Searcharxiv⌕ Search

arXiv subjects

Francesco D'Andrea

Publications and source records attributed to Francesco D'Andrea.

At least 37 records · Page 2Linked to original sources
arXiv2018

An equivariant pullback structure of trimmable graph C*-algebras

We prove that the graph C*-algebra $C^*(E)$ of a trimmable graph $E$ is $U(1)$-equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra $C^*(E'')$ and the C*-algebra of functions on a circle tensored with another subgraph C*-algebra $C^*(E')$. This allows us to unravel the structure and K-theory of the fixed-point subalgebra $C^*(E)^{U(1)}$ through the (typically simpler) C*-algebras $C^*(E')$, $C^*(E'')$ and $C^*(E'')^{U(1)}$. As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra $\mathcal{O}_2$ and the Toeplitz algebra $\mathcal{T}$. Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ and the quantum lens space $L_q^3(l; 1,l)$, respectively.

math.KT↗
arXiv2018

On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $δ:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_δ$ denote $H\otimes A$ with the right $A$-module structure induced by~$δ$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism \[Θ:H\otimes M\to (H\otimes A)_δ\otimes_AM\] that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $Θ$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $Θ$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_δ\otimes_AM$ is not an isomorphism.

math.RA↗
arXiv2017

Twist star products and Morita equivalence

We present a simple no-go theorem for the existence of a deformation quantization of a homogeneous space M induced by a Drinfel'd twist: we argue that equivariant line bundles on M with non-trivial Chern class and symplectic twist star products cannot both exist on the same manifold M. This implies, for example, that there is no symplectic star product on the complex projective spaces induced by a twist based on U(gl(n,C))[[h]] or any sub-bialgebra, for every n greater or equal than 2.

math.QA↗
arXiv2016

Pimsner algebras and circle bundles

We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with the examples of quantum weighted projective and lens spaces and theta-deformations.

math.QA↗
arXiv2016

Wick Rotation and Fermion Doubling in Noncommutative Geometry

In this paper, we discuss two features of the noncommmutative geometry and spectral action approach to the Standard Model: the fact that the model is inherently Euclidean, and that it requires a quadrupling of the fermionic degrees of freedom. We show how the two issues are intimately related. We give a precise prescription for the Wick rotation from the Euclidean theory to the Lorentzian one, eliminating the extra degrees of freedom. This requires not only projecting out mirror fermions, as has been done so far, and which leads to the correct Pfaffian, but also the elimination of the remaining extra degrees of freedom. The remaining doubling has to be removed in order to recover the correct Fock space of the physical (Lorentzian) theory. In order to get a Spin(1,3) invariant Lorentzian theory from a Spin(4) invariant Euclidean theory such an elimination must be performed after the Wick rotation. Differences between the Euclidean and Lorentzian case are described in detail, in a pedagogical way.

hep-th↗
arXiv2016

Some remarks on K-lattices and the Adelic Heisenberg Group for CM curves

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).

math.NT↗
arXiv2016

Non-Associative Geometry of Quantum Tori

We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a "principal bundle" construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold.

math.QA↗
arXiv2016

The Standard Model in Noncommutative Geometry and Morita equivalence

We discuss some properties of the spectral triple $(A_F,H_F,D_F,J_F,γ_F)$ describing the internal space in the noncommutative geometry approach to the Standard Model, with $A_F=\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$. We show that, if we want $H_F$ to be a Morita equivalence bimodule between $A_F$ and the associated Clifford algebra, two terms must be added to the Dirac operator; we then study its relation with the orientability condition for a spectral triple. We also illustrate what changes if one considers a spectral triple with a degenerate representation, based on the complex algebra $B_F=\mathbb{C}\oplus M_2(\mathbb{C})\oplus M_3(\mathbb{C})$.

math-ph↗
arXiv2015

Pythagoras Theorem in Noncommutative Geometry

After a review of the results in arXiv:1203.3184 [math-ph] about Pythagorean inequalities for products of spectral triples, I will present some new results and discuss classes of spectral triples and states for which equality holds.

math-ph↗
arXiv2015

Topics in Noncommutative Geometry

The leitmotiv of this review is noncommutative principal U(1)-bundles and associated line bundles. In the first part I give a brief introduction to Hopf-Galois theory and its applications, from field extensions to principal group actions. I then recall Woronowicz' definition of compact quantum group and the notion of noncommutative principal bundle. When the structure group is U(1), there is a construction due to Pimsner that allows to get the total space of a "bundle" (more precisely, a strongly graded C*-algebra) from the base space and a noncommutative "line bundle" (a self-Morita equivalence bimodule). As an example of this construction, I will discuss the U(1)-principal bundles of quantum lens spaces over quantum weighted projective space. The second part is a peek into the realm of nonassociative geometry: after a review of some properties of Hopf cochains and cocycles, I will discuss the theory of cochain quantization and its applications, from Albuquerque-Majid example of octonions, to "line bundles" on the noncommutative torus.

math.QA↗
arXiv2015

Quantum weighted projective and lens spaces

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules associated with the principal bundles and pair them with the Fredholm modules, thus proving their non-triviality.

math.QA↗
arXiv2014

Anti-selfdual Connections On The Quantum Projective Plane: Instantons

We study one-instantons, that is anti-selfdual connections with instanton number 1, on the quantum projective plane with orientation which is reversed with respect to the usual one. The orientation is fixed by a suitable choice of a basis element for the rank 1 free bimodule of top forms. The noncommutative family of solutions is foliated, each non-singular leaves being isomorphic to the quantum projective plane itself.

math.QA↗
arXiv2014

Matrix Geometries Emergent from a Point

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.

math-ph↗
arXiv2014

Modules over the Noncommutative Torus and Elliptic Curves

Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra $A_θ$ of the noncommutative torus. We show that such $A_θ$-modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve $E_τ$, under the condition that the deformation parameter $θ$ and the modular parameter $τ$ satisfy a non-trivial relation.

math.OA↗
arXiv2014

Spectral geometry with a cut-off: topological and metric aspects

Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on Connes distance associated to a spectral triple (A, H, D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov-Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of "state with finite moment of order 1" for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if P has finite rank. When P is a spectral projection of $D$, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejer probability distributions. Finally we apply the results to Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.

math-ph↗
arXiv2013

Quantum gauge symmetries in Noncommutative Geometry

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M_n(R), M_n(C) and M_n(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A_F=C+H+M_3(C) and A^{ev}=H+H+M_4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).

math.QA↗
arXiv2012

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.

math-ph↗