arXiv subjects

Paolo Aschieri

Publications and source records attributed to Paolo Aschieri.

At least 19 recordsLinked to original sources

arXiv2026

Bianchi identities in noncommutative geometry

We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.

math.QA↗

arXiv2024

Atiyah sequences of braided Lie algebras and their splittings

Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. For the $SU(2)$-principal bundle over the sphere $S^{4}_θ$ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra $so_θ(5,1)$ yields a five parameter family of splittings. On the principal $SO_θ(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere $S^{2n}_θ$, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric on the $S^{2n}_θ$. The corresponding Riemannian geometry of $S^{2n}_θ$ is worked out.

math.QA↗

arXiv2023

Metric compatibility and Levi-Civita Connections on Quantum Groups

Arbitrary connections on a generic Hopf algebra $H$ are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an $H$-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on $SL_q(2)$.

math.QA↗

arXiv2022

Braided Hopf algebras and gauge transformations II: $*$-structures and examples

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible $*$-structures, encompassing the quasitriangular case.

math.QA↗

arXiv2022

Noncommutative gauge and gravity theories and geometric Seiberg-Witten map

We give a pedagogical account of noncommutative gauge and gravity theories, where the exterior product between forms is deformed into a $\star$-product via an abelian twist (e.g. the Groenewold-Moyal twist). The Seiberg-Witten map between commutative and noncommutative gauge theories is introduced. It allows to express the action of noncommutative Einstein gravity coupled to spinor fields in terms of the usual commutative action with commutative fields plus extra interaction terms dependent on the noncommutativity parameter.

hep-th↗

arXiv2022

Braided Hopf algebras and gauge transformations

We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of the instanton bundle and of the orthogonal bundle on the quantum sphere $S^4_θ$.

math.QA↗

arXiv2021

Dispersion Relations in $κ$-Noncommutative Cosmology

We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist $-$giving $κ$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lemaître-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.

gr-qc↗

arXiv2021

Topological T-Duality for Twisted Tori

We apply the $C^*$-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative $C^*$-algebra with an action of ${\mathbb R}^n$. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier-Douady classes. We prove that any such solvmanifold has a topological T-dual given by a $C^*$-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these $C^*$-algebras rigorously describe the T-folds from non-geometric string theory.

hep-th↗

arXiv2021

On Curvature and Torsion in Courant Algebroids

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.

math.DG↗

arXiv2020

Cartan structure equations and Levi-Civita connection in noncommutative geometry

We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of $H$-equivariant $A$-bimodules, whose internal morphisms correspond to tensor fields. Vector fields and forms approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) $A$-module (not necessarily $A$-bimodule) connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary pseudo-Riemannian metrics is proven via a Koszul formula. The general theory includes Drinfeld twists of commutative geometries and also cotriangular Hopf algebras. It is illustrated with the example of the tensor square of Sweedler Hopf algebra which becomes a noncommutative Einstein manifold via a non-central metric.

math.QA↗

arXiv2020

The Gauge Group of a Noncommutative Principal Bundle and Twist Deformations

We study noncommutative principal bundles (Hopf-Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps. We study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

math.QA↗

arXiv2020

Quantum principal bundles on projective bases

The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study the quantization of principal bundles G -> G/P, where G is a semisimple group and P a parabolic subgroup.

math.QA↗

arXiv2019

Global Seiberg-Witten maps for U(n)-bundles on tori and T-duality

Seiberg-Witten maps are a well-established method to locally construct noncommutative gauge theories starting from commutative gauge theories. We revisit and classify the ambiguities and the freedom in the definition. Geometrically, Seiberg-Witten maps provide a quantization of bundles with connections. We study the case of U(n)-vector bundles on two-dimensional tori, prove the existence of globally defined Seiberg-Witten maps (induced from the plane to the torus) and show their compatibility with Morita equivalence.

hep-th↗

arXiv2018

Nonassociative differential geometry and gravity with non-geometric fluxes

We systematically develop the metric aspects of nonassociative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, and use it to construct preliminary steps towards a nonassociative theory of gravity on spacetime. We obtain explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in nonassociative Riemannian geometry on phase space, and write down Einstein field equations. We apply this formalism to construct R-flux corrections to the Ricci tensor on spacetime, and comment on the potential implications of these structures in non-geometric string theory and double field theory.

hep-th↗

arXiv2017

Observables and Dispersion Relations in k-Minkowski Spacetime

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

hep-th↗

arXiv2016

Noncommutative principal bundles through twist deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

math.QA↗

arXiv2016

Deformation quantization of principal bundles

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next we twist deform a subgroup of the group of authomorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

math.QA↗

arXiv2015

Triproducts, nonassociative star products and geometry of R-flux string compactifications

We elucidate relations between different approaches to describing the nonassociative deformations of geometry that arise in non-geometric string theory. We demonstrate how to derive configuration space triproducts exactly from nonassociative phase space star products and extend the relationship in various directions. By foliating phase space with leaves of constant momentum we obtain families of Moyal-Weyl type deformations of triproducts, and we generalize them to new triproducts of differential forms and of tensor fields. We prove that nonassociativity disappears on-shell in all instances. We also extend our considerations to the differential geometry of nonassociative phase space, and study the induced deformations of configuration space diffeomorphisms. We further develop general prescriptions for deforming configuration space geometry from the nonassociative geometry of phase space, thus paving the way to a nonassociative theory of gravity in non-geometric flux compactifications of string theory.

hep-th↗