Moduli space of logarithmic states in critical massive gravities

Abstract

We take new algebraic and geometric perspectives on the combinatorial results recently obtained on the partition functions of critical massive gravities conjectured to be dual to Logarithmic CFTs through the AdS3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}/LCFT2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} correspondence. We show that the partition functions of logarithmic states can be expressed in terms of Schur polynomials. Subsequently, we show that the moduli space of the logarithmic states is the symmetric product SnC2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^n\\left( {\\mathbb {C}}^2 \\right) $$\\end{document}. As the quotient of an affine space by the symmetric group, this orbifold space is shown to be described by Hilbert series that have palindromic numerators. The palindromic properties of the Hilbert series indicate that the orbifolds are Calabi-Yau, and allow for a new interpretation of the logarithmic state spaces in critical massive gravities as Calabi-Yau singular spaces.