Leibniz Gauge Theories and Infinity Structures

Bonezzi, Roberto; Hohm, Olaf

doi:10.1007/s00220-020-03785-2

Cited by 31 publications

(45 citation statements)

References 56 publications

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“…where the omni-Lie bracket [•, •] ol is given by (6), and (•, •) + is a symmetric V -valued pairing given by…”

Section: Embedding Tensors Omni-lie Algebras and Leibniz Algebrasmentioning

confidence: 99%

“…Leibniz algebras, embedding tensors and their associated tensor hierarchies provide a nice and efficient way to construct supergravity theories and further to higher gauge theories (see e.g. [6,7,18,24,51] and references therein for a rich physics literature on this subject, and see [27,28] for a math-friendly introduction on this subject). Recently, this topic has attracted much attention of the mathematical physics world.…”

Section: Introductionmentioning

confidence: 99%

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The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

Sheng

Tang

Zhu

2021

Commun. Math. Phys.

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In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an $$L_\infty $$ L ∞ -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$$_\infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_\infty $$ L ∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz$$_\infty $$ ∞ -algebras, and a functor further to that of $$L_\infty $$ L ∞ -algebras.

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“…where the omni-Lie bracket [•, •] ol is given by (6), and (•, •) + is a symmetric V -valued pairing given by…”

Section: Embedding Tensors Omni-lie Algebras and Leibniz Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

Sheng

Tang

Zhu

2021

Commun. Math. Phys.

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“…The idea of tensor hierarchy was introduced in [50] in the context of the dimensional reduction of DFT, then further formalized in [7, 51] and [8] as a higher gauge structure. See also work by [23] and [24].…”

Section: Review Of Proposals For Dft Geometrymentioning

confidence: 99%

The Puzzle of Global Double Field Theory: Open Problems and the Case for a Higher Kaluza‐Klein Perspective

Alfonsi

2021

Fortschritte der Physik

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The history of the geometry of Double Field Theory is the history of string theorists' effort to tame higher geometric structures. In this spirit, the first part of this paper will contain a brief overview on the literature of geometry of DFT, focusing on the attempts of a global description. In [1] we proposed that the global doubled space is not a manifold, but the total space of a bundle gerbe. This would mean that DFT is a field theory on a bundle gerbe, in analogy with ordinary Kaluza‐Klein Theory being a field theory on a principal bundle. In this paper we make the original construction by [1] significantly more immediate. This is achieved by introducing an atlas for the bundle gerbe. This atlas is naturally equipped with 2d‐dimensional local charts, where d is the dimension of physical spacetime. We argue that the local charts of this atlas should be identified with the usual coordinate description of DFT. In the last part we will discuss aspects of the global geometry of tensor hierarchies in this bundle gerbe picture. This allows to identify their global non‐geometric properties and explain how the picture of non‐abelian String‐bundles emerges. We interpret the abelian T‐fold and the Poisson‐Lie T‐fold as global tensor hierarchies.

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“…(See [11] for an extensive review of this 'higher gauge theory' aspect of ExFT, [12] for a review of L ∞ algebras, and [13] for a general theory of tensor hierarchies. )…”

Section: Pos(corfu2018)098mentioning

confidence: 99%

The many facets of exceptional field theory

Hohm¹,

Samtleben²

2019

Proceedings of Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2018)

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Exceptional field theory The U-duality transformations, acting in particular on the generalized metric, then include timelike dualities. It should be emphasized that such dualities, while somewhat unconventional, are of physical interest since they apply whenever one has timelike Killing vectors (or BPS solutions). In addition, theories with a 'Lorentzian generalized metric' are intriguing in that the latter may go through a phase in which the conventional metric components g mn become singular while the generalized metric is still perfectly regular. In this fashion, ExFT encodes truly 'non-geometric' configurations. Another application we will discuss here is the relation to the 'magic tables' that arose in the literature since the early days of supergravity. Specifically, there is an abundance of interesting, often exceptional groups arising in lower dimensions, with various intriguing interrelations and symmetries between them. Here we consider the 'magic triangle' discovered by Cremmer et. al. in compactifications to three dimensions [15]. We will show that the E 8(8) ExFT unifies all theories corresponding to the different entries of that table into a single Lagrangian that, moreover, realises the symmetry of the triangle following a simple group-theory argument. Finally, we review how the generalized type IIB supergravity theory that appeared recently in integrable deformations of AdS/CFT is naturally embedded in ExFT. The rest of this article is organized as follows. In sec. 2 we review ExFT with a focus on the E 6(6) and E 8(8) theories that will be employed later. In sec. 3 we introduce the magic triangle of Cremmer et. al. and explain how it is accommodated within the E 8(8) ExFT. We then turn to timelike dualities and Hull's M *-theories in sec. 4. In sec. 5 we discuss the embedding of generalized IIB supergravity theory, while we close with a brief outlook in sec. 6.

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Leibniz Gauge Theories and Infinity Structures

Cited by 31 publications

References 56 publications

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

The Puzzle of Global Double Field Theory: Open Problems and the Case for a Higher Kaluza‐Klein Perspective

The many facets of exceptional field theory

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