Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

Abstract

Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about nonperturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NP-saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the SU(N ) principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel 'fracton' saddle points, which turn out to be the fractionalized constituents of previously observed unstable 'uniton' saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of 't Hooft's renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

“…7 The same result is obtained in a continuum formulation of the gauge theory [33,34]. 8 Methods of trans-series and resurgence have been recently investigated in matrix models [37][38][39][40] and in quantum field theory [41][42][43][44].…”

Neutral bions in the ℂ $$ \mathbb{C} $$ P N −1 model

“…In order to reach deeper understanding on bions and the associated physics, it is of great importance to study examples in the low-dimensional models such as CP N −1 models [12][13][14], principal chiral models [16,19] and quantum mechanics [15,17,18]. In particular, the CP N −1 model in 1+1 dimensions has been studied for a long time as a toy model of the Yang-Mills theory in 3+1 dimensions [48], because of similarities between them such as dynamical mass gap, asymptotic freedom and the existence of instantons [49].…”

“…It is stressed byÜnsal and his collaborators that these configurations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], which are termed "bions", have two physical significances associated with two types of topologically trivial bion configurations called "magnetic (charged) bions" and "neutral bions", as seen in the following examples: in the weak-coupling regime (L ≪ 1/Λ QCD ) in QCD(adj.) on R 3 × S 1 , or in the U(1) N −1 center-symmetric phase [20][21][22][23][24][25][26][27][28][29], condensation of magnetic bions (zero topological charge and nonzero magnetic charge) causes the confinement [3][4][5][6][7][8][9].…”

“…This argument is also of importance in terms of the recent progress in large-N volume reduction [30][31][32][33][34][35][36][37][38][39][40]. On the other side, neutral bions (zero topological charge and zero magnetic charge) can be identified as the infrared renormalon [10][11][12][13][14][15][16][17][18][19][41][42][43]. Here imaginary ambiguities arising in bion's amplitude and those arising in non-Borelsummable perturbative series cancel against each other, and it is expected that full semiclassical expansion including perturbative and non-perturbative sectors, which is called "resurgent" expansion [44], leads to unambiguous and self-consistent definition of field theories in the same manner as the Bogomolny-Zinn-Justin (BZJ) prescription in quantum mechanics [45][46][47].…”

Fast summation of divergent series and resurgent transseries from Meijer- G approximants

“…Conventional resummation is, however, not sufficient in the presence of the so-called Stokes phenomenon where different asymptotic expansions hold in different regions of the plane made up of complexified expansion parameter values [17][18][19][20][21][22][23]. Thus the Stokes phenomenon requires generally distinct resummations in each of these regions.…”