Multi-instantons and multicuts

Mariño, Marcos; Schiappa, Ricardo; Weiss, Marlene

doi:10.1063/1.3097755

Cited by 85 publications

(126 citation statements)

References 36 publications

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“…For the case of the topological-string free energy, the general construction of its resurgent transseries-out of a nonperturbative extension of the holomorphic anomaly equations-was set-up in [4]; with the explicit example of the local P 2 toric Calabi-Yau threefold being fully worked out in [5]. These results were obtained based upon earlier stringy constructions [6][7][8][9][10][11][12][13][14][15], and have since led to a few further developments, e.g., [16,17]. In principle, this construction allows us to obtain fully nonperturbative results for the string-theoretic free energy, at any value of the string coupling constant.…”

Section: Introductionmentioning

confidence: 99%

“…Further, for the particular values of z which we shall consider, the Borel plane has a reflexive symmetry along the real axis, and there is a conjugation symmetry we should also take into account. 9 We follow the notation in [4,5]: roman letters (F , S zz ) indicate nonholomorphic quantities, curly letters (F, S zz ) indicate their holomorphic limits. The chosen frame is shown as a subscript separated by a semicolon (;).…”

Section: Perturbative and Nonperturbative Sectors Of Local Pmentioning

confidence: 99%

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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%

Section: Perturbative and Nonperturbative Sectors Of Local Pmentioning

confidence: 99%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

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“…Further motivation along these lines comes from the ODE/Integrable Model correspondence [66][67][68][69][70], which provides explicit mappings between monodromy operators in certain Schrödinger systems and Yang-Baxter operators in integrable models. We are also strongly motivated by the geometric relation between supersymmetric gauge theories, matrix models and topological strings [71][72][73][74][75], for which a rich web of resurgent structures has been comprehensively established both analytically and numerically [76][77][78][79][80][81][82][83][84][85][86]. There are surprisingly close parallels between the resurgent structures found in such theories for the partition function (or free energy) as a function of (at least) two parameters, g s and N , and the resurgent structure of the Schrödinger energy eigenvalue u( , N ), as a function of and the perturbative level number N .…”

Section: Jhep05(2017)087mentioning

confidence: 99%

Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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For a wide variety of quantum potentials, including the textbook 'instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and 'special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

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“…The application of the resurgence theory and the framework of the complexified path integral to quantum theories has recently been attracting a great deal of attention [1][2][3][4][5][6][7][8][9][10][11][12]. In the resurgence theory, all the perturbative series around nontrivial backgrounds are taken into account, and it is expected that a full semi-classical expansion in perturbative and non-perturbative sectors, which is called a "resurgent" trans-series, leads to unambiguous and self-consistent definition of quantum theories .…”

Section: Introductionmentioning

confidence: 99%

Nonperturbative contributions from complexified solutions inCPN−1models

et al. 2016

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We discuss the non-perturbative contributions from real and complex saddle point solutions in the CP 1 quantum mechanics with fermionic degrees of freedom, using the Lefschetz thimble formalism beyond the gaussian approximation. We find bion solutions, which correspond to (complexified) instanton-antiinstanton configurations stabilized in the presence of the fermonic degrees of freedom. By computing the one-loop determinants in the bion backgrounds, we obtain the leading order contributions from both the real and complex bion solutions. To incorporate quasi zero modes which become nearly massless in a weak coupling limit, we regard the bion solutions as well-separated instanton-antiinstanton configurations and calculate a complexified quasi moduli integral based on the Lefschetz thimble formalism. The non-perturbative contributions from the real and complex bions are shown to cancel out in the supersymmetric case and give an (expected) ambiguity in the non-supersymmetric case, which plays a vital role in the resurgent trans-series. For nearly supersymmetric situation, evaluation of the Lefschetz thimble gives results in precise agreement with those of the direct evaluation of the Schrödinger equation. We also perform the same analysis for the sine-Gordon quantum mechanics and point out some important differences showing that the sine-Gordon quantum mechanics does not correctly describe the 1d limit of the CP N −1 field theory of R × S

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Multi-instantons and multicuts

Cited by 85 publications

References 36 publications

Resurgence matches quantization

Resurgence matches quantization

Quantum geometry of resurgent perturbative/nonperturbative relations

Nonperturbative contributions from complexified solutions inCPN−1models

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