arXiv research

Develops noncommutative Cowen-Douglas theory for noncommuting operators.

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

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the …

This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…

Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…

Formulates quantum jet bundles over noncommutative algebras with connections and braiding.

Study noncommutative deformations of Calabi-Yau threefolds.

New spectral functionals related to noncommutative residue and torsion Dirac operators.

In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions us…

We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with ass…

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 interpretatio…

Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. I…

In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries ("quantizations") that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and argume…

Derives Atiyah sequence for noncommutative bundles.

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this…

Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and a vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a no…

In this paper we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a…

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Study the complexity of horizontality in 4-torus vector bundles.

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

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct the associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of the homotopy groups of Diff(Z).

The paper computes metrics and Einstein tensors on even-dimensional manifolds.

From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of topological and geometrical aspe…

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…

In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boun…

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fi…

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…

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…

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

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebra…

The paper proves new theorems about specific types of operator perturbations.

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…

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base…

A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…

Let $Λ$ be a finite abelian group. A dynamical system with transformation group $Λ$ is a triple $(A,Λ,α)$, consisting of a unital locally convex algebra $A$, the finite abelian group $Λ$ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of $Λ$ on $A$. In this paper we present a new, geometricall…

Noncommutatively deformed geometries, such as the noncommutative torus, do not exist generically. I showed in a previous paper that the existence of such a deformation implies compatibility conditions between the classical metric and the Poisson bivector (which characterizes the noncommutativity). Here I present anothe…

The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to classical geometers. The definition of vector field used is a specialisation of the Cart…

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…

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the $(2,2)$-dimensional supersphere $S^{2,2}$ by means of global projectors $p$ via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle ove…

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…

Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.

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 …

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in t…

New kernels capture both local and non-local interactions efficiently.