A note on perturbation series in supersymmetric gauge theories

Exact results in supersymmetric Chern-Simons and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N} = 2$\end{document} Yang-Mills theories can be used to examine the quantum behavior of observables and the structure of the perturbative series. For the U(2) × U(2) ABJM model, we determine the asymptotic behavior of the perturbative series for the partition function and write it as a Borel transform. Similar results are obtained for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N} = 2$\end{document} SU(2) super Yang-Mills theory with four fundamental flavors and in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N} = {2^* }$\end{document} super Yang-Mills theory, for the partition function as well as for the expectation values for Wilson loop and ’t Hooft loop operators (in the 0 and 1 instanton sectors). In all examples, one has an alternate perturbation series where the coefficient of the nth term increases as n!, and the perturbation series are Borel summable. We also calculate the expectation value for a Wilson loop operator in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N} = {2^* }$\end{document} SU(N ) theory at large N in different regimes of the ’t Hooft gauge coupling and mass parameter. For large masses, the calculation reproduces the running gauge coupling for the pure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N} = 2$\end{document} SYM theory.