabi-Yau geometries[J]. Physical Review Letters, 2015, 115(12): 121601.
    [11]
    KHARCHEV S, LEBEDEV D, SEMENOV-TIAN-SHANSKY M. Unitary representations of Uq(sl(2; R)), the modular double, and the multiparticle q-deformed Toda chains[J]. Communications in Mathematical Physics, 2002, 225(3): 573-609.
    [12]
    BULLIMORE M, KIM H C. The superconformal index of the (2, 0) theory with defects[J]. Journal of High Energy Physics, 2015, 2015(5): 48.
    [13]
    FADDEEV L D, KASHAEV R M. Quantum dilogarithm[C]// Fifty Years of Mathematical Physics: Selected Works of Ludwig Faddeev. Singapore: World Scientific, 2016: 502-509.
    [14]
    BARNES E W. The theory of the double gamma function[J]. Philosophical Transactions of the Royal Society of London, Series A, 1901, 196: 265-387.
    [15]
    BARNES E W. On the theory of multiple gamma function[J].Trans Cambridge Phil Soc, 1904, 19: 374-425.
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