Noncommutative Differential Geometry Research Articles

Monopoles, vortices, and kinks in the framework of noncommutative geometry

Non-commutative differential geometry allows a scalar field to be regarded as a gauge connection, albeit on a discrete space. We explain how the underlying gauge principle corresponds to the independence of physics on the choice of vacuum state, should it be non-unique. A consequence is that Yang-Mills-Higgs theory can be reformulated as a generalised Yang-Mills gauge theory on Euclidean space with a $Z_2$ internal structure. By extending the Hodge star operation to this non-commutative space, we are able to define the notion of self-duality of the gauge curvature form in arbitrary dimensions. It turns out that BPS monopoles, critically coupled vortices, and kinks are all self-dual solutions in their respective dimensions. We then prove, within this unified formalism, that static soliton solutions to the Yang-Mills-Higgs system exist only in one, two and three spatial dimensions.

An estimate of the Higgs boson mass in two loop approximation in a noncommutative differential geometry

An estimation of the Higgs boson mass is performed by numerically solving the renormalization group equations in the two loop approximation based on the condition $g^2=(5/3)g'^2=4\lambda$ for SU(2) $_{\rm \tiny L}$ , U(1) $_{\rm\tiny Y}$ gauge and the Higgs quartic coupling constants, respectively. This condition is introduced in the new scheme of our noncommutative differential geometry (NCG) for the reconstruction of the standard model. However, contrary to ${\rm SU(5)}$ GUT without supersymmetry, the grand unification of coupling constants is not realized in this scheme. The physical mass of the Higgs boson depends strongly on the top quark mass $m_{\rm\scriptsize top}$ through the Yukawa coupling of the top quark in the $\beta$ functions. The two loop effect lowers the numerical value calculated within the one loop approximation by several GeV. The Higgs boson mass varies from 150.93 GeV to 167.96 GeV corresponding to $169.47\,{\rm GeV}\leq m_{\rm\scriptsize top}\leq 181.00\,{\rm GeV}$ . We find $m_{\rm\tiny H}=158.90$ GeV for $ m_{\rm\scriptsize top}=175.01$ GeV and $m_{\rm\tiny H}=166.98$ GeV for $m_{\rm\scriptsize top}=180.37$ GeV.

R-matrix approach to differential calculus on quantum groups

New symmetries (quantum symmetries) formulated in the language of quantum groups are being used more and more in theoretical and mathematical physics. The present paper is devoted to a discussion of bicovariant differential calculus on quantum groups and quantum vector spaces, and is a sequel to a review published in this journal in 1995. A detailed exposition is given of the bicovariant theory of Woronowicz, which is now regarded as the basis for constructing the noncommutative differential geometry on quantum groups. The R -matrix approach is used to give a complete description of the differential calculus on the group GL q (N) and on linear quantum spaces. It is shown how the differential algebra on the group SL q (N) can be obtained (as a subalgebra) from the differential algebra on GL q (N). The problems of the bicovariant differential calculus on the groups SO q (N) and Sp q (2n) are discussed. Essential supplementary information on the general theory of quantum groups, not covered in the first part of the review, is also given.

Two Higgs Doublet Model in a Noncommutative Geometry on the Discrete Space M4 xZ4

The two Higgs doublet model with the symmetry of SU(3)c x SU(2)L x U(I)y is recon­ structed in a new scheme of a noncommutative differential geometry (NCG) on the discrete space M4 x Z4, which is a product space of Minkowski space and four point space. The characteristic point of this new scheme is to take the fermion field to be a vector in a 24dimensional space which contains all leptons and quarks. Corresponding to this specification, all gauge and Higgs boson fie~ds are represented in 24 x 24 matrix forms. We incorporate the Higgs boson fields hand h = r 2 h*r2 which contain two Higgs doublet hI and h2 trans forming as (2,1) and (2, -1) under SU(2)L x U(I), respectively. h and It are properly placed on the discrete space Z4. Owing to the new algebraic rules adopted in this article, we can obtain the necessary potential and interacting terms between these Higgs bosons, which are responsible for giving masses to the particles included. Up and down quarks have different masses through the vacuum expectation value of h2 and hI, respectively.

Toeplitz-Berezin quantization and non-commutative differential geometry

In this survey article we describe how the recent work in quantization in multi-variable complex geometry (domains of holomorphy, symmetric domains, tube domains, etc.) leads to interesting results and problems in C-algebras which can be viewed as examples of the “non-commutative geometry” in the sense of A. Connes. At the same time, one obtains new functional calculi (of pseudodifferential type) with possible applications to partial differential equations and group representations.

Differentials of Higher Order in Noncommutative Differential Geometry

In differential geometry, the notation dn f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher-order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra ΩA of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of ΩnA) as particular elements of what we call the ‘iterated frame algebra’ of order n, FnA, which is itself defined as the2n tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.

Non-Commutative Differential Geometry on Discrete Space M4 x ZN and Gauge Theory

The algebra of non-commutative differential geometry (NCG) on the discrete space $M_4\times Z\ma{N}$ previously proposed by the present author is improved to give the consistent explanation of the generalized gauge field as the generalized connection on $M_4\times Z\ma{N}$. The nilpotency of the generalized exterior derivative ${\mbf d}$ is easily proved. The matrix formulation where the generalized gauge field is denoted in matrix form is shown to have the same content with the ordinary formulation using {\mbf d}, which helps us understand the implications of the algebraic rules of NCG on $M_4\times Z\ma{N}$. The Lagrangian of spontaneously broken gauge theory which has the extra restriction on the coupling constant of the Higgs potential is obtained by taking the inner product of the generalized field strength. The covariant derivative operating on the fermion field determines the parallel transformation on $M_4\times Z\ma{N}$, which confirms that the Higgs field is the connection on the discrete space. This implies that the Higgs particle is a gauge particle on the same footing as the weak bosons. Thus, it is expected that the mass relation $m\ma{H}=\frac{4}{\sqrt{3}}m\ma{W}\sin\theta\ma{W}$ proposed by the present author holds without any correction in the same way as the mass relation $m\ma{W}=m\ma{Z}\cos\theta\ma{W}$. The Higgs kinetic and potential terms are regarded as the curvatures on $M_4\times Z\ma{N}$.

Connections on central bimodules in noncommutative differential geometry

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.

Super Yang-Mills Theories and Non-Commutative Geometry

It is shown that the scalar potential in super Yang-Mills theories without extra matter fields can be derived by eliminating auxiliary fields in the x-formalism of noncommutative differential geometry to be compared with the algebraic method of Chamseddine. A remark on Sogami's method applied to these theories is also made.

The Einstein action for algebras of matrix valued functions—Toy models

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi–Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M4(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi–Civita connection. In this model, no additional assumptions regarding metrics are made.

Non-commutative geometry, multiscalars, and the symbol map

Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum dynamics which ordinary tensor fields have with respect to classical hamiltonian dynamics.

Covariant noncommutative differential geometry

We use the R-matrix formalism of the quantum inverse scattering method to formulate the noncommutative differential geometry for a noncommutative analog of the algebra of functions on a linear space. The sources of nonuniqueness are pointed out. The case of super-spaces is briefly discussed. Bibliography: 23 titles.

Some models in differential geometry

Some variants of the axiomatics of the algebras of “vector fields” in models of noncommutative differential geometry are considered. In the case of a commutative model (the de Rham complex) a matrix analogue of the Kadomtsev-Petviashvili hierarchy is constructed. The corresponding Sato system is presented. The method of deformations of D-modules is used. Bibliography: 14 titles.

Differential structures in lie superalgebras

We develop a formal construction of an U-system as a fundamental concept of noncommutative differential geometry. By using the notion of “conditional differential” (an analog of the Hamiltonian mapping), we construct a series of brackets that generalize the classical Poisson brackets.

Schwinger terms and cohomology of pseudodifferential operators

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.

A DETAILED ACCOUNT OF ALAIN CONNES’ VERSION OF THE STANDARD MODEL IN NON-COMMUTATIVE DIFFERENTIAL GEOMETRY III

We describe in detail Alain Connes’ last presentation of the (classical level of the) standard model in noncommutative differential geometry, now free of the cumbersome adynamical fields which parasited the initial treatment. Accessorily, the theory is presented in a more transparent way by systematic use of the skew tensor-product structure, and of 2×2 matrices with 2×2 matrix-entries instead of the previous 4×4 matrices.

Noncommutative differential geometry related to the Young-Baxter equation

An analogue of the differential calculus associated with a unitary solution of the quantum Young-Baxter equation is constructed. An example of a ring sheaf is considered in which local solutions of the Young-Baxter quantum equation are defined but there is no global section. Bibliography: 13 titles.

Modified Reconstruction of Standard Model in Non-Commutative Differential Geometry

Sogami recently proposed the new idea to express Higgs particle as a kind of gauge particle by prescribing the generalized covariant derivative with gauge and Higgs fields operating on quark and lepton fields. The field strengths for both the gauge and Higgs fields are defined by the commutators of the covariant derivative by which he could obtain the Yang-Mills Higgs Lagrangian in the standard model. Inspired by Sogami's work, we present a modification of our previous scheme to formulate the spontaneously broken gauge theory in non-commutative geometry on the discrete space; Minkowski space multiplied by two points space by introducing the generation mixing matrix in operation of the generalized derivative on the more fundamental fields a_i(x,y) which compose the gauge and Higgs fields. The standard model is reconstructed according to the modified scheme, which does not yields not only any special relations between the particle masses but also the special restriction on the Higgs potential.

Non-Commutative Differential Geometry and Standard Model

We incorporate Sogami's idea in the standard model into our previous formulation of non-commutative differential geometry by extending the action of the extra exterior derivative operator on spinors defined over the discrete space-time; four dimensinal Minkovski space multiplyed by two point discrete space. The extension consists in making it possible to require that the operator become nilpotent when acting on the spinors. It is shown that the generalized field strength leads to the most general, gauge-invariant Yang-Mills-Higgs Lagrangian even if the extra exterior derivative operator is not nilpotent, while the fermionic part remains intact. The proof is given for a single Higgs model. The method is applied to reformulate the standard model by putting left-handed fermion doublets on the upper sheet and right-handed fermion singlets on the lower sheet with generation mixing among quarks being taken into account. We also present a matrix calculus of the method without referring to the discrete space-time.

SO(10) Grand Unified Theory in Non-Commutative Differential Geometry on the Discrete Space M4 xZN

SO(10) Grand Unified Theory in Non-Commutative Differential Geometry on the Discrete Space M4 xZN