Finite Noncommutative Geometry Research Articles

Random finite noncommutative geometries and topological recursion

Random finite noncommutative geometries and topological recursion

Spectral statistics of Dirac ensembles

In this paper, we find spectral properties in the large N limit of Dirac operators that come from random finite noncommutative geometries. In particular, for a Gaussian potential, the limiting eigenvalue spectrum is shown to be universal, regardless of the geometry, and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models, this convolution property is also evident. In order to prove these results, we show that a wide class of multi-trace multimatrix models have a genus expansion.

Spectral estimators for finite non-commutative geometries

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addresses the question of how to extract information about these geometries from the spectrum of the Dirac operator. Since the Dirac operator is a finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no sense and is replaced by measurements of the spectrum at a finite energy scale. The spectral dimension of the square of the Dirac operator is improved to provide a new spectral measure of the dimension of a space called the spectral variance. Similarly, the volume of a space can be computed from the spectrum once the dimension is known. Two methods of doing this are investigated: the well-known Dixmier trace and a recent improvement due to Abel Stern. Finally, the distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these techniques are tested on the explicit examples of the fuzzy spheres and fuzzy tori, which can be regarded as approximations of the usual Riemannian sphere and flat tori. Then they are applied to characterise some random fuzzy spaces using data generated by a Monte Carlo simulation.

Finite Noncommutative Geometries Related to $\mathbb {F}_{p}[x

It is known that irreducible noncommutative differential structures over $\mathbb {F}_{p}[x]$ are classified by irreducible monics m. We show that the cohomology $H_{\text {dR}}^{0}(\mathbb {F}_{p}[x]; m)=\mathbb {F}_{p}[g_{d}]$ if and only if Trace(m)≠ 0, where $g_{d}=x^{p^{d}}-x$ and d is the degree of m. This implies that there are ${\frac {p-1}{pd}}{\sum }_{k|d, p\nmid k}\mu _{M}(k)p^{\frac {d}{k}}$ such noncommutative differential structures (μM the Mobius function). Motivated by killing this zero’th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_{d}=\mathbb {F}_{p}[x]/(g_{d})$ as well as their inherited bicovariant differential calculi Ω(Ad;m). We show that Ad = Cd ⊗χA1 is a cocycle extension where $C_{d}=A_{d}^{\psi }$ is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra Bd of dimension $\frac {1}{d} {\sum }_{k | d} \phi (k) p^{\frac {d}{k}}$ (ϕ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, $A_{1}\cong \mathbb {F}_{p}(\mathbb {Z}/p\mathbb {Z})$, the algebra of functions on the finite group $\mathbb {Z}/p\mathbb {Z}$, and we show dually that $\mathbb {F}_{p}\mathbb {Z}/p\mathbb {Z}\cong \mathbb {F}_{p}[L]/(L^{p})$ for a ‘Lie algebra’ generator L with eL group-like, using a truncated exponential. By contrast, A2 over $\mathbb {F}_{2}$ is a cocycle modification of $\mathbb {F}_{2}((\mathbb {Z}/2\mathbb {Z})^{2})$ and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.

Quantum Isometries of the Finite Noncommutative Geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.

Finding the standard model of particle physics a combinatorial problem

We present a combinatorial problem which consists in finding irreducible Krajewski diagrams from finite geometries. This problem boils down to placing arrows into a quadratic array with some additional constraints. The Krajewski diagrams play a central role in the description of finite noncommutative geometries. They allow to localise the standard model of particle physics within the set of all Yang–Mills–Higgs models. Program summary Title of program:ko0 and ko6 Catalogue identifier:ADZZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:4875 No. of bytes in distributed program, including test data, etc.:29 933 Distribution format:tar.gz Programming language used:C++ Computer:Simple PC Operating system:Any, where C++ runs RAM:Depending on the global parameters ‘DIMENSION’ and ‘MAX_LEVEL’, a few KB to some MB Classification:5, 11.1 Nature of problem:Automatised classification of almost-commutative geometries via Krajewski diagrams. Solution method:Combinatorial generation of all possible Krajewski diagrams and identification of the irreducible ones. Equivalence classes of irreducibles = classification. Running time:Depending on the global parameters ‘DIMENSION’ and ‘MAX_LEVEL’, running time reaches from seconds ( 3 × 3 ) to weeks ( 6 × 6 ) on a standard PC.

Why the Standard Model

The Standard Model is based on the gauge invariance principle with gauge group U ( 1 ) × SU ( 2 ) × SU ( 3 ) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four-dimensional continuum by a finite noncommutative geometry F . The raison d’être for F is to correct the K -theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K -theoretic dimension six and show that the dimension (per generation) is a square of an integer k . Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k = 4 ) with the correct quantum numbers for all fields. The spectral action applied to the product M × F delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input).

Ginsparg-Wilson Relation and Admissibility Condition in Noncommutative Geometry

Ginsparg-Wilson relation and admissibility condition have the key role to construct lattice chiral gauge theories. They are also useful to define the chiral structure in finite noncommutative geometries or matrix models. We discuss their usefulness briefly.

Index Theorem in Finite Noncommutative Geometry

Noncommutative (NC) geometry is interesting since it is related to string theories and matrix models,1)–3) and it may capture some nature of quantum gravity. It is also a candidate for a new regularization of quantum field theory.4) Study about topological aspects of gauge theory on it is important since compactification of extra dimensions with nontrivial index in string theory can realize chiral gauge theory in our spacetime. Ultimately, we hope to realize such a mechanism dynamically, for instance, in IIB matrix model5) where spacetime structure has been studied intensively.6),7) Unusual properties of NC geometry may also provide a solution of strong CP problem and baryon asymmetry of the universe. Index theorem plays a key role in these studies. While it can be proved in theories with infinite degrees of freedom,8),9) it becomes a nontrivial issue in finite cases. This problem was solved in lattice gauge theory by using Ginsparg-Wilson (GW) relation,10)–13) and this idea has been successfully extended to the NC geometry. We have provided a general prescription to construct a GW Dirac operator with coupling to non-vanishing gauge field backgrounds on general finite NC geometries.14) Owing to the GW relation, an index theorem can be proved even for finite NC geometries. The index takes only integer values by construction, and it is shown to become the corresponding topological charge as the number of degrees of freedom is properly taken to infinity. While explicit construction has been provided for the fuzzy 2sphere14),15) and for the NC torus,16),17) it is possible for other cases, which will be reported in future publication. As a topologically nontrivial configuration on the fuzzy 2-sphere, we constructed ’t Hooft-Polyakov (TP) monopole configuration.18),19) We further presented a mechanism for dynamical generation of a nontrivial index, by showing that the TP monopole configurations are stabler than the topologically trivial sector in the YangMills-Chern-Simons matrix model.20) However, in order to obtain non-zero indices for these configurations, one needs to introduce a projection operator in the definition of the index. We gave an interpretation for the projection operator, and extended the index theorem to general configurations which do not necessarily satisfy the equation of motion.21) Since the U(2) gauge theory on the fuzzy sphere

Noncommutative geometry and the standard model with neutrino mixing

We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F-- whose product with classical 4-dimensional space-time gives the standard model coupled with gravity--into a space of KO-dimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in "by hand" but a conceptual understanding of its structure and a classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The spectral action of our joint work with Chamseddine now automatically generates the full standard model coupled with gravity with neutrino mixing and see-saw mechanism for neutrino masses. The predictions of the Weinberg angle and the Higgs scattering parameter at unification scale are the same as in our joint work but we also find a mass relation (to be imposed at unification scale).

Gravity and the standard model with neutrino mixing

We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.

Ginsparg-Wilson Relation, Topological Invariants, and Finite Noncommutative Geometry(Quantum Field Theories: Fundamental Problems and Applications)

Ginsparg-Wilson Relation, Topological Invariants, and Finite Noncommutative Geometry(Quantum Field Theories: Fundamental Problems and Applications)

Path integral quantisation of finite noncommutative geometries

We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries: the two-point space, and the matrix geometry M 2( C) . On the first, the graviton is described by a Higgs field, and on the second, it is described by a gauge field. We start with the partition function and calculate the propagator and Greens functions for the gravitons. The expectation values of distances are evaluated and we discover that distances shrink with increasing graviton excitations. We find that adding fermions reduces the effects of the gravitational field. A comparison is also made with Rovelli’s canonical quantisation approach, and with his idea of spectral path integrals. We include a brief discussion on the quantisation of a Riemannian manifold.

DISTANCES IN NONCOMMUTATIVE GEOMETRY

In this review, we give examples of explicitely computed distances including classical geometries, finite noncommutative geometries and applications to particle physics.

Linear connections on extended space-time

A modification of Kaluza–Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a nontrivial extension to the total geometry of a linear connection on space–time places severe restrictions on the structure of the noncommutative factor. A counter-example is given.

Graded non-commutative geometries

We present a short exposition of graded finite non-commutative geometries. The theory that serves as an example is based on the algebra of matrices M n N . This non-commutative algebra replaces the algebra of functions on a manifold. Consequently, vector fields (differentiations), forms and connections are constructed. The gauge theory can be introduced without the notion of internal manifold. We discuss some physical application, the similarities with the standard model, and the graded version of this geometry.