A representation of general translation-invariant star products in the algebra of M(C) = lim_N\to \infty M_N (C) is introduced which results in the Moyal-Weyl-Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so called star-cohomology, with groups H^k_*(C), is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg-Witten map for the Moyal case. Employing the Chern-Weil theory via the integral classes of H^k_*(Z) a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang-Mills theory. Thereby we study the mentioned Yang-Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of *-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the *-cohomology.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawa's method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $β$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coi…
A BRST-cohomological analysis of Seiberg-Witten maps and results on gauge anomalies in noncommutative Yang-Mills theories with general gauge groups are reviewed.
We propose an exact expression for the unintegrated form of the star gauge invariant axial anomaly in an arbitrary even dimensional gauge theory. The proposal is based on the inverse Seiberg-Witten map and identities related to it, obtained earlier by comparing Ramond-Ramond couplings in different decsriptions. The integrated anomalies are expressed in terms of a simplified version of the Elliott formula involving the noncommutative Chern character. These anomalies, under the Seiberg-Witten transformation, reduce to the ordinary axial anomalies. Compatibility with existing results of anomalies…
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.
Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^μ(\equiv θ^{μν}p_ν) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.
We study chiral symmetries of fermionic non commutative dipole theories. By using Fujikawa's approach we obtain explicit expressions of the anomalies for Dirac and chiral fermions in 2 and 4 dimensions.
We exploit the Seiberg-Witten maps for fields and currents in a U(1) gauge theory relating the noncommutative and commutative (usual) descriptions to obtain the O(θ) structure of the commutator anomalies in noncommutative electrodynamics. These commutators involve the (covariant) current-current algebra and the (covariant) current-field algebra. We also establish the compatibility of the anomalous commutators with the noncommutative covariant anomaly through the use of certain consistency conditions derived here.
We discuss nonplanar anomalies in noncommutative gauge theories. In particular we show that a nonplanar anomaly exists when the external noncommutative momentum is zero and that it leads to a non-conservation of the associated axial charge. In the case of nonplanar local anomalies, a cancellation of the anomaly can be achieved by a Green-Schwarz mechanism. In an example of D3 branes placed on orbifold singularity that leads to a chiral theory, the mechanism involves twisted RR fields which propagate with zero noncommutative momentum. Global anomalies are not cancelled and, in particular, the d…
Gravitational axial and chiral anomalies in a noncommutative space are examined through the explicit perturbative computation of one-loop diagrams in various dimensions. The analysis depend on how gravity is coupled to noncommutative matter fields. Delbourgo-Salam computation of the gravitational axial anomaly contribution to the pion decay into two photons, is studied in detail in this context. In the process we show that the two-dimensional chiral pure gravitational anomaly does not receive noncommutative corrections. Pure gravitational chiral anomaly in 4k+2 dimensions with matter fields be…
We discuss both the UV and IR origin of the one-loop triangle gauge anomalies for noncommutative nonabelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only come from planar triangle diagrams, the non-planar triangle contributions giving rise to no breaking of the Ward identies. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free.
Chiral gauge anomalies on noncommutative Minkowski Space-time are computed and have their origin elucidated. The consistent form and the covariant form of the anomaly are obtained. Both Fujikawa's method and Feynman diagram techniques are used to carry out the calculations
Invariant (nonplanar) anomaly of noncommutative QED is reexamined. It is found that just as in ordinary gauge theory UV regularization is needed to discover anomalies, in noncommutative case, in addition, an IR regularization is also required to exhibit existence of invariant anomaly. Thus resolving the controversy in the value of invariant anomaly, an expression for the unintergrated anomaly is found. Schwinger terms of the current algebra of the theory are derived.
Using cohomological methods we discuss several issues related to chiral anomalies in noncommutative U(N) YM theories in any even dimension. We show that for each dimension there is only one solution of the WZ consistency condition and that there cannot be any reducible anomaly, nor any mixed anomaly when the gauge group is a product group. We also clarify some puzzling aspects of the issue of the anomaly when chiral fermions are in the adjoint representation.
The explicit form of non-Abelian noncommutative supersymmetric (SUSY) chiral anomaly is calculated, the Wess-Zumino consistency condition is verified and the correspondence of the Yang-Mills sector to the previously obtained results is shown. We generalize the Seiberg-Witten map to the case of N=1 SUSY Yang-Mills theory and calculations up to the second order in the noncommutativity parameter are done.
The Konishi anomalies for noncommutative N=1 supersymmetric U(1) gauge theory arising from planar and nonplanar diagrams are calculated. Whereas planar Konishi anomaly is the expected \star-deformation of the commutative anomaly, nonplanar anomaly reflects the important features of nonplanar diagrams of noncommutative gauge theories, such as UV/IR mixing and the appearance of nonlocal open Wilson lines. We use the planar and nonplanar Konishi anomalies to calculate the effective superpotential of the theory. In the limit of vanishing |Θp|, with Θthe noncommutativity parameter, the noncommutati…
We derive maps relating currents and their divergences in non-abelian U(N) noncommutative gauge theory with the corresponding expressions in the ordinary (commutative) description. For the U(1) theory, in the slowly-varying-field approximation, these maps are also seen to connect the star-gauge-covariant anomaly in the noncommutative theory with the standard Adler--Bell--Jackiw anomaly in the commutative version. For arbitrary fields, derivative corrections to the maps are explicitly computed up to O(θ^2).
Being motivated by physical applications (as the phi^4 model) we calculate the heat kernel coefficients for generalised Laplacians on the Moyal plane containing both left and right multiplications. We found both star-local and star-nonlocal terms. By using these results we calculate the large mass and strong noncommutativity expansion of the effective action and of the vacuum energy. We also study the axial anomaly in the models with gauge fields acting on fermions from the left and from the right.
Translation-invariant noncommutative gauge theories are discussed in the setting of matrix modeled gauge theories. Using the matrix model formulation the explicit form of consistent anomalies and consistent Schwinger terms for translation-invariant noncommutative gauge theories are derived.
Anomalies arising from nonplanar triangle diagrams of noncommutative gauge theory are studied. Local chiral gauge anomalies for both noncommutative U(1) and U(N) gauge theories with adjoint matter fields are shown to vanish. For noncommutative QED with fundamental matters, due to UV/IR mixing a finite anomaly emerges from the nonplanar contributions. It involves a generalized $\star$-product of gauge fields.
Consistent Yang--Mills anomalies $\int\om_{2n-k}^{k-1}$ ($n\in\N$, $ k=1,2, \ldots ,2n$) as described collectively by Zumino's descent equations $δ\om_{2n-k}^{k-1}+\dd\om_{2n-k-1}^{k}=0$ starting with the Chern character $Ch_{2n}=\dd\om_{2n-1}^{0}$ of a principal $\SU(N)$ bundle over a $2n$ dimensional manifold are considered (i.e.\ $\int\om_{2n-k}^{k-1}$ are the Chern--Simons terms ($k=1$), axial anomalies ($k=2$), Schwinger terms ($k=3$) etc.\ in $(2n-k)$ dimensions). A generalization in the spirit of Connes' noncommutative geometry using a minimum of data is found. For an arbitrary graded d…
These are the notes of a lecture given during the summer school "Geometric and Topological Methods for Quantum Field Theory", Villa de Leyva, Colombia, july 11 - 29, 2005. We review basic facts concerning gauge anomalies and discuss the link with the Connes-Moscovici index formula in noncommutative geometry.
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…
We discuss some exact Seiberg--Witten-type maps for noncommutative electrodynamics. Their implications for anomalies in different (noncommutative and commutative) descriptions are also analysed.
In this paper, we present a general modified dispersion relation derived from q-deformed noncommutative theory and apply it to the ultrahigh energy cosmic ray and the TeV-photon paradoxes--threshold anomalies. Our purpose is not only trying to solve these puzzles by noncommutative theory but also to support noncommutative theory through the coincidence of the region in the parameter space for resolving the threshold anomalies with the one from the q-deformed noncommutative theory.
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.
Motivated by possible scalar-leptoquark explanations of the recently reported $B$-decay anomalies, we investigate whether the required leptoquarks can be accommodated within models based on noncommutative geometry (NCG). The models considered have the gauge structure of Pati-Salam models, $SU(4)\times SU(2)_L\times SU(2)_R$, with gauge coupling unification at a single scale. In one of the models, we find a unique scalar leptoquark with quantum numbers $(3,1,-\frac{1}{3})_{321}$, originating from a complex multiplet $(6,1,1)_{422}$, which can potentially explain the $B$-decay anomalies if its m…
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.
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.
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.
We show that U(N) Yang-Mills theory on noncommutative Minkowski space-time can be renormalized, in a BRS invariant way, at the one-loop level, by multiplicative dimensional renormalization of its coupling constant, its gauge parameter and its fields. It is shown that the Slavnov-Taylor equation, the gauge-fixing equation and the ghost equation hold, up to order $\hbar$, for the MS renormalized noncommutative U(N) Yang-Mills theory. We give the value of the pole part of every 1PI diagram which is UV divergent.
We consider a U(2) Yang-Mills theory on M x S_F^2 where M is an arbitrary noncommutative manifold and S_F^2 is a fuzzy sphere spontaneously generated from a noncommutative U(N) Yang-Mills theory on M, coupled to a triplet of scalars in the adjoint of U(N). Employing the SU(2)-equivariant gauge field constructed in arXiv:0905.2338, we perform the dimensional reduction of the theory over the fuzzy sphere. The emergent model is a noncommutative U(1) gauge theory coupled adjointly to a set of scalar fields. We study this model on the Groenewald-Moyal plane and find that, in certain limits, it admi…
We consider 10-dimensional super Yang-Mills theory with topological terms compactified on a noncommutative torus. We calculate supersymmetry algebra and derive BPS energy spectra from it. The cases of d-dimensional tori with d=2,3,4 are considered in full detail. SO(d,d|Z)-invariance of the BPS spectrum and relation of new results to the previous work in this direction are discussed.
We show that there are one-loop IR singularities arising from UV/IR mixing in noncommutative SU(N) Yang-Mills theory defined by means of the $θ$-exact Seiberg-Witten map. This is in spite of the fact that there are no ordinary U(1) gauge fields in the theory and this is at variance with the noncommutative U(N) case, where the two-point part of the effective action involving the ordinary SU(N) fields do not suffer from those one-loop IR singularities.
Seiberg-Witten maps and a recently proposed construction of noncommutative Yang-Mills theories (with matter fields) for arbitrary gauge groups are reformulated so that their existence to all orders is manifest. The ambiguities of the construction which originate from the freedom in the Seiberg-Witten map are discussed with regard to the question whether they can lead to inequivalent models, i.e., models not related by field redefinitions.
By introducing an appropriate parent action and considering a perturbative approach, we establish, up to fourth order terms in the field and for the full range of the coupling constant, the equivalence between the noncommutative Yang-Mills-Chern-Simons theory and the noncommutative, non-Abelian Self-Dual model. In doing this, we consider two different approaches by using both the Moyal star-product and the Seiberg-Witten map.
