Applications Of Noncommutative Geometry Research Articles

Applications of noncommutative geometry in function theory and mathematical physics

Applications of noncommutative geometry in function theory and mathematical physics

A Value Region Problem for Continued Fractions and Discrete Dirac Equations

Motivated by applications in noncommutative geometry we prove several value range estimates for even convergents and tails, and odd reverse sequences of Stieltjes type continued fractions with bounded ratio of consecutive elements, and show how those estimates control growth of solutions of a system of discrete Dirac equations.

Braided Cartan calculi and submanifold algebras

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten–Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over the full underlying algebra. Furthermore, equivariant covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmetric braided monoidal category we argue that Drinfel’d twist deformation corresponds to gauge equivalences of braided Cartan calculi. The notions of equivariant covariant derivative and metric are compatible with the Drinfel’d functor as well. Moreover, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation.

Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.

Finite-matrix formulation of gauge theories on a non-commutative torus with twisted boundary conditions

We present a novel finite-matrix formulation of gauge theories on a non-commutative torus. Unlike the previous formulation based on a map from a square matrix to a field on a discretized torus with periodic boundary conditions, our formulation is based on the algebraic characterization of the configuration space. This enables us to describe the twisted boundary conditions in terms of finite matrices and hence to realize the Morita equivalence at a fully regularized level. Matter fields in the fundamental representation turn out to be represented by rectangular matrices for twisted boundary conditions analogously to the matrix spherical harmonics on the fuzzy sphere with the monopole background. The corresponding Ginsparg-Wilson Dirac operator defines an index, which can be used to classify gauge field configurations into topological sectors. We also perform Monte Carlo calculations for the index as a consistency check. Our formulation is expected to be useful for applications of non-commutative geometry to various problems related to topological aspects of field theories and string theories.