Working with Nonassociative Geometry and Field Theory

Barnes, Gwendolyn Elizabeth; Schenkel, Alexander; Szabo, Richard J.

doi:10.48550/arxiv.1601.07353

Cited by 12 publications

(22 citation statements)

References 26 publications

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“…This violation is well-known in twisted noncommutative differential geometry, see e.g. [1,9,16]. For trivial R-matrix, the right-hand sides of these covariant derivative equations vanish identically and we recover the usual Bianchi identity in classical geometry.…”

Section: Braided Curvaturementioning

confidence: 51%

“…This paper arose as part of an ongoing programme to formulate a nonassociative theory of gravity that is inspired by scenarios in closed non-geometric string theory, see e.g. [9,16,18]. We argued in [28] that the L ∞ -algebra formalism should provide the natural receptacle to capture the failure of closure and covariance of field equations under nonassociative gauge transformations.…”

Section: Braided Noncommutative Gravitymentioning

confidence: 99%

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Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

et al. 2021

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We define a new homotopy algebraic structure, that we call a braided $$L_\infty $$ L ∞ -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

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Section: Braided Curvaturementioning

confidence: 51%

Section: Braided Noncommutative Gravitymentioning

confidence: 99%

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

et al. 2021

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“…This has led to attempts to treat noncommutative gravity as a deformation of a 'gauge theory', that is, in the first order formalism for general relativity: One uses the Einstein-Cartan principal bundle formulation, with star-gauge symmetry, where the corresponding variational principle is based on the Palatini action functional [10,22,34,37,41]. However, diffeomorphisms can never be implemented as star-gauge symmetries, in any formalism, and instead the noncommutative theory of gravity is invariant under a 'twisted' action of infinitesimal diffeomorphisms [16,19].…”

Section: Non-geometric Backgrounds and Noncommutative Gravitymentioning

confidence: 99%

“…the exterior algebra of differential forms on Ê 1,d−1 gives the noncommutative differential calculus discussed in Section 3.1; for the holonomic coframe L ∂ν dx µ = 0, so that indeed dx µ ∧ ⋆ dx ν = dx µ ∧ dx ν and λ ⋆ dx µ = dx µ ⋆ λ = λ • dx µ for λ ∈ Ω 0 (Ê 1,d−1 ). 22 Similarly, for the holonomic frame…”

Section: Drinfel'd Twist Deformation Theorymentioning

confidence: 99%

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Braided Symmetries in Noncommutative Field Theory

Giotopoulos,

Szabo

2021

Preprint

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We give a pedagogical introduction to L ∞ -algebras and their uses in organising the symmetries and dynamics of classical field theories, as well as of the conventional noncommutative gauge theories that arise as low-energy effective field theories in string theory. We review recent developments which formulate field theories with braided gauge symmetries as a new means of overcoming several obstacles in the standard noncommutative theories, such as the restrictions on gauge algebras and matter fields. These theories can be constructed by using techniques from Drinfel'd twist deformation theory, which we review in some detail, and their symmetries and dynamics are controlled by a new homotopy algebraic structure called a 'braided L ∞ -algebra'. We expand and elaborate on several novel theoretical issues surrounding these constructions, and present three new explicit examples: the standard noncommutative scalar field theory (regarded as a braided field theory), a braided version of BF theory in arbitrary dimensions (regarded as a higher gauge theory), and a new braided version of noncommutative Yang-Mills theory for arbitrary gauge algebras.

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G 2-structures and quantization of non-geometric M-theory backgrounds

Kupriyanov

Szabo

2017

J. High Energ. Phys.

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Abstract:We describe the quantization of a four-dimensional locally non-geometric Mtheory background dual to a twisted three-torus by deriving a phase space star product for deformation quantization of quasi-Poisson brackets related to the nonassociative algebra of octonions. The construction is based on a choice of G 2 -structure which defines a nonassociative deformation of the addition law on the seven-dimensional vector space of Fourier momenta. We demonstrate explicitly that this star product reduces to that of the three-dimensional parabolic constant R-flux model in the contraction of M-theory to string theory, and use it to derive quantum phase space uncertainty relations as well as triproducts for the nonassociative geometry of the four-dimensional configuration space. By extending the G 2 -structure to a Spin(7)-structure, we propose a 3-algebra structure on the full eight-dimensional M2-brane phase space which reduces to the quasi-Poisson algebra after imposing a particular gauge constraint, and whose deformation quantisation simultaneously encompasses both the phase space star products and the configuration space triproducts. We demonstrate how these structures naturally fit in with previous occurences of 3-algebras in M-theory.

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Working with Nonassociative Geometry and Field Theory

Cited by 12 publications

References 26 publications

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

Braided Symmetries in Noncommutative Field Theory

G 2-structures and quantization of non-geometric M-theory backgrounds

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