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Black holes and general Freudenthal transformations

  • Borsten, L.1
  • Duff, M. J.2, 3, 4
  • Fernández-Melgarejo, J. J.5
  • Marrani, A.6, 7
  • Torrente-Lujan, E.5
  • 1 Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland , Dublin 4 (Ireland)
  • 2 Imperial College London, Theoretical Physics, Blackett Laboratory, London, SW7 2AZ, U.K. , London (United Kingdom)
  • 3 University of Oxford, Mathematical Institute, Andrew Wiles Building, Woodstock Road, Radcliffe Observatory Quarter, Oxford, OX2 6GG, U.K. , Oxford (United Kingdom)
  • 4 Texas A&M University, Institute for Quantum Science and Engineering and Hagler Institute for Advanced Study, College Station, TX, 77840, U.S.A. , College Station (United States)
  • 5 Universidad de Murcia, Fisica Teorica, Departamento de Física, Campus de Espinardo, Murcia, E-30100, Spain , Murcia (Spain)
  • 6 Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, via Panisperna 89A, Roma, I-00184, Italy , Roma (Italy)
  • 7 Università di Padova, and INFN, sezione di Padova, Dipartimento di Fisica e Astronomia “Galileo Galilei”, via Marzolo 8, Padova, I-35131, Italy , Padova (Italy)
Published Article
Journal of High Energy Physics
Publication Date
Jul 15, 2019
DOI: 10.1007/JHEP07(2019)070
Springer Nature


We study General Freudenthal Transformations (GFT) on black hole solutions in Einstein-Maxwell-Scalar (super)gravity theories with global symmetry of type E7. GFT can be considered as a 2-parameter, a, b ∈ ℝ, generalisation of Freudenthal duality: x→xF=ax+bx˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x\to {x}_F= ax+b\tilde{x} $$\end{document}, where x is the vector of the electromagnetic charges, an element of a Freudenthal triple system (FTS), carried by a large black hole and x˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{x} $$\end{document} is its Freudenthal dual. These transformations leave the Bekenstein-Hawking entropy invariant up to a scalar factor given by a2 ± b2. For any x there exists a one parameter subset of GFT that leave the entropy invariant, a2 ± b2 = 1, defining the subgroup of Freudenthal rotations. The Freudenthal plane defined by spanℝ{x,x˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{x} $$\end{document}} is closed under GFT and is foliated by the orbits of the Freudenthal rotations. Having introduced the basic definitions and presented their properties in detail, we consider the relation of GFT to the global symmetries or U-dualities in the context of supergravity. We consider explicit examples in pure supergravity, axion-dilaton theories and N = 2, D = 4 supergravities obtained from D = 5 by dimensional reductions associated to (non-degenerate) reduced FTS’s descending from cubic Jordan Algebras.

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