Commutative Theory Research Articles

On the quantum equivalence of commutative and noncommutative Chern–Simons theories at higher orders

We continue our investigation of the quantum equivalence between commutative and noncommutative Chern–Simons theories by computing the complete set of two-loop quantum corrections to the correlation function of a pure open Wilson line and an open Wilson line with a field strength insertion, on the noncommutative side in a covariant gauge. The conjectured perturbative equivalence between the free commutative theory and the apparently interacting noncommutative one requires that the sum of these corrections vanish, and herein we exhibit the remarkable cancellations that enforce this. From this computation we speculate on the form of a possible all-order result for this simplest nonvanishing correlator of gauge invariant observables.

On degenerations of modules

We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0→Z → φ ψ M⊕Z→N→0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.

On lattice-ordered commutative semigroups

Decompositions of elements into intersections of primal elements and into intersections of p-components are studied in certain lattice-ordered commutative semigroups, by making use of the new development in commutative ideal theory without finiteness conditions, due to Fuchs-Heinzer-Olberding [7]. Several results concerning ideals can be phrased as theorems in ‘abstract ideal theory’.

Analyticity and forward dispersion relations in noncommutative quantum field theory

We derive the analytical properties of the elastic forward scattering amplitude of two scalar particles from the axioms of the noncommutative quantum field theory. For the case of only space–space noncommutativity, i.e., θ 0 i =0, we prove the dispersion relation which is similar to the one in commutative quantum field theory. The proof in this case is based on the existence of the analog of the usual microcausality condition and uses the Lehmann–Symanzik–Zimmermann (LSZ) or equivalently the Bogoliubov–Medvedev–Polivanov (BMP) reduction formalisms. The existence of the latter formalisms is also shown. We remark on the general noncommutative case, θ 0 i ≠0, as well as on the nonforward scattering amplitude and mention their peculiarities.

Renormalizability of non(anti)commutative gauge theories with Script N = 1/2 supersymmetry

Non(anti)commutative gauge theories are supersymmetric Yang-Mills and matter system defined on a deformed superspace whose coordinates obey non(anti)commutative algebra. We prove that these theories in four dimensions with N=1/2 supersymmetry are renormalizable to all orders in perturbation theory. Our proof is based on operator analysis and symmetry arguments. In a case when the Grassman-even coordinates are commutative, deformation induced by non(anti)commutativity of the Grassman-odd coordinates contains operators of dimension-four or higher. Nevertheless, they do not lead to power divergences in a loop diagram because of absence of operators Hermitian-conjugate to them. In a case when the Grassman-even coordinates are noncommutative, the ultraviolet-infrared mixing makes the theory renormalizable by the planar diagrams, and the deformed operators are not renormalized at all. We also elucidate relation at quantum level between non(anti)commutative deformation and N=1/2 supersymmetry. We point out that the star product structure dictates a specific relation for renormalization among the deformed operators.

Renormalization of the energy–momentum tensor in non-commutative scalar field theory

We consider the one-loop renormalization of dimension four composite operators and the energy–momentum tensor in non-commutative φ 4 scalar field theory. Proper operator bases are constructed and it is proved that the bare composite operators are expressed via renormalized ones, with the help of a mixing matrix, whose explicit form is calculated. The corresponding matrix elements turn out to differ from the commutative theory. The canonically defined energy–momentum tensor is not finite and must be replaced by the “improved” one, in order to provide finiteness. The suitable “improving” terms are found.

Commutative and noncommutative \U0001d4a9 = 2 SYM in (2 + 1) from wrapped D6-branes

We give the supergravity duals of commutative and noncommutative non-Abelian gauge theories with \U0001d4a9 = 2 in 2 + 1 dimensions. The moduli space on the Coulomb branch of these theories is studied using supergravity.

A distance between orbits that controls commutator estimates and invertibility of operators

A distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits, thus proving a version of Shneiberg's theorem for orbital methods. The same machinery is used to extend the celebrated Rochberg–Weiss commutator theorem to the setting of orbital methods. It is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element.

Multivariable Nehari problem and interpolation

In this paper we obtain a multivariable Nehari type theorem for Hankel operators, extending the classical result as well as Treil–Volberg generalization. The result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by Treil and Volberg, and recently generalized by Biswas, Foiaş, and Frazho. An extension of I.S. Iokhvidov–Ky Fan theorem is obtained and used to prove a multivariable Adamian–Arov–Krein type theorem for generalized Hankel operators. As consequences, we obtain Carathéodory and Nevanlinna–Pick type interpolation results for “meromorphic” operators on Fock spaces (resp. operator-valued meromorphic functions on the unit ball of C n ). The Nehari type results are used to obtain descriptions of Hankel operators acting on Fock spaces or weighted Fock spaces (multivariable Dirichlet type spaces) and to solve new “weighted” interpolation problems (e.g. Sarason) for noncommutative analytic Toeplitz algebras, which have consequences to the operator-valued analytic interpolation in the unit ball of C n . Moreover, using central intertwining liftings, we obtain explicit formulas for certain solutions of the weighted interpolation problems. As another application of the generalized noncommutative commutant lifting theorem, we derive an abstract interpolation problem for multipliers. This is used to solve a left tangential Nevanlinna–Pick type interpolation problem with operatorial argument for noncommutative analytic Toeplitz algebras.

Quasi-closed primary components in abelian group rings

All groups considered in this work are assumed to be multiplicatively written abelian groups and all rings are commutative with identity of prime characteristic p for some fixed prime p. We follow essentially throughout the notation and terminology to the abelian group theory of the excellent classical monographs of L. Fuchs [8]. All topological references are to the p-adic topology. For G a group and R a ring, RG will denote a group ring with a normed Sylow p-subgroup S(RG), which is in the focus of our interest. The notations and terminology from the commutative group algebras theory of the nice book of G. Karpilovsky [10] will be followed. This paper is a supplement and a generalization to our previous articles [6, 7]. The main purpose that motivates the present research is the global investigation of the quasicompleteness of S(RG) for large R and G on their minimal restrictions. In particular, as corollaries to our main results, we will obtain well-documented facts in [7] and other given by us in [6]. Before proving the central theorems, we need in the sequel some assertions stated in the following paragraph.

On the integral closure of a half-factorial domain

A half-factorial domain (HFD), R, is an atomic integral domain where given any two products of irreducible elements of R: α 1α 2⋯α n=β 1β 2⋯β m then n= m. As a natural generalization of unique factorization domains (UFD), one wishes to investigate which “good” properties of UFDs that HFDs possess. In particular, it has been conjectured that the integral closure of a half-factorial domain is again a HFD (see Non-Noetherian Commutative Ring Theory, Mathematics and its applications, Vol. 520, Kluwer, Dordrecht, 2000, pp. 97–115. for example). In this paper we produce an example that demonstrates that the integral closure of a HFD does not even have to be atomic. We shall investigate the failure of this conjecture closely and highlight some cases where the conjecture does indeed hold.

Lattice perturbation theory in noncommutative geometry and parity anomaly in 3d noncommutative QED

We formulate lattice perturbation theory for gauge theories in noncommutative geometry. We apply it to three-dimensional noncommutative QED and calculate the effective action induced by Dirac fermions. In particular "parity invariance" of a massless theory receives an anomaly expressed by the noncommutative Chern-Simons action. The coefficient of the anomaly is labelled by an integer depending on the lattice action, which is a noncommutative counterpart of the phenomenon known in the commutative theory. The parity anomaly can also be obtained using Ginsparg-Wilson fermions, where the masslessness is guaranteed at finite lattice spacing. This suggests a natural definition of the lattice-regularized Chern-Simons theory on a noncommutative torus, which could enable nonperturbative studies of quantum Hall systems.

Various Facets of Rings Between D[X] and K[X

Since the circulation, in 1974, of the first draft of “The construction D + XD S [X], J. Algebra 53 (1978), 423–439” a number of variations of this construction have appeared. Some of these are: The generalized D + M construction, the A + (X)B[X] construction, with X a single variable or a set of variables, and the D + I construction (with I not necessarily prime). These constructions have proved their worth not only in providing numerous examples and counter examples in commutative ring theory, but also in providing statements that often turn out to be forerunners of results on general pullbacks. The aim of this paper will be to discuss these constructions and the remarkable uses they have been put to. I will concentrate more on the A + XB[X] construction, its basic properties and examples arising from it.

The weighted commutant lifting theorem in the coupling approach

A simple coupling argument is seen to provide an alternate proof of the weighted commutant lifting theorem of Biswas, Foias and Frazho (which includes, as a particular case, the abstract Nehari theorem of Treil and Volberg).

Induced commutative Chern Simons term from noncommutativity in planar systems

We test the consistency of the use of a noncommutative theory description for charged particles in a strong magnetic field, by deriving the induced Chern-Simons (CS) term for an external Abelian gauge field in 2+1 dimensions. In this description, the system is modeled by a noncommutative matter field coupled to a U(1) noncommutative gauge field, related to the original, commutative one, by a Seiberg-Witten transformation. We show that an Abelian CS term for the commutative gauge field is indeed induced, and moreover that it matches the result of previous commutative field theory calculations.

Open-String Actions and Noncommutativity Beyond the Large-BLimit

In the limit of large, constant B-field (the ``Seiberg-Witten limit''), the derivative expansion for open-superstring effective actions is naturally expressed in terms of the symmetric products *n. Here, we investigate corrections around the large-B limit, for Chern-Simons couplings on the brane and to quadratic order in gauge fields. We perform a boundary-state computation in the commutative theory, and compare it with the corresponding computation on the noncommutative side. These results are then used to examine the possible role of Wilson lines beyond the Seiberg-Witten limit. To quadratic order in fields, the entire tree-level amplitude is described by a metric-dependent deformation of the *2 product, which can be interpreted in terms of a deformed (non-associative) version of the Moyal * product.

Scaling properties of the perturbative Wilson loop in two-dimensional noncommutative Yang-Mills theory

Commutative Yang-Mills theories in $1+1$ dimensions exhibit an interesting interplay between geometrical properties and $U(N)$ gauge structures: in the exact expression of a Wilson loop with n windings a nontrivial scaling intertwines n and N. In the noncommutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger gauge group $U(\ensuremath{\infty}).$ We perform an explicit perturbative calculation of such a loop up to $\mathcal{O}{(g}^{6});$ rather surprisingly, we find that in the contribution from the crossed graphs (the genuine noncommutative terms) the scaling we mentioned occurs for large n and N in the limit of maximal noncommutativity $\ensuremath{\theta}=\ensuremath{\infty}.$ We present arguments in favor of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series.

D1/D5 system and Wilson loops in (non)commutative gauge theories

We study the behavior of the Wilson loop in the (5+1)-dimensional supersymmetric Yang–Mills theory with the presence of the solitonic object. Using the dual string description of the Yang–Mills theory that is given by the D1/D5 system, we estimate the Wilson loops both in the temporal and spatial cases. For the case of the temporal loop, we obtain the velocity dependent potential. For the spatial loop, we find that the area law is emerged due to the effect of the D1-branes. Further, we consider D1/D5 system in the presence of the constant B field. It is found that the Wilson loop obeys the area law for the effect of the noncommutativity.

The Renormalizability of Gauge Theory on a Non-Commutative Plane

We perform a non-perturbative study of pure gauge theory in a two dimensional non-commutative (NC) space. On the lattice, it is equivalent to the twisted Eguchi-Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large-N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. Based on a Morita equivalence, the large-N double scaling limit corresponds to the continuum limit of NC gauge theory, so the observed large-N scaling demonstrates the non-perturbative renormalizability of this NC field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter.

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.