Related papers: Geometric Dirac operator on the fuzzy sphere
The product of a non-commutative matrix spectral triple with a simple two-dimensional internal space is considered. This is interpreted as a non-commutative spacetime that contains one charged Dirac fermion and its antiparticle. The inner…
We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and…
The spectral torsion is defined by three vector fields and Dirac operators and the noncommutative residue. Motivated by the spectral torsion and the one form rescaled Dirac operator, we give some new spectral torsion which is the extension…
The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$…
We determine the spectrum of Kostant's cubic Dirac operator $D^{1/3}$ on locally symmetric Lorentzian manifolds of the form $\Gamma\backslash {\rm Osc}_1$, where ${\rm Osc}_1$ is the four-dimensional oscillator group and $\Gamma\subset {\rm…
We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally…
We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called string…
After recalling Snyder's idea of using vector fields over a smooth manifold as `coordinates on a noncommutative space', we discuss a two dimensional toy-model whose `dual' noncommutative coordinates form a Lie algebra: this is the well…
We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey-de Witt coefficients and make explicit calculations…
Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite…
Assume that the compact Riemannian spin manifold $(M^n,g)$ admits a $G$-structure with characteristic connection $\nabla$ and parallel characteristic torsion ($\nabla T=0$), and consider the Dirac operator $D^{1/3}$ corresponding to the…
A building block of noncommutative geometry is the observation that most of the geometric information of a compact riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to…
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(\Omega; \mathbb{C}^4)$, where $\Omega \subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth…
In the last decades, many mathematicians have studied the {\em curl operator} on compact (both with or without empty boundary) three-manifolds, mainly the behaviour of its spectrum and some iso\-pe\-ri\-me\-tric problems associated with it.…
This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global $L$--functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the id\`ele class space, we place a…
We show that the first five of the axioms we had formulated on spectral triples suffice (in a slightly stronger form) to characterize the spectral triples associated to smooth compact manifolds. The algebra, which is assumed to be…
Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it…
We show that the spectral theory of the Dirac operator $D = i\delsl-\sigma(x) -i\pi(x)\gam_5$ in a static background $(\sigma(x),\pi(x))$ in 1+1 space-time dimensions, is underlined by a certain generalization of supersymmetric quantum…
Spectral triples over noncommutative principal $\T^n$-bundles are studied, extending recent results about the noncommutative geometry of principal U(1)-bundles. We relate the noncommutative geometry of the total space of the bundle with the…
We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold.…