Commutative Field Research Articles

The role of the Seiberg–Witten field redefinition in renormalization of noncommutative chiral electrodynamics

It has been conjectured in the literature that renormalizability of the θ-expanded noncommutative gauge theories improves when one takes into account full nonuniqueness of the Seiberg–Witten expansion which relates noncommutative (‘high-energy’) with commutative (‘low-energy’) fields. In order to check this conjecture we analyze renormalizability of the θ-expanded noncommutative chiral electrodynamics by quantizing the action which contains all terms implied by this nonuniqueness. After renormalization we arrive at a different theory, characterized by different relations between the coupling constants: this means that the θ-expanded noncommutative chiral electrodynamics is not renormalizable.

Real elements in T n (K)

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL n (K).

Noncommutative gravity at second order via Seiberg-Witten map

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter \theta. We thus express the action at order \theta^n+2 in terms of noncommutative fields of order at most \theta^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most \theta^n. This in particular provides the explicit expression of the action at order \theta^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.

Renormalization of the Commutative Scalar Theory with Harmonic Term to All Orders

The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.

Large affine spaces of non-singular matrices

Let $\mathbb {K}$ be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of $\operatorname {M}_n(\mathbb {K})$ consisting only of non-singular matrices must have a dimension less than or equal to $\binom {n}{2}$. Here, we classify, up to equivalence, the subspaces whose dimension equals $\binom {n}{2}$. This is done by classifying, up to similarity, all the $\binom {n}{2}$-dimensional linear subspaces of $\operatorname {M}_n(\mathbb {K})$ consisting of matrices with no non-zero invariant vector, reinforcing a classical theorem of Gerstenhaber. Both classifications only involve the quadratic structure of the field $\mathbb {K}$.

AdS-inspired noncommutative gravity on the Moyal plane

We consider noncommutative gravity on a space with canonical noncommutativity that is based on the commutative MacDowell-Mansouri action. Gravity is treated as gauge theory of the noncommutative $SO(1,3)_\star$ group and the Seiberg-Witten (SW) map is used to express noncommutative fields in terms of the corresponding commutative fields. In the commutative limit the noncommutative action reduces to the Einstein-Hilbert action plus the cosmological term and the topological Gauss-Bonnet term. After the SW expansion in the noncommutative parameter the first order correction to the action, as expected, vanishes. We calculate the second order correction and write it in a manifestly gauge covariant way.

Extended gravity theories from dynamical noncommutativity

In this paper we couple noncommutative vielbein gravity to scalar fields. Noncommutativity is encoded in a $$\star $$ -product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg–Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein–Hilbert and Klein–Gordon actions are organized in successive powers of the noncommutativity parameter $$\theta ^{AB}$$ .

Spaces of matrices with a sole eigenvalue

Let 𝕂 be an arbitrary (commutative) field and be an algebraic closure of it. Let V be a linear subspace of M n (𝕂), with n ≥ 3. We show that if every matrix of V has at most one eigenvalue in 𝕂, then . If every matrix of V has a sole eigenvalue in and , we show that V is similar to the space of all upper-triangular matrices with equal diagonal entries, except if n = 3 and 𝕂 has characteristic 3, or if n = 4 and 𝕂 has characteristic 2. In both of those special cases, we classify the exceptional solutions up to similarity.

NONCOMMUTATIVE FIELD THEORY: NUMERICAL ANALYSIS WITH THE FUZZY DISK

The fuzzy disk is a discretization of the algebra of functions on the two-dimensional disk using finite matrices which preserves the action of the rotation group. We define a φ4 scalar field theory on it and analyze numerically three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already present in the commutative scalar field theory and a nonuniform ordered phase as noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disk seems to be better. We have also performed the limits for the theory in the cases of the theory going to the commutative plane or commutative disk. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g of prime order p > 3 if and only if p divides exactly one of the integers q − 1, q, q + 1, q2 + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p = 3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p = 3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.

A Point Model for the Free Cyclic Submodules over Ternions

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T 2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.

Large affine spaces of matrices with rank bounded below

Let K be an arbitrary (commutative) field with at least three elements, and let n, p and r be positive integers with r⩽min(n,p). In a recent work [11], we have proved that an affine subspace of Mn,p(K) containing only matrices of rank greater than or equal to r must have a codimension greater than or equal to (r+12). Here, we classify, up to equivalence, these subspaces of minimal codimension (r+12). This uses our recent classification [10] of the affine subspaces of Mr(K) contained in GLr(K) which have the maximal dimension (r2).

The linear preservers of non-singularity in a large space of matrices

Let K be an arbitrary (commutative) field, and V be a linear subspace of Mn(K) such that codim V<n-1. Using a recent generalization of a theorem of Atkinson and Lloyd [9], we show that every linear embedding of V into Mn(K) which strongly preserves non-singularity must be M↦PMQ or M↦PMTQ for some pair (P,Q) of non-singular matrices of Mn(K), unless n=3, codim V=1 and K≃F2. This generalizes a classical theorem of Dieudonné with a similar strategy of proof. Weak linear preservers are also discussed, as well as the exceptional case of a hyperplane of M3(F2).

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T 2. This set of points can be represented as a set of planes in the projective space over F 6. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.

Locally nilpotent derivations with a PID ring of constants

Let K \mathcal {K} be a commutative field of characteristic zero, A \mathcal {A} be a domain containing K \mathcal {K} and ∂ \partial be a locally nilpotent K \mathcal {K} -derivation of A \mathcal {A} . We give in this paper a description of the differential K \mathcal {K} -algebra ( A , ∂ ) (\mathcal {A},\partial ) under the assumptions that the ring of constants A ∂ \mathcal {A}^{\partial } of ∂ \partial is a PID, ∂ \partial is fixed point free and its special fibers are reduced.

Index-modules and applications

Let K be a commutative field, \({A\subseteq K}\) be a Dedekind ring and V be a K-vector space. For any pair of A-lattices R ≠ 0 and S of V, we define an A-submodule \({\left[R : S\right]^{\prime}_{A}}\) of K, their A-index-module. Once the basic properties of these modules are stated, we show that this notion can be used to recover more usual ones: the group-index, the relative invariant, the Fitting ideal of R/S when \({S\subseteq R}\), and the generalized index of Sinnott. As an example, we consider the following situation. Let F/k be a finite abelian extension of global function fields, with Galois group G, and degree g. Let ∞ be a place of k which splits completely in F/k. Let \({{\mathcal O}_{F}}\) be the ring of functions of F, which are regular outside the places of F sitting over ∞. Then one may use Stark units to define a subgroup \({\mathcal E_F}\) of \({{\mathcal O}_{F}^{\times}}\), the group of units of \({\mathcal O_F}\). We use the notion of index-module to prove that for every nontrivial irreducible rational character ψ of G, the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb Z\left(\mathcal O_F^\times/\mathcal E_F\right)}\) and the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb ZCl(\mathcal O_F)}\) have the same order.

Noncommutative duplicate and Leibniz algebras

Let K be a commutative field of characteristic zero. Let A be a finite-dimensional algebra over K, not necessarily commutative and D(A) be the noncommutative duplicate of A. We give here necessary and sufficient conditions for D(A) to be a Leibniz algebra.

Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

The pexiderized Goła̧b–Schinzel functional equation

Let X be a linear space over a commutative field K . We characterize a general solution f , g , h , k : X → K of the pexiderized Goła̧b–Schinzel equation f ( x + g ( x ) y ) = h ( x ) k ( y ) , as well as real continuous solutions of the equation.

Affine Tallini Sets and Grassmannians

Abstract In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr(h, 𝔸) of index h of an affine space 𝔸 of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families of maximal singular subspaces of Gr(h, 𝔸) and, till now, their result represents the only known characterization of Gr(h, 𝔸). If K is a commutative field and 𝔸 has finite dimension m, then the image under the well known Plücker morphism is a proper subset of PG(M, K), , called the affine Grassmannian of the h-subspaces of 𝔸. The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of from the point-line geometry point of view. More precisely, we prove that if a projective space over a skew-field K contains an Affine Tallini Set Δ satisfying suitable axioms on “perp” of lines, then the skew-field K is forced to be a commutative field and Δ is an affine Grassmannian, up to projections. Furthermore, several results concerning Affine Tallini Sets are stated and proved.