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A primer on resurgent transseries and their asymptotics

A primer on resurgent transseries and their asymptotics
A primer on resurgent transseries and their asymptotics
The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries---usually described mathematically via alien calculus---are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.
hep-th, hep-lat, hep-ph, math-ph, math.MP, quant-ph
0370-1573
Aniceto, Inês
0061ca0c-1ad8-4510-9b12-008e5c27a7ea
Başar, Gökçe
a2e893e6-43f4-4cdf-89a0-cb47ae27f4d1
Schiappa, Ricardo
7b7a5b7d-7ecc-454e-ba14-51c2adc24c5f
Aniceto, Inês
0061ca0c-1ad8-4510-9b12-008e5c27a7ea
Başar, Gökçe
a2e893e6-43f4-4cdf-89a0-cb47ae27f4d1
Schiappa, Ricardo
7b7a5b7d-7ecc-454e-ba14-51c2adc24c5f

Aniceto, Inês, Başar, Gökçe and Schiappa, Ricardo (2019) A primer on resurgent transseries and their asymptotics. Physics Reports. (doi:10.1016/j.physrep.2019.02.003).

Record type: Review

Abstract

The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries---usually described mathematically via alien calculus---are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.

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a primer - Accepted Manuscript
Restricted to Repository staff only until 22 February 2021.
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More information

Accepted/In Press date: 2 February 2019
e-pub ahead of print date: 22 February 2019
Additional Information: 192 pages, 76 plots in 41 figures, jheppub-nosort.sty Prepared for submission to JHEP
Keywords: hep-th, hep-lat, hep-ph, math-ph, math.MP, quant-ph

Identifiers

Local EPrints ID: 424138
URI: https://eprints.soton.ac.uk/id/eprint/424138
ISSN: 0370-1573
PURE UUID: 31e320f0-cc22-4f34-85f1-3847e2e92b84

Catalogue record

Date deposited: 05 Oct 2018 11:30
Last modified: 13 Mar 2019 17:59

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