Induced gauge theory on a noncommutative space

We consider a scalar φ⁴ theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1-loop effective action, using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model. PACS 11.10.Nx; 11.15.-q     

The Action Functional for Moyal Planes

Modulo some natural generalizations to noncompact spaces, we show in this Letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the expected result, i.e. the noncommutative Yang–Mills action associated with the Moyal product. In particular, we show that Moyal gauge theory naturally fits into the rigorous framework of noncommutative geometry.     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000     

Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism

We consider noncommutative gauge theory defined by means of Seiberg–Witten maps for an arbitrary semisimple gauge group. We compute the one-loop UV divergent matter contributions to the gauge field effective action to all orders in the noncommutative parameters θ. We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group. We use path-integral methods in the framework of dimensional regularisation and consider arbitrary invertible Seiberg–Witten maps that are linear in the matter fields. Surprisingly, it turns out that the UV divergent parts of the matter contributions are proportional to the noncommutative Yang–Mills action where traces are taken over the representation of the matter fields; this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.     

From Large N Matrices to the Noncommutative Torus

We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the C ^*-algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of noncommutative gauge theories using finite dimensional matrix models, and it shows precisely how the continuum limits of these models must be taken. We clarify various aspects of Morita equivalence using this formalism and describe some applications to noncommutative Yang–Mills theory. Received: 24 December 1999 / Accepted: 7 October 2000     

String Geometry and the Noncommutative Torus

We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive Dirac fermion fields. Received: 26 October 1998/ Accepted: 9 April 1999     

Renormalisation of <Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript>-Theory on Noncommutative ℝ<Superscript>4</Superscript> in the Matrix Base

We prove that the real four-dimensional Euclidean noncommutative ⁴-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative ⁴. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.     

A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant d=4 operator F _{μ ν} ²(x) to all orders of perturbation theory in pure Yang–Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other d=4 gauge variant operators, which we identify explicitly. We determine the mixing matrix Z to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix Γ derived from Z. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.     

Instantons and Yang–Mills Flows on Coset Spaces

We consider the Yang–Mills flow equations on a reductive coset space G/H and the Yang–Mills equations on the manifold \mathbbR×G/H{\mathbb{R}\times G/H}. On non-symmetric coset spaces G/H one can introduce geometric fluxes identified with the torsion of the spin connection. The condition of G-equivariance imposed on the gauge fields reduces the Yang–Mills equations to f⁴{\phi^4}-kink equations on \mathbbR{\mathbb{R}}. Depending on the boundary conditions and torsion, we obtain solutions to the Yang–Mills equations describing instantons, chains of instanton–anti-instanton pairs or modifications of gauge bundles. For Lorentzian signature on \mathbbR×G/H{\mathbb{R}\times G/H}, dyon-type configurations are constructed as well. We also present explicit solutions to the Yang–Mills flow equations and compare them with the Yang–Mills solutions on \mathbbR×G/H{\mathbb{R}\times G/H}.     

Renormalization of the Asymptotically Expanded Yang–Mills Spectral Action

We study renormalizability aspects of the spectral action for the Yang–Mills system on a flat 4-dimensional background manifold, focusing on its asymptotic expansion. Interpreting the latter as a higher-derivative gauge theory, a power-counting argument shows that it is superrenormalizable. We determine the counterterms at one-loop using zeta function regularization in a background field gauge and establish their gauge invariance. Consequently, the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action.     

One-loop beta functions for the orientable non-commutative Gross–Neveu model

We compute at the one-loop order the β-functions for a renormalisable non-commutative analog of the Gross–Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The β-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.     

A Noncommutative Generalization of the Free-Field Yang–Mills Equations

The purpose of this paper is to propose a noncommutative generalization of a gauge connection and the free-field Yang–Mills equations. The paper draws upon the techniques proposed by Heller et al. for the noncommutative generalization of the Einstein field equations.     

Non-minimal pp-wave Einstein–Yang–Mills–Higgs model: color cross-effects induced by curvature

Non-minimal interactions in the pp-wave Einstein–Yang–Mills–Higgs (EYMH) model are shown to give rise to color cross-effects analogous to the magneto-electricity in the Maxwell theory. In order to illustrate the significance of these color cross-effects, we reconstruct the effective (associated, color, and color-acoustic) metrics for the pp-wave non-minimal seven-parameter EYMH model with parallel gauge and scalar background fields. Then these metrics are used as hints for obtaining explicit exact solutions of the non-minimally extended Yang–Mills and Higgs equations for the test fields propagating in the vacuum interacting with curvature. The influence of the non-minimal coupling on the test particle motion is interpreted in terms of the so-called trapped surfaces, introduced in the Analog Gravity theory.     

UV/IR duality in noncommutative quantum field theory

We review the construction of renormalizable noncommutative Euclidean ϕ⁴-theories based on the UV/IR duality covariant modification of the standard field theory, and how the formalism can be extended to scalar field theories defined on noncommutative Minkowski space.     

Deformation Quantization and the Baum–Connes Conjecture

 Alternative titles of this paper would have been ``Index theory without index' or ``The Baum–Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C ^* -algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl–Moyal quantization on manifolds, C ^* -algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum–Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum–Connes conjecture in terms of twisted Weyl–Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum–Connes Conjecture actually suggests a strategy for quantizing such singular spaces. Received: 30 April 2002 / Accepted: 2 October 2002 Published online: 17 April 2003 RID="⋆" ID="⋆" Supported by a Fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW). Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One

We study correlation functions in topologically twisted , d=4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b _2,+(X)>0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over =1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU(N) , d=4 supersymmetric Yang–Mills theory. Received: 30 March 1998 / Accepted: 17 April 1998     

The ADHM Construction and Non-local Symmetries of the Self-dual Yang–Mills Equations

We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat \mathbbR⁴{\mathbb{R}^4}. We show how all such connections lie in the orbit of the flat connection on \mathbbR⁴{\mathbb{R}^4} under the action of non-local symmetries of the self-dual Yang–Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang–Mills equations that are analogous to harmonic maps of finite type.     

U(1) gauge field theory on κ-Minkowski space

This study of U(1) gauge field theory on the kappa-deformed Minkowski space-time extends previous work on gauge field theories on this type of noncommutative space-time.We construct the conserved gauge current, fix part of the ambiguities in the Seiberg-Witten map and obtain an effective U(1) action invariant under the action of the undeformed Poincare group. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.