Commutative Manifolds Research Articles

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.

Commutative deformations of general relativity: nonlocality, causality, and dark matter

Hopf algebra methods are applied to study Drinfeld twists of (3+1)-diffeomorphisms and deformed general relativity on commutative manifolds. A classical nonlocality length scale is produced above which microcausality emerges. Matter fields are utilized to generate self-consistent Abelian Drinfeld twists in a background independent manner and their continuous and discrete symmetries are examined. There is negligible experimental effect on the standard model of particles. While baryonic twist producing matter would begin to behave acausally for rest masses above {sim }1–10 TeV, other possibilities are viable dark matter candidates or a right-handed neutrino. First order deformed Maxwell equations are derived and yield immeasurably small cosmological dispersion and produce a propagation horizon only for photons at or above Planck energies. This model incorporates dark matter without any appeal to extra dimensions, supersymmetry, strings, grand unified theories, mirror worlds, or modifications of Newtonian dynamics.

Electrodynamics from noncommutative geometry

Within the framework of Connes’ noncommutative geometry, the notion of an almost commutative manifold can be used to describe field theories on compact Riemannian spin manifolds. The most notable example is the derivation of the Standard Model of high energy physics from a suitably chosen almost commutative manifold. In contrast to such a non-abelian gauge theory, it has long been thought impossible to describe an abelian gauge theory within this framework. The purpose of this paper is to improve on this point. We provide a simple example of a commutative spectral triple based on the two-point space and show that it yields a U(1)-gauge theory. Then we slightly modify the spectral triple such that we obtain the full classical theory of electrodynamics on a curved background manifold.

Non-constant non-commutativity in 2d field theories and a new look at fuzzy monopoles

We write down scalar field theory and gauge theory on two-dimensional non-commutative spaces M with non-vanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of M going to (i) a commutative manifold M 0 having non-vanishing curvature and (ii) the non-commutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the Kähler structure in the limit (i) and identify the symplectic two-form with the volume two-form. As an example, we take M to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the non-commutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The non-commutative gauge field strength transforms in the usual manner, but the same is not, in general, true for the associated potentials. We develop a perturbation scheme to obtain the expression for gauge transformations about limits (i) and (ii). We also obtain the lowest order Seiberg–Witten map to write down corrections to the commutative field equations and show that solutions to Maxwell theory on M 0 are stable under inclusion of lowest order non-commutative corrections. The results are applied to the example of non-commutative AdS 2.

Co)bordism Groups in Quantum PDEs

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prastaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A m 1 1×···×A m s s , with A≡A 1×···×A s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0≤p≤m−1) are canonically associated to any QPDE, Ek ⊂ Ĵk m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0≤p≤3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.