[Paper Review] Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

Inês Aniceto, Jorge G. Russo+1 more|arXiv (Cornell University)|Oct 21, 2014

Black Holes and Theoretical Physics66 references8 citations

TL;DR

This paper applies resurgent analysis to localize observables in supersymmetric gauge theories, using exact matrix model results from localization to precisely determine the Borel plane singularities of free energy and partition functions in 3D Chern–Simons and ABJM theories, and 4D N=2 Yang–Mills theories. It establishes a complete correspondence between the large-order factorial growth of perturbative coefficients and nonperturbative instanton/renormalon effects via exact Borel transform structures.

ABSTRACT

Localization methods have recently led to a plethora of new exact results in supersymmetric gauge theories, as certain observables may be computed in terms of matrix integrals. These can then be evaluated by making use of standard large N techniques, or else via perturbative expansions in the gauge coupling. Either approximation often leads to observables given in terms of asymptotic series, which need to be properly defined in order to obtain nonperturbative results. At the same time, resurgent analysis has recently been successfully applied to several problems, e.g., in quantum, field and string theories, precisely to overcome this issue and construct nonperturbative answers out of asymptotic perturbative expansions. The present work uses exact results from supersymmetric localization to address the resurgent structure of the free energy and partition function of Chern-Simons and ABJM gauge theories in three dimensions, and of N=2 supersymmetric Yang-Mills theories in four dimensions. For each case, the complete structure of Borel singularities is exactly determined, and the relation of these singularities with the large-order behavior of (multi-instanton) perturbative expansions is made fully precise.

Motivation & Objective

• To understand the nonperturbative structure of supersymmetric gauge theories using perturbative data.
• To determine the complete Borel plane singularity structure of free energy and partition functions in 3D and 4D supersymmetric theories.
• To establish a precise link between large-order factorial divergence in perturbative expansions and nonperturbative effects such as instantons and renormalons.
• To apply resurgence techniques to exact results from supersymmetric localization, particularly matrix model reductions.
• To provide a nonperturbative definition of observables via resummation of asymptotic series using exact Borel transform data.

Proposed method

• Utilizes exact results from supersymmetric localization to express observables as matrix integrals.
• Analyzes the large-order behavior of perturbative expansions in terms of factorial growth of coefficients.
• Constructs the Borel transform of the perturbative series to expose singularities in the complex Borel plane.
• Identifies Borel singularities at positions $ s_n = -n^2 2A = -(4 ilde{ au}n)^2 $, corresponding to instanton/renormalon contributions.
• Computes Laurent expansions of the Borel transform around each singularity using exact expressions involving Barnes G-functions and zeta values.
• Applies lateral Borel resummation to resolve Stokes phenomena and extract unambiguous nonperturbative contributions.

Experimental results

Research questions

• RQ1 How do the Borel singularities of the perturbative free energy in 3D Chern–Simons and ABJM theories relate to their nonperturbative instanton sectors?
• RQ2 What is the exact structure of the Borel transform singularities in 4D N=2 supersymmetric Yang–Mills theories?
• RQ3 How does the large-order factorial growth of perturbative coefficients encode information about instanton and renormalon effects?
• RQ4 Can the full nonperturbative structure of localized observables be reconstructed from their asymptotic perturbative series via resurgence?
• RQ5 What is the precise role of Stokes lines and lateral Borel resummations in defining physical observables in these theories?

Key findings

• The Borel transform of the free energy in 3D Chern–Simons and ABJM theories exhibits poles at $ s_n = -n^2 2A = -(4 ilde{ au}n)^2 $, with exact residues involving Barnes G-functions and zeta values.
• For 4D N=2 theories, the Borel transform singularities are located at $ s_n = -n^2 2A $, and the Laurent coefficients are computed explicitly in terms of $ M $, $ ilde{ au} $, and special functions.
• The coefficients $ f_ u^{(n)} $ in the Laurent expansion near $ s_n = -n^2 2A $ are found to be proportional to $ i ilde{ au}^{(2n+1)/2} $, with explicit algebraic and zeta-function dependence.
• The singularities at $ s_n $ are shown to be directly responsible for the factorial growth $ F_n acksim n! $ in the perturbative coefficients, confirming the resurgence relation.
• The lateral Borel resummations across Stokes lines yield exponentially small nonperturbative corrections, with precise expressions involving $ ext{e}^{2M^2(1+ ilde{ au})} $, $ G $-functions, and polygamma functions.
• The full nonperturbative answer is reconstructed by summing over all instanton sectors and their associated Borel singularities, with exact expressions provided for the first few orders in the Laurent expansion.