Instanton contributions to the ABJM free energy from quantum M2 branes

Abstract

We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at large N and fixed level k. Using supersymmetric localization, such instanton contribution was found earlier to take the form FinstNk=−sin22πk−1exp−2π2Nk+.…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {F}^{inst}\\left(N,k\\right)=-{\\left({\\sin}^2\\frac{2\\pi }{k}\\right)}^{-1}\\exp \\left(-2\\pi \\sqrt{\\frac{2N}{k}}\\right)+.\\dots $$\\end{document} The exponent comes from the action of an M2 brane instanton wrapped on S3/ℤk, which represents the M-theory uplift of the ℂP1 instanton in type IIA string theory on AdS4 × ℂP3. The IIA string computation of the leading large k term in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor sin22πk−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\left({\\sin}^2\\frac{2\\pi }{k}\\right)}^{-1} $$\\end{document} is reproduced by the 1-loop term in the M2 brane partition function expanded near the S3/ℤk instanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization.