The Heterotic-Ricci Flow and Its Three-Dimensional Solitons

Abstract

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}, for a complete Riemannian metric and a three-form H on a manifold M. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion H. When κ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 0$$\\end{document} the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} and κ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa >0$$\\end{document} the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form H. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining, in particular, the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton and leave as open the construction of a Heterotic soliton with non-constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature.