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Geometry and 2-Hilbert space for nonassociative magnetic translations

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Abstract

We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth distributions of magnetic charge, and dually of closed strings in locally non-geometric flux backgrounds, which naturally allows for representations of nonassociative magnetic translation operators. We show how one can use the 2-Hilbert space of sections of a bundle gerbe in a putative framework for canonical quantisation. We define a parallel transport on bundle gerbes on \({\mathbb {R}}^d\) and show that it naturally furnishes weak projective 2-representations of the translation group on this 2-Hilbert space. We obtain a notion of covariant derivative on a bundle gerbe and a novel perspective on the fake curvature condition.

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Notes

  1. For the construction of the 2-Hilbert space of sections it is important to work with categorified line bundles instead of principal bundles, since it allows the existence of non-invertible morphisms.

  2. A section of a Hermitian vector bundle is parallel if it is annihilated by the covariant derivative.

  3. This space is finite-dimensional since the dimension of the space of parallel sections of a vector bundle on a path-connected manifold is bounded from above by the rank of the vector bundle.

  4. See [13] for the corresponding definitions.

  5. The element \(\imath \) encodes the coherence 2-isomorphism \(\omega (1)\Longrightarrow \text {id}\).

  6. See [31] for a treatment of the Dirac monopole field in this setting.

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Acknowledgements

We thank Marco Benini and Lennart Schmidt for helpful discussions. This work was supported by the COST Action MP1405 ‘Quantum Structure of Spacetime’, funded by the European Cooperation in Science and Technology (COST). The work of S.B. was partially supported by the RTG 1670 ‘Mathematics Inspired by String Theory and Quantum Field Theory’. The work of L.M. was supported by the Doctoral Training Grant ST/N509099/1 from the UK Science and Technology Facilities Council (STFC). The work of R.J.S. was supported in part by the STFC Consolidated Grant ST/P000363/1 ‘Particle Theory at the Higgs Centre’.

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Authors and Affiliations

  1. Fachbereich Mathematik, Bereich Algebra und Zahlentheorie, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Severin Bunk

  2. Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh, EH14 4AS, UK

    Lukas Müller  & Richard J. Szabo

  3. Maxwell Institute for Mathematical Sciences, Edinburgh, UK

    Lukas Müller  & Richard J. Szabo

  4. The Higgs Centre for Theoretical Physics, Edinburgh, UK

    Lukas Müller  & Richard J. Szabo

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  1. Severin Bunk
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  2. Lukas Müller
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Correspondence to Lukas Müller .

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Bunk, S., Müller , L. & Szabo, R.J. Geometry and 2-Hilbert space for nonassociative magnetic translations. Lett Math Phys 109, 1827–1866 (2019). https://doi.org/10.1007/s11005-019-01160-4

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  • DOI: https://doi.org/10.1007/s11005-019-01160-4

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