arXiv subjects

Paolo Aschieri

Publications and source records attributed to Paolo Aschieri.

At least 37 records · Page 2Linked to original sources

arXiv2014

Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $θ$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $θ$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $θ\to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $θ$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.

hep-th↗

arXiv2014

Noncommutative connections on bimodules and Drinfeld twist deformation

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra H is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra). We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily H-equivariant. The theory canonically lifts to the tensor product structure.

math.QA↗

arXiv2013

Constitutive relations, off shell duality rotations and the hypergeometric form of Born-Infeld theory

We review equivalent formulations of nonlinear and higher derivatives theories of electromagnetism exhibiting electric-magnetic duality rotations symmetry. We study in particular on shell and off shell formulations of this symmetry, at the level of action functionals as well as of equations of motion. We prove the conjecture that the action functional leading to Born-Infeld nonlinear electromagnetism, that is duality rotation invariant off shell and that is known to be a root of an algebraic equation of fourth order, is a hypergeometric function.

hep-th↗

arXiv2013

Constitutive relations and Schroedinger's formulation of nonlinear electromagnetic theories

We present a systematic study of nonlinear and higher derivatives extensions of electromagnetism. We clarify when action functionals S[F] can be explicitly obtained from arbitrary (not necessarily self-dual) nonlinear equations of motion. We show that the "Deformed twisted self-duality condition" proposal originated in the context of supergravity counterterms is actually the general framework needed to discuss self-dual theories starting from a variational principle. We generalize to nonlinear and higher derivatives theories Schroedinger formulation of Born-Infeld theory, and for the latter, and more in general for nonlinear theories, we derive a closed form expression of the corresponding deformed twisted self-duality conditions. This implies that the hypergeometric expression entering these duality conditions and leading to Born-Infeld theory satisfies a hidden quartic equation.

hep-th↗

arXiv2012

Extended gravity theories from dynamical noncommutativity

In this paper we couple noncommutative (NC) vielbein gravity to scalar fields. Noncommutativity is encoded in a star product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg-Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein-Hilbert and Klein-Gordon actions are organized in successive powers of the noncommutativity parameter θ^{AB}.

hep-th↗

arXiv2012

Twisting all the way: from algebras to morphisms and connections

Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules, where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist F of H we then quantize (deform) H to H^F, A to A_\star and correspondingly the category of left H-modules and A-bimodules to the category of left H^F-modules and A_\star-bimodules. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A_\star-bimodule connections. Their curvatures and those on tensor product modules are also determined.

math.QA↗

arXiv2012

Extended gravity from noncommutativity

We review the first order theory of gravity (vierbein formulation) on noncommutative spacetime studied in [1, 2]. The first order formalism allows to couple the theory to fermions. This NC action is then reinterpreted (using the Seiberg-Witten map) as a gravity theory on commutative spacetime that contains terms with higher derivatives and higher powers of the curvature and depend on the noncommutativity parameter θ. When the noncommutativity is switched off we recover the usual gravity action coupled to fermions. The first nontrival corrections to the usual gravity action coupled to fermions are presented in a manifest Lorentz invariant form.

gr-qc↗

arXiv2012

Noncommutative gravity at second order via Seiberg-Witten map

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter θ. We thus express the action at order θ^n+2 in terms of noncommutative fields of order at most θ^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most θ^n. This in particular provides the explicit expression of the action at order θ^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.

hep-th↗

arXiv2012

Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map

We use the Seiberg-Witten map (SW map) to expand noncommutative gravity coupled to fermions in terms of ordinary commuting fields. The action is invariant under general coordinate transformations and local Lorentz rotations, and has the same degrees of freedom as the commutative gravity action. The expansion is given up to second order in the noncommutativity parameter θ. A geometric reformulation and generalization of the SW map is presented that applies to any abelian twist. Compatibility of the map with hermiticity and charge conjugation conditions is proven. The action is shown to be real and invariant under charge conjugation at all orders in θ. This implies the bosonic part of the action to be even in θ, while the fermionic part is even in θ for Majorana fermions.

hep-th↗

arXiv2011

Non-commutative Einstein equations and Seiberg-Witten map

The Seiberg--Witten map is a powerful tool in non-commutative field theory, and it has been recently obtained in the literature for gravity itself, to first order in non-commutativity. This paper, relying upon the pure-gravity form of the action functional considered in Ref. 2, studies the expansion to first order of the non-commutative Einstein equations, and whether the Seiberg--Witten map can lead to a solution of such equations when the underlying classical geometry is Schwarzschild.

hep-th↗

arXiv2009

Noncommutative Gravity Solutions

We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented, including the fundamental theorem of noncommutative Riemannian geometry. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that are compatible with the curved spacetime metric.

hep-th↗

arXiv2009

Star Product Geometries

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. This way symmetry principles can be implemented. We review two main examples [15]-[18]: a) general covariance in noncommutative spacetime. This leads to a noncommutative gravity theory. b) Symplectomorphims of the algebra of observables associated to a noncommutative configuration space. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e., we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. New results concerning noncommutative gravity include the Cartan structural equations for the torsion and curvature tensors, and the associated Bianchi identities. Concerning scalar field theories the deformed algebra of classical and quantum observables has been understood in terms of a twist within the algebra.

math.QA↗

arXiv2009

Noncommutative supergravity in D=3 and D=4

We present a noncommutative D=3, N=1 supergravity, invariant under diffeomorphisms, local U(1,1) noncommutative \star-gauge transformations and local \star-supersymmetry. Its commutative limit is the usual D=3 pure supergravity, without extra fields. A noncommutative deformation of D=4, N=1 supergravity is also obtained, reducing to the usual simple supergravity in the commutative limit. Its action is invariant under diffeomorphisms and local GL(2,C) \star-gauge symmetry. The supersymmetry of the commutative action is broken by noncommutativity. Local \star-supersymmetry invariance can be implemented in a noncommutative D=4, N=1 supergravity with chiral gravitino and complex vierbein.

hep-th↗

arXiv2009

Noncommutative D=4 gravity coupled to fermions

We present a noncommutative extension of Einstein-Hilbert gravity in the context of twist-deformed space-time, with a $\star$-product associated to a quite general triangular Drinfeld twist. In particular the $\star$-product can be chosen to be the usual Groenewald-Moyal product. The action is geometric, invariant under diffeomorphisms and centrally extended Lorentz $\star$-gauge transformations. In the commutative limit it reduces to ordinary gravity, with local Lorentz invariance and usual real vielbein. This we achieve by imposing a charge conjugation condition on the noncommutative vielbein. The theory is coupled to fermions, by adding the analog of the Dirac action in curved space. A noncommutative Majorana condition can be imposed, consistent with the $\star$-gauge transformations. Finally, we discuss the noncommutative version of the Mac-Dowell Mansouri action, quadratic in curvatures.

hep-th↗

arXiv2008

Duality Rotations in Nonlinear Electrodynamics and in Extended Supergravity

We review the general theory of duality rotations which, in four dimensions, exchange electric with magnetic fields. Necessary and sufficient conditions in order for a theory to have duality symmetry are established. A nontrivial example is Born-Infeld theory with n abelian gauge fields and with Sp(2n,R) self-duality. We then review duality symmetry in supergravity theories. In the case of N=2 supergravity duality rotations are in general not a symmetry of the theory but a key ingredient in order to formulate the theory itself. This is due to the beautiful relation between the geometry of special Kaehler manifolds and duality rotations.

hep-th↗

arXiv2008

Dynamical noncommutativity and Noether theorem in twisted phi^*4 theory

A \star-product is defined via a set of commuting vector fields X_a = e_a^μ(x) \partial_μ, and used in a phi^*4 theory coupled to the e_a^μ(x) fields. The \star-product is dynamical, and the vacuum solution phi =0, e_a^μ(x)=delta_a^μreproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy-momentum and angular momentum tensors are explicitly derived.

hep-th↗

arXiv2007

Twisting all the way: from Classical Mechanics to Quantum Fields

We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. In particular commutation relations among creation and annihilation operators are deduced.

hep-th↗

arXiv2007

Lectures on Hopf Algebras, Quantum Groups and Twists

Lead by examples we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vectorfields). The examples of the quantum sl(2) Lie algebra and of the quantum (twisted) Poincare Lie algebra iso_θ(3,1) are presented.

hep-th↗