Exact quantization conditions for the relativistic Toda lattice

Hatsuda, Yasuyuki; Mariño, Marcos

doi:10.1007/jhep05(2016)133

Cited by 62 publications

(186 citation statements)

References 102 publications

(265 reference statements)

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“…Based upon earlier constructions [19][20][21][22][23][24][25][26][27][28][29][30], it was proposed in [31] that there is a precise correspondence between the spectral theory of operators obtained by quantizing the mirror curve, and topological string theory on the toric Calabi-Yau geometry. This proposal has since led to many recent developments, e.g., [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] (see [52] for an introduction). In addition, the topological string/spectral theory correspondence of [31] provides a nonperturbative definition of the topological-string partition function on toric Calabi-Yau geometries.…”

Section: Introductionmentioning

confidence: 99%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

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Abstract:The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi-Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local P 2 toric Calabi-Yau threefold, the present work shows how the Borel-Padé-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure.

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Section: Introductionmentioning

confidence: 99%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

Self Cite

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“…Remarkably, the grand potential (2.6) automatically implements the correct scheme to make contact with the NO partition function and the Painlevé III 3 equation. Similarly the analysis of [27] shows that (2.6) also implements the correct scheme in the context of SU (2) Toda. In (3.6) we denote by P(t(µ, ), m, ) the polynomial part of the grand potential, which reads…”

Section: The Topological String Computationmentioning

confidence: 73%

“…In the scaling limit (3.4) the conjecture of [24] makes contact with the SU (2) quantum Toda as shown in [27]. However to make contact with Painlevé III 3 equation we have to consider the scaling (3.2).…”

Section: The Four-dimensional Limitmentioning

confidence: 99%

“…Therefore we are studying precisely the opposite regime with respect to the one considered in the standard geometric engineering limit (3.4) and in the context of Toda systems [4,27]. To compute the spectral determinant (2.5) we have to perform the shift…”

Section: The Topological String Computationmentioning

confidence: 99%

“…More precisely, different phases of the four-dimensional theory in the gravitational background parameters can be obtained by different scaling limits of the refined topological string. In [27] the authors specified the conjecture of [24] in the four-dimensional limit corresponding to the Nekrasov-Shatashvili (NS) regime ( 2 = 0) and the resulting spectral problem was found to coincide with the quantum Toda system found in [4].…”

Section: Introductionmentioning

confidence: 99%

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The short pulse equation by a Riemann–Hilbert approach

2017

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We explore a new connection between Seiberg-Witten theory and quantum statistical systems by relating the dual partition function of SU (2) Super Yang-Mills theory in a self-dual Ω-background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé III 3 τ function. In addition we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local P 1 × P 1 geometry.

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Exact quantization conditions, toric Calabi-Yau and non-perturbative topological string

Sun

Wang

Huang

2017

J. High Energ. Phys.

103

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We establish the precise relation between the Nekrasov-Shatashvili (NS) quantization scheme and Grassi-Hatsuda-Mariño conjecture for the mirror curve of arbitrary toric Calabi-Yau threefold. For a mirror curve of genus g, the NS quantization scheme leads to g quantization conditions for the corresponding integrable system. The exact NS quantization conditions enjoy a self S-duality with respect to Planck constanth and can be derived from the Lockhart-Vafa partition function of non-perturbative topological string. Based on a recent observation on the correspondence between spectral theory and topological string, another quantization scheme was proposed by Grassi-Hatsuda-Mariño, in which there is a single quantization condition and the spectra are encoded in the vanishing of a quantum Riemann theta function. We demonstrate that there actually exist at least g nonequivalent quantum Riemann theta functions and the intersections of their theta divisors coincide with the spectra determined by the exact NS quantization conditions. This highly nontrivial coincidence between the two quantization schemes requires infinite constraints among the refined Gopakumar-Vafa invariants. The equivalence for mirror curves of genus one has been verified for some local del Pezzo surfaces. In this paper, we generalize the correspondence to higher genus, and analyze in detail the resolved C 3 /Z 5 orbifold and several SU (N ) geometries. We also give a proof for some models ath = 2π/k. arXiv:1606.07330v2 [hep-th]

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Exact quantization conditions for the relativistic Toda lattice

Cited by 62 publications

References 102 publications

Resurgence matches quantization

Resurgence matches quantization

The short pulse equation by a Riemann–Hilbert approach

Exact quantization conditions, toric Calabi-Yau and non-perturbative topological string

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