Resurgence and Lefschetz thimble in three-dimensional $\mathcal{N}=2$ supersymmetric Chern–Simons matter theories

Abstract

We study a certain class of supersymmetric (SUSY) observables in 3d N = 2 SUSY Chern-Simons (CS) matter theories and investigate how their exact results are related to the perturbative series with respect to coupling constants given by inverse CS levels. We show that the observables have nontrivial resurgent structures by expressing the exact results as a full transseries consisting of perturbative and non-perturbative parts. As real mass parameters are varied, we encounter Stokes phenomena at an infinite number of points, where the perturbative series becomes non-Borel-summable due to singularities on the positive real axis of the Borel plane. We also investigate the Stokes phenomena when the phase of the coupling constant is varied. For these cases, we find that the Borel ambiguities in the perturbative sector are canceled by those in nonperturbative sectors and end up with an unambiguous result which agrees with the exact result even on the Stokes lines. We also decompose the Coulomb branch localization formula, which is an integral representation for the exact results, into Lefschetz thimble contributions and study how they are related to the resurgent transseries. We interpret the non-perturbative effects appearing in the transseries as contributions of complexified SUSY solutions which formally satisfy the SUSY conditions but are not on the original path integral contour. * Electronic address: toshiaki.fujimori018(at)

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

“…We will shortly see that Stokes phenomenon and an asymptotic perturbative expansion are present also in our case although both very different in nature from the analysis of [17].…”

“…In the extremely thorough analysis of [17] the authors noted that as the real mass parameter is increased, or equivalently the argument of the coupling g = 1/k is varied, more and more non-perturbative thimbles cross the perturbative one and Stokes phenomenon take place, presented in their Figure 7 and 9. These jumps are directly correlated with the jumps in the resummation of the asymptotic expansion for small g 1.…”

“…In [17] 2 the authors performed the complete resurgent analysis and thimble decomposition for 3-d N = 2 Chern-Simons matter theories showing that, as one varies the argument of the coupling g = 1/k, the thimble decomposition of the path-integral exhibits Stokes phenomenon. As expected the ambiguities in resummation of the Borel transform are directly related to the jump in thimbles attached to non-perturbative saddles.…”

Lefschetz thimbles and quantum phases in zero-dimensional bosonic models

“…[30][31][32][33][34][35] the Lefschetz-thimble approach has been employed to study bosonic quantum field theories, and in Refs. [36][37][38][39][40][41][42][43] models including fermions were studied. The relevance of Lefschetz thimbles in the context of semi-classical expansion in asymptotically free quantum field theories is discussed in Refs.…”

Phase structure of the twisted SU(3)/U(1)2 flag sigma model on ℝ × S1

“…Research on the adiabatic circle compactification is also of great influence on the attempt to give nonperturbative definition of 2-dimensional field theories based on the resurgence theory [36,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. Resurgence theory, which was originally developed for the ordinary differential equation, has been applied to a broad area of theoretical physics including, quantum mechanics [53][54][55][56][57][58][59][60][61][62], string theory [63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] and supersymmetric field theory [79][80][81][82][83] (see more references in [84,...…”