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Constitutive Relations, Off Shell Duality Rotations and the Hypergeometric Form of Born-Infeld Theory

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Breaking of Supersymmetry and Ultraviolet Divergences in Extended Supergravity

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 153))

Abstract

We review equivalent formulations of nonlinear and higher derivatives theories of electromagnetism exhibiting electric-magnetic duality rotations symmetry. We study in particular on shell and off shell formulations of this symmetry, at the level of action funcitonals as well as of equations of motion. We prove the conjecture that the action functional leading to Born-Infeld nonlinear electromagnetism, that is duality rotation invariant off shell and that is known to be a root of an algebraic equation of fourth order, is a hypergeometric function.

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Notes

  1. 1.

    Nonlinear and higher derivatives theories of electromagnetism admit one (or more) dimensionful coupling constant(s) \(\lambda \).

  2. 2.

    The factor 2 is due to the convention \(\frac{\partial {F_{\rho \sigma }}}{\partial F_{\mu \nu }}=\delta ^\mu _\rho \delta ^\nu _\sigma \,\) adopted in [7] and in the review [37]. It will be used throughout the paper.

  3. 3.

    Note that (2.16) (the integrated form of the more restrictive self-duality condition \(F\widetilde{F}+G\widetilde{G}\)) also follows in a straightforward manner by repeating the passages in [7] but with \(G\) the functional derivatives of the action rather than the partial derivatives of the lagrangian [13, 37]. This makes a difference for nonlinear theories which also contain terms with derivatives of \(F\).

  4. 4.

    In a general nonlinear theory the Hamiltonian depends on the magnetic field \({\overrightarrow{B}}\) and on the electric displacement \(\overrightarrow{D}=\frac{\delta S[F]}{\delta {\overrightarrow{E}}}\), that rotate into each other under the duality (2.11), \(\Big ({}^{~{\overrightarrow{B}}'}_{-{\overrightarrow{D}}'}\Big )= \left( \begin{array}{lr} \cos {\alpha } &{} {-\sin {\alpha }}\\ {\sin {\alpha }} &{}{\mathrm{cos}{\alpha }} \end{array}\right) \Big ({}^{~{\overrightarrow{B}}}_{-{D}}\Big )\). Since the composite fields \({\overrightarrow{B}}^2+{\overrightarrow{D}}^2\) and \((\overrightarrow{B}\times \overrightarrow{D})^2\) are duality invariant, Hamiltonians that depend upon these combinations and their derivatives are trivially duality invariant and lead to duality symmetric theories.

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Acknowledgments

P.A. acknowledges the hospitality of Max-Planck-Institut für Gravitationsphysik Albert-Einstein-Institut during commencement of the present work, and the hospitality of Galileo Galilei Institute during its completion. This work is supported by the ERC Advanced Grant no. 226455, Supersymmetry, Quantum Gravity and Gauge Fields (SUPERFIELDS).

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Authors and Affiliations

  1. Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, INFN, Sezione di Torino, Gruppo collegato di Alessandria, Viale T. Michel 11, 15121, Alessandria, Italy

    Paolo Aschieri

  2. Physics Department, Theory Unit, CERN, CH 1211, 23, Geneva, Switzerland

    Sergio Ferrara

  3. INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044, Frascati, Italy

    Sergio Ferrara

  4. Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA, 90095-1547, USA

    Sergio Ferrara

  5. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476, Golm, Germany

    Stefan Theisen

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  1. Paolo Aschieri
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Correspondence to Paolo Aschieri .

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  1. Laboratori Nazionali di Frascati, National Institute of Nuclear Physics, Frascati RM, Italy

    Stefano Bellucci

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Aschieri, P., Ferrara, S., Theisen, S. (2014). Constitutive Relations, Off Shell Duality Rotations and the Hypergeometric Form of Born-Infeld Theory. In: Bellucci, S. (eds) Breaking of Supersymmetry and Ultraviolet Divergences in Extended Supergravity. Springer Proceedings in Physics, vol 153. Springer, Cham. https://doi.org/10.1007/978-3-319-03774-5_2

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