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Instances of higher geometry in field theory

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Abstract

Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some basic aspects of algebroid structures on bundles and (differential graded) Q-manifolds, we briefly discuss their relation to \((\alpha )\) the Batalin–Vilkovisky quantization of topological sigma models, \((\beta )\) higher gauge theories and generalized global symmetries and \((\gamma )\) tensor gauge theories, where the universality of their form and properties in terms of graded geometry is highlighted.

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Notes

  1. In this mini-review, the word “higher” is preferred. One reason is that it is sometimes used here as an umbrella term for a variety of different—but often closely related—ideas, including generalised complex geometry, differential geometry of homotopy algebras or algebroids, graded supergeometry and noncommutative geometry, although focus will be on a small subset due to length restrictions.

  2. It is useful to recall that all gauge field theories are constrained Hamiltonian systems.

  3. Other versions corresponding to the same notion exist, for example based on Leibniz-type brackets (a.k.a. Dorfman bracket) [7, 8] or foliations [9].

  4. This and some relaxed structures thereof were used in [26] for membrane sigma models.

  5. We note that E-covariant derivatives on a different vector bundle V can also be defined.

  6. We mainly refer here to ’t Hooft anomalies, which can obstruct the gauging of a symmetry, see e.g. [57] for a recent review. The same reasoning also applies of course to ABJ anomalies.

  7. One should not take this as having literally any relation to the matching of scattering amplitudes between gravity and Yang–Mills.

  8. It is amusing to note that the more symmetric slots one adds to a tensor, the more antisymmetric coordinates are introduced to describe them in terms of Q-manifolds. I thank Peter Schupp for emphasizing this.

  9. This means that Lagrangians whose nonlinearity is due to being an algebraic functional of the free kinetic and theta terms are included, but Yang–Mills type theories or full general relativity have not been fully described in this form yet.

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Acknowledgements

Work supported by the Croatian Science Foundation Project “New Geometries for Gravity and Spacetime” (IP-2018-01-7615).

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  1. Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000, Zagreb, Croatia

    Athanasios Chatzistavrakidis

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S.I.: Noncommutativity and Physics. Guest editors: George Zoupanos, Konstantinos Anagnostopoulos, Peter Schupp.

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Chatzistavrakidis, A. Instances of higher geometry in field theory. Eur. Phys. J. Spec. Top. 232, 3705–3713 (2023). https://doi.org/10.1140/epjs/s11734-023-00839-z

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