Resurgent Analysis of the Witten Laplacian  in One Dimension—II

Getmanenko, Alexander

doi:10.1619/fesi.56.121

Cited by 7 publications

(6 citation statements)

References 27 publications

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“…A general discussion of all-orders cancellations in real trans-series is elegantly described in [79], also see other examples in [80,81]. Some aspects of resurgence in supersymmetric quantum mechanics from a complex WKB point of view is also discussed in [82,83]. The contribution of the complex bions to the vacuum energy for the tilted-DW system is…”

Section: Complex Saddles and Resurgencementioning

confidence: 99%

Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence

Behtash

Dunne

Schäfer

et al. 2017

Annals of Mathematical Sciences and Applications

123

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We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on R 3 × S 1 , we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

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Section: Complex Saddles and Resurgencementioning

confidence: 99%

Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence

Behtash

Dunne

Schäfer

et al. 2017

Annals of Mathematical Sciences and Applications

123

View full text Add to dashboard Cite

show abstract

“…The resurgence theory [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58], which was originally discussed in the study of ordinary differential equations, has been investigated in various contexts including matrix models and supersymmetric gauge theories . From the viewpoint of the resurgence theory, it has been conjectured in four-dimensional (4D) QCD(adj.)…”

Section: Introductionmentioning

confidence: 99%

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

Fujimori

Kamata

Misumi

et al. 2019

J. High Energ. Phys.

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We derive the semiclassical contributions from the real and complex bions in the twodimensional CP N −1 sigma model on R × S 1 with a twisted boundary condition. The bion configurations are saddle points of the complexified Euclidean action, which can be viewed as bound states of a pair of fractional instantons with opposite topological charges. We first derive the bion solutions by solving the equation of motion in the model with a potential which simulates an interaction induced by fermions in the CP N −1 quantum mechanics. The bion solutions have quasi-moduli parameters corresponding to the relative distance and phase between the constituent fractional instantons. By summing over the Kaluza-Klein modes of the quantum fluctuations around the bion backgrounds, we find that the effective action for the quasi-moduli parameters is renormalized and becomes a function of the dynamical scale (or the renormalized coupling constant). Based on the renormalized effective action, we obtain the semiclassical bion contribution in a weak coupling limit by making use of the Lefschetz thimble method. We find that the non-perturbative contribution vanishes in the supersymmetric case and it has an imaginary ambiguity which is consistent with the expected infrared renormalon ambiguity in non-supersymmetric cases. This is the first explicit result indicating the relation between the complex bion and the infrared renormalon. * Electronic address: toshiaki.fujimori018(at)

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“…In the exact-WKB analysis [129,130,[132][133][134][135][136][137][138][139][140][141][142][143][144][145][146], which has been studied mainly by mathematicians, one investigates the properties of solutions to certain differential equations using the Borel summation. 3 Our main interest is in its application to the Schrödinger equation,…”

Section: Jhep12(2020)114mentioning

confidence: 99%

“…In this section, we review the exact WKB method [50,129,130,[132][133][134][135][136][137][138][139][140][141][142][143][144] and the related techniques, including Borel resummation, Stokes curves and monodromy matrices. 9 For simplicity, we focus on the one-dimensional Schrödinger equation, and assume that the potential V (x) doesn't include , i.e.…”

Section: Exact Wkbmentioning

confidence: 99%

On exact-WKB analysis, resurgent structure, and quantization conditions

Sueishi

Kamata

Misumi

et al. 2020

J. High Energ. Phys.

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There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

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Resurgent Analysis of the Witten Laplacian in One Dimension—II

Cited by 7 publications

References 27 publications

Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence

Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

On exact-WKB analysis, resurgent structure, and quantization conditions

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