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Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories

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Abstract

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang–Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.

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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    A. Rod Gover

  2. Institut für Mathematik, Universität Zürich-Irchel, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Emanuele Latini

  3. INFN, Laboratori Nazionali di Frascati, CP 13, 00044, Frascati, Italy

    Emanuele Latini

  4. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Emanuele Latini

  5. INFN, Sezione di Bologna, via Irnerio 46, 40126, Bologna, Italy

    Emanuele Latini

  6. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Andrew Waldron

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  1. A. Rod Gover
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  2. Emanuele Latini
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Communicated by P. T. Chruściel

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Gover, A.R., Latini, E. & Waldron, A. Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories. Commun. Math. Phys. 341, 667–697 (2016). https://doi.org/10.1007/s00220-015-2490-x

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