arXiv research

The paper proves a Connes trace theorem for curved noncommutative tori.

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group o…

A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this paper, we exte…

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

As the second part of the sequel, we investigate the variation of rearrangement operators (more precisely, the spectral functions behind) arising in the study of modular geometry on noncommutative (two) tori. We initiate a systematic approach by introducing transformations corresponding to basic operations in calculus,…

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…

We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed.…

We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theore…

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed L…

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two fun…

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time hea…

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three …

We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient f…

The paper introduces a trilinear functional to recover torsion in spectral triples.

Combines noncommutative geometry and spectral theory for new Weyl laws.

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function …

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a com…

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P= - [g^{μν} u(x)\partial_μ\partial_ν+ v^ν(x)\partial_ν+ w(x)]$ where $u,\,v^ν,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. F…

We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the …

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…

The curvature of the noncommutative torus $T^2_θ$ ($θ$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twist…

There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double covering of the quantum group $SO_q(3)$. This work is devoted to a single general …

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of th…

We give a new proof of the rearrangement lemma that works for all dimensions and all heat coefficients in the study of modular geometry on noncommutative tori. The building blocks of the spectral functions are landed in a hypergeometric family knowns as Lauricella functions of type $D$. We investigate the differential …

Develops noncommutative Cowen-Douglas theory for noncommuting operators.

A reverse Riesz estimate and spectral gap imply a Poincaré inequality.

Constructs noncommutative spaces for D-branes on complex algebraic spaces.

D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.

Promotes spectral functionals to noncommutative fields and proves a theorem.

Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metri…

Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.

Defines and proves generalized noncommutative residue theorems for specific dimensions.

The paper explores noncommutative geometry of frame bundles using C*-algebras.

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are co…

Develops Riemannian geometry for noncommutative super surfaces.

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kern…

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential g…

Intrinsic formulation of noncommutative geometry for quantum gravity.

The paper constructs a noncommutative bracket on surface groups and proves it's Hamiltonian.

We study Finsler black holes induced from Einstein gravity as possible effects of quantum spacetime noncommutativity. Such Finsler models are defined by nonholonomic frames not on tangent bundles but on (pseudo) Riemannian manifolds being compatible with standard theories of physics. We focus on noncommutative deformat…

We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We presen…

Derives Atiyah sequence for noncommutative bundles.

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and co…

Generalizes soft noncommutative schemes to flag varieties.

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further…

Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.

The paper extends Lie bracket to noncommutative geometry using differential operators.