Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
Eric Cagnache, Thierry Masson, Jean-Christophe Wallet
备注：
23 pages, 2 figures. Improved and enlarged version. Some references have been added and updated. Two subsections and a discussion on the appearence of Higgs fiels in noncommutative gauge theories have been added
Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra $M_n({\mathbb{C}})$ and the algebra of matrix valued functions $C^\infty(M)\otimes M_n({\mathbb{C}})$. The UV/IR mixing problem of the resulting Yang-Mills-Higgs models is also discussed.
We explore a differential calculus on the algebra of smooth functions on a manifold. The former is `noncommutative' in the sense that functions and differentials do not commute, in general. Relations with bicovariant differential calculus on certain quantum groups and stochastic calculus are discussed. A similar differential calculus on a superspace is shown to be related to the Batalin-Vilkovisky antifield formalism.
We perform a Noether analysis for a description of translation transformations in 4D kappa-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. Taking properly into account the properties of the differential calculus we arrive at an explicit formula for the conserved charges. We also propose a choice of basis for the 5D calculus which leads to an intuitive description of time derivatives.
In this paper we try to construct noncommutative Yang-Mills theory for generic Poisson manifolds. It turns out that the noncommutative differential calculus defined in an old work is exactly what we need. Using this calculus, we generalize results about the Seiberg-Witten map, the Dirac-Born-Infeld action, the matrix model and the open string quantization for constant B field to non-constant background with H=0.
In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define ``Instanton number'' by the size of $B_α$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes' theorem for th…
We build a differential calculus for subalgebras of the Moyal algebra on R^4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of 1-forms which allows to realize the complex of forms as a tensor product of the noncommutative subalgebras with the external algebra Lambda^*.
In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita's theorem, stating that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivati…
We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.
A noncommutative version of the (anti-) self-dual Yang-Mills equations is shown to be related via dimensional reductions to noncommutative formulations of the generalized (SO(3)/SO(2)) nonlinear Schrodinger (NS) equations, of the super-Korteweg- de Vries (super-KdV) as well as of the matrix KdV equations. Noncommutative extensions of their linear systems and bicomplexes associated to conserved quantities are discussed.
It is well known that for a given Poisson structure one has infinitely many star products related through the Kontsevich gauge transformations. These gauge transformations have an infinite functional dimension (i.e., correspond to an infinite number of degrees of freedom per point of the base manifold). We show that on a symplectic manifold this freedom may be almost completely eliminated if one extends the star product to all tensor fields in a covariant way and impose some natural conditions on the tensor algebra. The remaining ambiguity either correspond to constant renormalizations to the …
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such…
We perform a Noether analysis for a description of translation transformations in 4D $κ$-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. The techniques that some of us had previously developed (hep-th/0607221) for a description of translation transformations based on a 4D differential calculus turn out to be applicable without any modification, and they allow us to show that the basis usually adopted for the 5D calculus does not take into account certain aspects of the structure of time translations in $κ$-Minkowski. We propose a change of basi…
Following the construction of the $κ$-Minkowski space from the bicrossproduct structure of the $κ$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $κ$-Minkowski space and and we briefly discuss the main properties of its differential calculi.
We compute the one-loop 1PI contributions to all the propagators of the noncommutative N=1, 2, 4 super Yang-Mills (SYM) U(1) theories defined by the means of the theta-exact Seiberg-Witten (SW) map in the Wess-Zumino gauge. Then we extract the UV divergent contributions and the noncommutative IR divergences. We show that all the quadratic noncommutative IR divergences add up to zero in each propagator.
A complete Fock space representation of the covariant differential calculus on quantum space is constructed. The consistency criteria for the ensuing algebraic structure, mapping to the canonical fermions and bosons and the consequences of the new algebra for the statistics of quanta are analyzed and discussed. The concept of statistical transmutation between bosons and fermions is introduced.
We compute here the Yang-Mills effective action on Moyal space by integrating over the scalar fields in a noncommutative scalar field theory with harmonic term, minimally coupled to an external gauge potential. We also explain the special regularisation scheme chosen here and give some links to the Schwinger parametric representation. Finally, we discuss the results obtained: a noncommutative possibly renormalisable Yang-Mills theory.
We introduce the new notion of epsilon-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras. We define and study the associated notion of epsilon-derivation-based differential calculus, which generalizes the derivation-based differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of epsilon-graded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also t…
Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of he resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A com…
The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of ${\mathbb{Z}}_2$-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related ${\mathbb{Z}}_2$-graded version of it, that the derivation based differential calculus is a suitable framework to construct Ya…
In this paper, we show that various noncommutative integrable equations can be derived from noncommutative anti-self-dual Yang-Mills equations in the split signature, which include noncommutative versions of Korteweg-de Vries, Non-Linear Schroedinger, N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. U(1) part of gauge groups for the original Yang-Mills equations play crucial roles in noncommutative extension of Mason-Sparling's celebrated discussion. The present results would be strong evidences for noncommutative Ward's conjecture and imply that these noncommutative integrable e…
Recent studies of the $AdS_4/CFT_3$ correspondence involve the construction of a peculiar supersymmetric gauge theory on the worldvolume of multiple M2s branes as a boundary field theory. Under suitable conditions the quantum theory becomes a noncommutative supersymmetric YM-CS gauge theory which call for an study of its renormalized perturbative corrections. As a preliminary step to more general consideration, the modification of the ${\cal N}=3,2,1$ supersymmetric YM-CS gauge theory due to noncommutativity of spatial coordinates is proposed. We carry out the one-loop renormalization and a no…
We generalize to topologically non-trivial gauge configurations the description of the Einstein-Yang-Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle is an endomorphism bundle, we construct a PSU(N)-principal bundle for which it is an associated bundle. The so-called internal fluctuations of the spectral triple are parametrized by…
We present Backlund transformations for the noncommutative anti-self-dual Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasideterminants and belong to a noncommutative version of the Atiyah-Ward ansatz. In commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.
We present Backlund transformations for the noncommutative anti-self-dual Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasideterminants. We also explain the origins of all of the ingredients of the Backlund transformations within the framework of noncommutative twistor theory. In particular we show that the generated solutions belong to a noncommutative version of the Atiyah-Ward ansatz.
We discuss extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang-Mills equations. We give wide class of exact solutions by solving a Riemann-Hilbert problem for the Atiyah-Ward ansatz and present Backlund transformations for the G=U(2) noncommutative anti-self-dual Yang-Mills equations. We find that one kind of noncommutative determinants, quasideterminants, play crucial roles in the construction of noncommutative solutions. We also discuss reduction of a noncommutative anti-self-dual Yang-Mills equatio…
We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the cas…
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres in Connes and Landi [8] and in Connes and Dubois Violette [7], by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives by a quotient procedure a meaningful calculus on the sphere. In this cal…
The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expr…
We apply noncommutative geometry to a system of N parallel D-branes, which is interpreted as a quantum space. The Dirac operator defining the quantum differential calculus is identified to be the supercharge for strings connecting D-branes. As a result of the calculus, Connes' Yang-Mills action functional on the quantum space reproduces the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. Several features that may look ad hoc in a noncommutative geometric construction are shown to have very natural physical or geometric origin in the D…
Connes' view at Yang-Mills theories is reviewed with special emphasis on the gauge invariant scalar product. This landscape is shown to contain Chamseddine and Connes' noncommutative extension of general relativity restricted to flat space-time, if the top mass is between 172 and 204 GeV. Then the Higgs mass is between 188 and 201 GeV.
It is known that noncommutative Yang-Mills is equivalent to IIB matrix model with a noncommutative background, which is interpreted as a twisted reduced model. In noncommutative Yang-Mills, long range interactions can be seen in nonplanar diagrams after integrating high momentum modes. These interactions can be understood as block-block interactions in the matrix model. Using this relation, we consider long range interactions in noncommutative Yang-Mills associated with fermionic backgrounds. Exchanges of gravitinos, which couple to a supersymmetry current, are examined.
Noncommutative tori are among historically the oldest and by now the most developed examples of noncommutative spaces. Noncommutative Yang-Mills theory can be obtained from string theory. This connection led to a cross-fertilization of research in physics and mathematics on Yang-Mills theory on noncommutative tori. One important result stemming from that work is the link between T-duality in string theory and Morita equivalence of associative algebras. In this article we give an overview of the basic results in differential geometry of noncommutative tori. Yang-Mills theory on noncommutative t…
Topological structure of translation-invariant noncommutative Yang-Mills theories are studied by means of a cohomology theory, so called star-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg-Witten map.
A BRST-cohomological analysis of Seiberg-Witten maps and results on gauge anomalies in noncommutative Yang-Mills theories with general gauge groups are reviewed.
D0-brane theory on a torus with a nonvanishing B field is embedded into a string theory in the weak coupling limit. It is shown that the usual supersymmetric Yang-Mills theory on a noncommutative torus can not be the whole story. The Born-Infeld action survives the noncommutative torus limit.
We describe some recent progress in our understanding of Yang-Mills theories formulated on noncommutative spaces and in particular how to formulate the standard model on such spaces.
We study a class of noncommutative geometries that give rise to dimensionally reduced Yang-Mills theories. The emerging geometries describe sets of copies of an even dimensional manifold. Similarities to the D-branes in string theory are discussed.
It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume.
We analyze in detail the recursive construction of the Seiberg-Witten map and give an exhaustive description of its ambiguities. The local BRST cohomology for noncommutative Yang-Mills theory is investigated in the framework of the effective commutative Yang-Mills type theory. In particular, we show how some of the conformal symmetries get obstructed by the noncommutative deformation.
We present an action for noncommutative supersymmetric Yang-Mills theory in ten-dimensions, and confirm its invariance under supersymmetry. We next add higher-order derivative terms to such a noncommutative supersymmetric action. These terms contain fields as high as the quartic order. This resulting action can be regarded as supersymmetric generalization of noncommutative non-Abelian Dirac-Born-Infeld action. Some ambiguities related to field redefinitions are also clarified.
We prove by explicit calculation that Feynman graphs in noncommutative Yang-Mills theory made of repeated insertions into itself of arbitrarily many one-loop ghost propagator corrections are renormalizable by local counterterms. This provides a strong support for the renormalizability conjecture of that model.
Studied are the moduli spaces of Yang-Mills connections on finitely generated projective modules associated with noncommutative flows. It is actually shown that they are homeomorphic to those on the dual modules associated with the dual noncommutative flows. Moreover the result is also affirmative in the case of multiflows. As an important application, computed are the moduli spaces of the instanton bundles over the noncommutative Euclidean 4-space with respect to the canonical action of space translations without using the ADHM-construction.
Following the spirit of S-matrix program, we proposed a modified Britto-Cachazo-Feng-Witten recursion relation for tree amplitudes of noncommutative U(N) Yang-Mills theory. Starting from three-point amplitudes, one can use this modified BCFW recursion relation to compute or analyze color-ordered tree amplitudes without relying on any detail information of noncommutative Yang-Mills theory. After clarifying the color structure of noncommutative tree amplitudes, we wrote down the noncommutative analogies of U(1)-decoupling, Kleiss-Kuijf and Bern-Carrasco-Johansson relations for color-ordered tree…
We show that the recently proposed formulation of noncommutative N=2 Super Yang-Mills theory implies that the commutative and noncommutative effective coupling constants τ(u) and τ_{nc}(u) coincide. We then introduce a key relation which allows to find a nontrivial solution of such equation, thus fixing the form of the low-energy effective action. The dependence on the noncommutative parameter arises from a rational function deforming the Seiberg-Witten differential.
We show that twisted reduced models can be interpreted as noncommutative Yang-Mills theory. Based upon this correspondence, we obtain noncommutative Yang-Mills theory with D-brane backgrounds in IIB matrix model. We propose that IIB matrix model with D-brane backgrounds serve as a concrete definition of noncommutative Yang-Mills. We investigate D-instanton solutions as local excitations on D3-branes. When instantons overlap, their interaction can be well described in gauge theory and AdS/CFT correspondence. We show that IIB matrix model gives us the consistent potential with IIB supergravity w…
In built noncommutativity of supermembranes with central charges in eleven dimensions is disclosed. This result is used to construct an action for a noncommutative supermembrane where interesting topological terms appear. In order to do so, we first set up a global formulation for noncommutative Yang Mills theory over general symplectic manifolds. We make the above constructions following a pure geometrical procedure using the concept of connections over Weyl algebra bundles on symplectic manifolds. The relation between noncommutative and ordinary supermembrane actions is discussed.
A U(1) BF-Yang-Mills theory on noncommutative ${\mathbb{R}}^4$ is presented and in this formulation the U(1) Yang-Mills theory on noncommutative ${\mathbb{R}}^4$ is seen as a deformation of the pure BF theory. Quantization using BRST symmetry formalism is discussed and Feynman rules are given. Computations at one-loop order have been performed and their renormalization studied. It is shown that the U(1) BFYM on noncommutative ${\mathbb{R}}^4$ is asymptotically free and its UV-behaviour in the computation of the $β$-function is like the usual SU(N) commutative BFYM and Yang Mills theories.
A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra, and of the algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.
We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourrier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is {\it one loop renormalizable}, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed.
Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for GL(1,R) connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole-Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the KP and a modified KP equation. Furthermore, a differential calculus associated with t…
In this letter we derive the Seiberg-Witten map for noncommutative super Yang-Mills theory in Wess-Zumino gauge. Following (and using results of) hep-th/0108045 we split the observer Lorentz transformations into a covariant particle Lorentz transformation and a remainder which gives directly the Seiberg-Witten differential equations. These differential equations lead to a theta-expansion of the noncommutative super Yang-Mills action which is invariant under commutative gauge transformations and commutative observer Lorentz transformation, but not invariant under commutative supersymmetry trans…
We formulate noncommutative self-dual N=4 supersymmetric Yang-Mills theory in D=2+2 dimensions. As in the corresponding commutative case, this theory can serve as the possible master theory of all the noncommutative supersymmetric integrable models in lower dimensions. As a by-product, noncommutative self-dual N=2 supersymmetric Yang-Mills theory is obtained in D=2+2. We also perform a dimensional reduction of the N=2 theory further into N=(2,2) in D=1+1, as a basis for more general future applications. As a typical example, we show how noncommutative integrable matrix N=(1,0) supersymmetric K…
In this note a lately proposed gravity dual of noncommutative Yang-Mills theory is derived from the relations, recently suggested by Seiberg and Witten, between closed string moduli and open string moduli. The only new input one needs is a simple form of the running string tension as a function of energy. This derivation provides convincing evidence that string theory integrates with the holographical principle, and demonstrates a direct link between noncommutative Yang-Mills theory and holography.
In this paper, we derive the universal (cut-off-independent) part of the holographic entanglement entropy in the noncommutative Yang-Mills theory and examine its properties in detail. The behavior of the holographic entanglement entropy as a function of a scale of the system changes drastically between large noncommutativity and small noncommutativity. The strong subadditivity inequality for the entanglement entropies in the noncommutative Yang-Mills theory is modified in large noncommutativity. The behavior of entropic $c$-function defined by means of the entanglement entropy also changes dra…
Considering the scattering of massless open strings attached to a D2-brane living in the $B$ field background, we show that corresponding scattering upto the order of $\a'^2$ is exactly given by the gauge theory on noncommutative background, which is characterized by the Moyal bracket.
We apply the noncommutative fields method to the three-dimensional non-Abelian gauge theory. We find that, first, implementing the noncommutativity between the canonical momenta implies in generation of the non-Abelian Chern-Simons term, second, if one introduces the noncommutativity between the field operators, the higher derivative terms would arise.
We present some evidence that noncommutative Yang-Mills theory in two dimensions is not invariant under area preserving diffeomorphisms, at variance with the commutative case. Still, invariance under linear unimodular maps survives, as is proven by means of a fairly general argument.
