Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

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

doi:10.1016/j.geomphys.2016.04.005

Cited by 24 publications

(64 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Physically consistent models with novel properties in the context of quantum mechanics were constructed in [25] using this formalism, and of Euclidean scalar quantum field theory in [23]. To extend these considerations to more complicated field theories, a general systematic formalism was developed in [6,7] for differential geometry on noncommutative and nonassociative spaces internal to the representation category of any quasi-Hopf algebra, generalizing and extending earlier work [15,8,2]. This is the starting point for the present contribution.…”

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 94%

“…The purpose of this contribution is to unpack and make explicit the somewhat abstract categorical constructions of [6,7] in a less formal language that we hope will be palatable to a larger audience. We focus on the special case of most physical relevance: the cochain twist quantization of a classical manifold; this construction is reviewed in Section 2.…”

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 99%

“…We further restrict to trivial vector bundles over these noncommutative and nonassociative spaces with diagonal action of the pertinent Hopf algebra of symmetries of the non-geometric background. This simplification enables us to give very explicit "local" descriptions of the noncommutative and nonassociative geometry while still retaining generic features and indicating how the general formalism of [6,7] may be applied to constructions of physically viable field theories; in particular, we give concrete realizations of the pertinent bimodule operations for homomorphism bundles. In Section 3 we apply this framework to obtain explicit expressions for connections and their curvatures on noncommutative and nonassociative vector bundles.…”

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 99%

“…E = M × C n → M with bundle projection given by projecting on the first factor. For a discussion of generic vector bundles see [6,7].…”

Section: Vector Bundlesmentioning

confidence: 99%

“…and refer to [6,7] for further details. The -tensor product V ⊗ W is the ordinary tensor product of vector spaces equipped with the H F -action…”

Section: Homomorphism Bundlesmentioning

confidence: 99%

See 4 more Smart Citations

Perspectives on Nonassociative Geometry

Szabo¹,

Barnes²,

Schenkel³

2016

Proceedings of Proceedings of the Corfu Summer Institute 2015 — PoS(CORFU2015)

Self Cite

View full text Add to dashboard Cite

We review aspects of our formalism for differential geometry on noncommutative and nonassociative spaces which arise from cochain twist deformation quantization of manifolds. We work in the simplest setting of trivial vector bundles and flush out the details of our approach providing explicit expressions for all bimodule operations, and for connections and curvature. As applications, we describe the constructions of physically viable action functionals for Yang-Mills theory and Einstein-Cartan gravity on noncommutative and nonassociative spaces, as first steps towards more elaborate models relevant to non-geometric flux deformations of geometry in closed string theory.

show abstract

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 94%

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 99%

Section: G E Barnes a Schenkel And R J Szabomentioning

confidence: 99%

“…E = M × C n → M with bundle projection given by projecting on the first factor. For a discussion of generic vector bundles see [6,7].…”

Section: Vector Bundlesmentioning

confidence: 99%

“…and refer to [6,7] for further details. The -tensor product V ⊗ W is the ordinary tensor product of vector spaces equipped with the H F -action…”

Section: Homomorphism Bundlesmentioning

confidence: 99%

See 3 more Smart Citations

Perspectives on Nonassociative Geometry

Szabo¹,

Barnes²,

Schenkel³

2016

Proceedings of Proceedings of the Corfu Summer Institute 2015 — PoS(CORFU2015)

Self Cite

View full text Add to dashboard Cite

show abstract

Global Double Field Theory is Higher Kaluza‐Klein Theory

Alfonsi

2020

Fortschritte der Physik

View full text Add to dashboard Cite

Kaluza-Klein Theory states that a metric on the total space of a principal bundle P → M, if it is invariant under the principal action of P, naturally reduces to a metric together with a gauge field on the base manifold M. We propose a generalization of this Kaluza-Klein principle to higher principal bundles and higher gauge fields. For the particular case of the abelian gerbe of Kalb-Ramond field, this Higher Kaluza-Klein geometry provides a natural global formulation for Double Field Theory (DFT). In this framework the doubled space is the total space of a higher principal bundle and the invariance under its higher principal action is exactly a global formulation of the familiar strong constraint. The patching problem of DFT is naturally solved by gluing the doubled space with a higher group of symmetries in a higher category. Locally we recover the familiar picture of an ordinary para-Hermitian manifold equipped with Born geometry. Infinitesimally we recover the familiar picture of a higher Courant algebroid twisted by a gerbe (also known as Extended Riemannian Geometry). As first application we show that on a torus-compactified spacetime the Higher Kaluza-Klein reduction gives automatically rise to abelian T-duality, while on a general principal bundle it gives rise to non-abelian T-duality. As final application we define a natural notion of Higher Kaluza-Klein monopole by directly generalizing the ordinary Gross-Perry one. Then we show that under Higher Kaluza-Klein reduction, this monopole is exactly the NS5-brane on a 10d spacetime. If, instead, we smear it along a compactified direction we recover the usual DFT monopole on a 9d spacetime.

show abstract

Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

Vacaru

Veliev

Bubuianu³

2021

Fortschritte der Physik

View full text Add to dashboard Cite

We elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non-integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi-Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R-flux background. Generalized Moyal-Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/momentum variables like in (non) commutative Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N-connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N-adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N-adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R-flux deformations of vacuum Einstein equations for the case of N-adapted linear connections and symmetric and nonsymmetric metric structures. Introduction, Motivations and GoalsWe initialize a research on nonassociative nonholonomic geometry induced by R-flux backgrounds in string gravity and classical and quantum models on phase spaces modelled as (co) tangent Lorentz bundles endowed with nonholonomic (non-integrable, equivalently, anholonomic) structures and nonsymmetric metrics. Such a nonholonomic geometric formalism was elaborated for various types of (non) commutative gravity and matter field theories, modified (relativistic) geometric flows, and classical and quantum information theories, see details in a series of our previous works [1][2][3][4][5][6][7][8][9][10][11] and references therein. In this article, nonassociative geometric and physical models are formulated in a form which is accessible to researchers in particle physics and modern cosmology. There are provided nonholonomic generalizations of two recent models of nonassociative geometry and vacuum gravity due to [12] and, with

show abstract

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

Cited by 24 publications

References 30 publications

Perspectives on Nonassociative Geometry

Perspectives on Nonassociative Geometry

Global Double Field Theory is Higher Kaluza‐Klein Theory

Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

Contact Info

Product

Resources

About