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

Sun, Kaiwen; Wang, Xin; Huang, Min–xin

doi:10.1007/jhep01(2017)061

Cited by 40 publications

(108 citation statements)

References 162 publications

(339 reference statements)

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“…This conjecture has been verified in many examples. In particular, in the higher genus case, it has been checked in [13,21,22,99]. As we will now show, our quantization conditions (4.4), (4.3) are limiting cases of this conjecture for a particular family of toric CY manifolds.…”

Section: The Ts/st Correspondencementioning

confidence: 65%

A Solvable Deformation of Quantum Mechanics

Grassi¹,

Mariño

2019

SIGMA

100

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The conventional Hamiltonian H = p 2 + V N (x), where the potential V N (x) is a polynomial of degree N , has been studied intensively since the birth of quantum mechanics. In some cases, its spectrum can be determined by combining the WKB method with resummation techniques. In this paper we point out that the deformed Hamiltonian H = 2 cosh(p) + V N (x) is exactly solvable for any potential: a conjectural exact quantization condition, involving well-defined functions, can be written down in closed form, and determines the spectrum of bound states and resonances. In particular, no resummation techniques are needed. This Hamiltonian is obtained by quantizing the Seiberg-Witten curve of N = 2 Yang-Mills theory, and the exact quantization condition follows from the correspondence between spectral theory and topological strings, after taking a suitable fourdimensional limit. In this formulation, conventional quantum mechanics emerges in a scaling limit near the Argyres-Douglas superconformal point in moduli space. Although our deformed version of quantum mechanics is in many respects similar to the conventional version, it also displays new phenomena, like spontaneous parity symmetry breaking.where V N (x) is a polynomial of degree N . When N = 2 (the harmonic oscillator), the spectral problem was solved in the 1920s, but for potentials of higher degree there is no obvious analytic solution. On the contrary, the Hamiltonian (1.1) with N ≥ 3 has become the textbook testing ground for approximation techniques, like stationary perturbation theory or the WKB method. Starting in the 1970s, it was realized that these approximation methods can be sometimes upgraded by using resummation techniques, providing in this way a close analogue of exact solutions. In particular, exact or uniform versions of the WKB method lead to quantization conditions for anharmonic oscillators [3,4,10,25,61,96,102,103,109,110,111]. These conditions require Borel-Écalle resummations of the WKB periods and include non-perturbative corrections to the perturbative quantization condition of Dunham [33], which in turn is a generalization to all orders in of the Bohr-Sommerfeld quantization condition.1 There are some peculiar quantum mechanical models with periodic potentials where spontaneous parity symmetry breaking is possible, see for instance [107]. However this is a very different setup from the one considered in this paper. 2 We would like to warn the reader that in this work "instantons" refers to both quantum mechanical instantons and gauge theory instantons. We hope this will not lead to confusion.

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Section: The Ts/st Correspondencementioning

confidence: 65%

A Solvable Deformation of Quantum Mechanics

Grassi¹,

Mariño

2019

SIGMA

100

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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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“…Interestingly enough, in the analysis [25,38,39,43,44] an integrable structure [45,46] originating from the polymer matrix model [47] was utilized. This relation is generalized to many other geometries [48][49][50][51][52][53][54][55][56] and many other superconformal Chern-Simons matrix models [57][58][59][60][61]. Besides, interesting relations such as the Wronskian relation [62] (with the chiral projections interpreted as the orientifold [63][64][65]) or the q-Painlevé equation [66] implying an integrable structure [67], were also proposed.…”

Section: Resultsmentioning

confidence: 97%

ABJM matrix model and 2D Toda lattice hierarchy

Furukawa

Moriyama

2019

J. High Energ. Phys.

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It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.

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

Cited by 40 publications

References 162 publications

A Solvable Deformation of Quantum Mechanics

A Solvable Deformation of Quantum Mechanics

Resurgence matches quantization

ABJM matrix model and 2D Toda lattice hierarchy

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