1. |
- Bossard, Guillaume, et al.
(författare)
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Extended geometry of magical supergravities
- 2023
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Ingår i: Journal of High Energy Physics (JHEP). - : Springer. - 1126-6708 .- 1029-8479. ; :5
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Tidskriftsartikel (refereegranskat)abstract
- We provide, through the framework of extended geometry, a geometrisation of the duality symmetries appearing in magical supergravities. A new ingredient is the general formulation of extended geometry with structure group of non-split real form. A simple diagrammatic rule for solving the section constraint by inspection of the Satake diagram is derived.
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2. |
- Bossard, Guillaume, et al.
(författare)
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Generalised diffeomorphisms for E9
- 2017
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Ingår i: Physical Review D - Particles, Fields, Gravitation and Cosmology. - : American Physical Society. - 2470-0010 .- 2470-0029. ; 96, s. 106022-
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Tidskriftsartikel (refereegranskat)abstract
- We construct generalised diffeomorphisms for E9 exceptional field theory. The trans- formations, which like in the E8 case contain constrained local transformations, close when acting on fields. This is the first example of a generalised diffeomorphism alge- bra based on an infinite-dimensional Lie algebra and an infinite-dimensional coordi- nate module. As a byproduct, we give a simple generic expression for the invariant tensors used in any extended geometry. We perform a generalised Scherk–Schwarz reduction and verify that our transformations reproduce the structure of gauged supergravity in two dimensions. The results are valid also for other affine algebras.
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3. |
- Carbone, Lisa, et al.
(författare)
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Generators and relations for (generalised) Cartan type superalgebras
- 2019
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Ingår i: Journal of Physics: Conference Series. - : IOP Publishing. - 1742-6588 .- 1742-6596. ; 1194:1
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Konferensbidrag (refereegranskat)abstract
- In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(n-1,0) = s (1|n) can be constructed by adding a "gray" node to the Dynkin diagram of An-1 = s (n), corresponding to an odd null root. The Cartan superalgebras constitute a difierent class, where the simplest example is Wpnq, the derivation algebra of the Grassmann algebra on n generators. Here we present a novel construction of Wpnq, from the same Dynkin diagram as A(n-1,0), but with additional generators and relations.
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4. |
- Carbone, Lisa, et al.
(författare)
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Generators and relations for Lie superalgebras of Cartan type
- 2019
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Ingår i: Journal of Physics A. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 52:5
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Tidskriftsartikel (refereegranskat)abstract
- We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A(r), D-r or E-r. Then W(A(n-1)) and S(A(n-1)) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A(r)) and S(A(r)) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level >= -2. The analogous definitions of the algebras in the D- and E-series are given. In the latter case the full set of defining relations is conjectured.
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5. |
- Cederwall, Martin, 1961, et al.
(författare)
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Extended geometries
- 2018
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Ingår i: Journal of High Energy Physics. - : Springer. - 1029-8479 .- 1126-6708. ; 2018:2
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Tidskriftsartikel (refereegranskat)abstract
- We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional (“ancillary”) gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-)action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces (the internal sector of) all previously considered cases of double and exceptional field theories.
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6. |
- Cederwall, Martin, 1961, et al.
(författare)
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L∞ algebras for extended geometry
- 2019
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Ingår i: Journal of Physics: Conference Series. - : IOP Publishing. - 1742-6588 .- 1742-6596. ; 1194:1
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Konferensbidrag (refereegranskat)abstract
- Extended geometry provides a unified framework for double geometry, exceptional geometry, etc., i.e., for the geometrisations of the string theory and M-theory dualities. In this talk, we will explain the structure of gauge transformations (generalised diffeomorphisms) in these models. They are generically infinitely reducible, and arise as derived brackets from an underlying Borcherds superalgebra or tensor hierarchy algebra. The infinite reducibility gives rise to an L∞structure, the brackets of which have universal expressions in terms of the underlying superalgebra.
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7. |
- Cederwall, Martin, 1961, et al.
(författare)
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L∞ Algebras for Extended Geometry from Borcherds Superalgebras
- 2019
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 369:2, s. 721-760
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Tidskriftsartikel (refereegranskat)abstract
- We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1) , which is a double extension of the structure algebra gr. The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.
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8. |
- Cederwall, Martin, 1961, et al.
(författare)
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Superalgebras, constraints and partition functions
- 2015
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Ingår i: Journal of High Energy Physics (JHEP). - : Springer. - 1126-6708 .- 1029-8479. ; :8
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Tidskriftsartikel (refereegranskat)abstract
- We consider Borcherds superalgebras obtained from semisimple finite-dimensional Lie algebras by adding an odd null root to the simple roots. The additional Serre relations can be expressed in a covariant way. The spectrum of generators at positive levels are associated to partition functions for a certain set of constrained bosonic variables, the constraints on which are complementary to the Serre relations in the symmetric product. We give some examples, focusing on superalgebras related to pure spinors, exceptional geometry and tensor hierarchies, of how construction of the content of the algebra at arbitrary levels is simplified.
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9. |
- Cederwall, Martin, 1961, et al.
(författare)
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Teleparallelism in the algebraic approach to extended geometry
- 2022
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Ingår i: Journal of High Energy Physics. - : Springer. - 1029-8479 .- 1126-6708.
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Tidskriftsartikel (refereegranskat)abstract
- Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered L-infinity structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field strengths and Bianchi identities, and identify these as elements of the tensor hierarchy algebra. The field strengths arise as generalised torsion, so the naturally occurring complex in the L-infinity algebra is ... <- torsion BI's <- torsion <- vielbein <- diffeomorphism parameters <- ... In order to obtain equations of motion, which are not in this complex, (pseudo-)actions, quadratic in torsion, are given for a large class of models. This requires considering the dual complex. We show how local invariance under the compact subgroup locally defined by a generalised metric arises as a "dual gauge symmetry" associated with a certain torsion Bianchi identity, generalising Lorentz invariance in the teleparallel formulation of gravity. The analysis is performed for a large class of finite-dimensional structure groups, with E-5 as a detailed example. The continuation to infinite-dimensional cases is discussed.
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10. |
- Cederwall, Martin, 1961, et al.
(författare)
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Tensor Hierarchy Algebra Extensions of Over-Extended Kac–Moody Algebras
- 2022
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 389:1, s. 571-620
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Tidskriftsartikel (refereegranskat)abstract
- Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation L1. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.
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