A primer on resurgent transseries and their asymptotics

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 largeorder 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.

Renormalons in integrable field theories

“…where n 0 is a non-negative integer. By a standard argument [5][6][7], the exponentially small correction (3.7) due to the condensate of O gives the following contribution to the large order behavior of the coefficients a n appearing in (3.2):…”

“…The relation between instantons and large-order behavior has led to many beautiful results and it has evolved into the theory of resurgence, which provides a universal structure linking perturbative and nonperturbative sectors in quantum theories (see e.g. [4,5] for a presentation of instanton-induced large order behavior, and [6,7] for reviews of the theory of resurgence).…”

Quasinormal modes of black holes and Borel summation

“…This is why the Borel summation is important both in mathematics and in physics. See [31,32] in more detail.…”

“…This method is important in physics because we can probe nonperturbative corrections from perturbative series. We refer the reader to an educational lecture note [31] and to a comprehensive review [32] on this topic.…”

Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $$ \mathcal{N} $$ = 2 field theories

“…In recent years Ecalle's resurgence theory [1] has been applied to an ever growing set of problems where the common denominator is the asymptotic nature of the perturbative expansion and how we can exploit its lack of convergence to reconstruct non-perturbative results out of perturbative data. For some recent introductions see [2,3] and references therein.…”