Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

Hatsuda, Yasuyuki; Sciarappa, Antonio; Zakany, Szabolcs

doi:10.1007/jhep11(2018)118

Cited by 8 publications

(12 citation statements)

References 85 publications

(261 reference statements)

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“…The results presented in this Appendix are not new and can be found in various textbook, as well as in [75,Sec. 2] where they also discuss them in relation to gauge theory.…”

Section: B Perturbative Study Of Quantum Mechanical Potentialsmentioning

confidence: 92%

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Quantum spectral problems and isomonodromic deformations

Bershtein,

Gavrylenko,

Grassi

2021

Preprint

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We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2 × 2 linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé III 3 equation. Then we extend the analysis to the 2particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (ǫ 1 + ǫ 2 = 0). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations (ǫ 2 = 0).In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the ǫ 2 → 0 limit of the blowup relations from the regularized action functional and CFT arguments.

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“…The results presented in this Appendix are not new and can be found in various textbook, as well as in [75,Sec. 2] where they also discuss them in relation to gauge theory.…”

Section: B Perturbative Study Of Quantum Mechanical Potentialsmentioning

confidence: 92%

“…We underline the stable digits. The numerical result reported in the last line is performed as Appendix B.1 , see also [74,75].…”

Section: Case #mentioning

confidence: 99%

Quantum spectral problems and isomonodromic deformations

Bershtein,

Gavrylenko,

Grassi

2021

Preprint

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“…[9][10][11][12][13][14][15][16]. Some recent papers which discuss quantum periods and their relation to four dimensional N = 2 theories are [8,[17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Quantum Periodsmentioning

confidence: 99%

“…(7.10) 25 Since ϑi = 0 passes through these singularities, each of them only contributes 1 2 µ, γ Ω(γ) log(1 − σcan(γ)X ϑ γ ). 26 As we will discuss later, we know from other methods that the red dot on the positive real axis represent the central charges for 2 distinct hypermultiplets: γ [0,1,−1] and γ [0, 1,1] .…”

Section: Instanton Counting Versus Borel Summationmentioning

confidence: 99%

Exact WKB methods in SU(2) Nf = 1

Grassi

Hao

Neitzke

2022

J. High Energ. Phys.

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We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) $$ \mathcal{N} $$ N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

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“…Relativistic generalization of Toda class integrable models are related to 5-dimensional N=1 supersymmetric gauge theory compactificated on a circle [54], the partition function of the Omega background deformed gauge theory is computed by K-theoretic localization formula [36,48]. Some recent discussions of this connection are [55][56][57][58]. It would be helpful to study the relativistic-DTV potential from the perspective of gauge theory, a version of relativistic Heun/van Diejen operator indeed shows up in supersymmetric field theory [59].…”

Section: Jhep08(2020)070mentioning

confidence: 99%

Spectra of elliptic potentials and supersymmetric gauge theories

2020

J. High Energ. Phys.

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We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to the strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.

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Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

Cited by 8 publications

References 85 publications

Quantum spectral problems and isomonodromic deformations

Quantum spectral problems and isomonodromic deformations

Exact WKB methods in SU(2) Nf = 1

Spectra of elliptic potentials and supersymmetric gauge theories

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