The framework of emergent gravity arising from Yang-Mills matrix models is developed further, for general noncommutative branes embedded in R^D. The effective metric on the brane turns out to have a universal form reminiscent of the open string metric, depending on the dynamical Poisson structure and the embedding metric in R^D. A covariant form of the tree-level equations of motion is derived, and the Newtonian limit is discussed. This points to the necessity of branes in higher dimensions. The quantization is discussed qualitatively, which singles out the IKKT model as a prime candidate for a quantum theory of gravity coupled to matter. The Planck scale is then identified with the scale of N=4 SUSY breaking. A mechanism for avoiding the cosmological constant problem is exhibited.
We construct exact solutions to noncommutative gravity following the formulation of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for generating cosmological electromagnetic fields in an expanding space-time background, and also leads to multipole-like fields surrounding black holes. Exact solutions to noncommutative Einstein-Maxwell theory can give rise to first order corrections to the metric tensor, as well as to the electromagnetic fields. This leads to first order shifts in the…
Fermions coupled to Yang-Mills matrix models are studied from the point of view of emergent gravity. We show that the simple matrix model action provides an appropriate coupling for fermions to gravity, albeit with a non-standard spin connection. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action and a dilaton-like term, at least for on-shell geometries. This explains and precisely reproduces the UV/IR mixing for fermions in noncommutative gauge theory, extending recent results for scalar fields. It also explains why some UV/IR mixing…
Matrix models of Yang-Mills type lead to an emergent gravity theory, which may not require fine-tuning of a cosmological constant. We find cosmological solutions of Friedmann-Robertson-Walker type. They generically have a big bounce, and an early inflation-like phase with graceful exit. The mechanism is purely geometrical, no ad-hoc scalar fields are introduced. The solutions are stabilized through vacuum fluctuations and are thus compatible with quantum mechanics. This leads to a Milne-like universe after inflation, which appears to be in remarkably good agreement with observation and may pro…
We show that the matrix-model action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4-dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric G_{ab}, which is determined by a dynamical Poisson structure. The emergent gravity is intimately related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. This leads to a class of metrics which contains the physical degrees of freedom of gravitational waves, and allows to rec…
We investigate a relation of the contravariant geometry to the emergent gravity from noncommutative gauge theories. We give a refined formulation of the contravariant gravity and provide solutions to the contravariant Einstein equation. We linearize the equation around background solutions, including curved ones. A noncommutative gauge theory on the Moyal plane can be rewritten as an ordinary gauge theory on a curved background via the Seiberg-Witten map, which is known as the emergent gravity. We show that this phenomenon also occurs for a gauge theory on a noncommutative homogeneous Kähler b…
We study the coupling of fermions to Yang-Mills matrix models in the framework of emergent noncommutative gravity. It is shown that the matrix model action provides an appropriate coupling for fermions to gravity, albeit with a non-standard spin-connection. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action for on-shell geometries plus a dilaton-like term. This result explains UV/IR mixing as a gravity effect. It also illuminates why UV/IR mixing remains even in supersymmetric models, except in the N=4 case.
The indirect evidence for at least a dozen massive white dwarfs violating the Chandrasekhar mass-limit is considered to be one of the wonderful discoveries in astronomy for more than a decade. Researchers have already proposed a diverse amount of models to explain this astounding phenomenon. However, each of these models always carries some drawbacks. On the other hand, noncommutative geometry is one of the best replicas of quantum gravity, which is yet to be proved from observations. Madore introduced the idea of a fuzzy sphere to describe a formalism of noncommutative geometry. This article …
Starting from a standard noncommutative gauge theory and using the Seiberg-Witten map we propose a new version of a noncommutative gravity. We use consistent deformation theory starting from a free gauge action and gauging a killing symmetry of the background metric to construct a deformation of the gauge theory that we can relate with gravity. The result of this consistent deformation of the gauge theory is nonpolynomial in A_μ. From here we can construct a version of noncommutative gravity that is simpler than previous attempts. Our proposal is consistent and is not plagued with the problems…
We study fermions coupled to Yang-Mills matrix models from the point of view of emergent gravity. The matrix model Dirac operator provides an appropriate coupling for fermions to the effective gravitational metric for general branes with nontrivial embedding, albeit with a non-standard spin connection. This generalizes previous results for 4-dimensional matrix models. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action of the effective metric, as well as additional terms which couple the Poisson tensor to the Riemann tensor, and a dila…
We showed before that self-dual electromagnetism in noncommutative (NC) spacetime is equivalent to self-dual Einstein gravity. This result implies a striking picture about gravity: Gravity can emerge from electromagnetism in NC spacetime. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy spacetime. We elucidate in some detail why electromagnetism in NC spacetime should be a theory of gravity. In particular, we show that NC electromagnetism is realized through the Darboux theorem as a diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H…
We present a brief review of selected topics in noncommutative field theories ranging from its revival in string theory, its influence on quantum field theories, its possible experimental signatures and ending with some applications in gravity and emergent gravity.
We argue that a field theory defined on noncommutative (NC) spacetime should be regarded as a theory of gravity, which we refer to as the emergent gravity. A whole point of the emergent gravity is essentially originated from the basic property: A NC spacetime is a (NC) phase space. This fact leads to two important consequences: (I) A NC field theory can basically be identified with a matrix model or a large N field theory where NC fields can be regarded as master fields of large N matrices. (II) NC fields essentially define vector (tetrad) fields. So they define a gravitational metric of some …
A introductory review to emergent noncommutative gravity within Yang-Mills Matrix models is presented. Space-time is described as a noncommutative brane solution of the matrix model, i.e. as submanifold of \R^D. Fields and matter on the brane arise as fluctuations of the bosonic resp. fermionic matrices around such a background, and couple to an effective metric interpreted in terms of gravity. Suitable tools are provided for the description of the effective geometry in the semi-classical limit. The relation to noncommutative gauge theory and the role of UV/IR mixing is explained. Several type…
We study metric-compatible Poisson structures in the semi-classical limit of noncommutative emergent gravity. Space-time is realized as quantized symplectic submanifold embedded in R^D, whose effective metric depends on the embedding as well as on the Poisson structure. We study solutions of the equations of motion for the Poisson structure, focusing on a natural class of solutions such that the effective metric coincides with the embedding metric. This leads to i-(anti-) self-dual complexified Poisson structures in four space-time dimensions with Lorentzian signature. Solutions on manifolds w…
We address issues on the origin of gravity and the cosmological constant problem based on a recent understanding about the correspondence between noncommutative field theory and gravity. We suggest that the cosmological constant problem can be resolved in a natural way if gravity emerges from a gauge theory in noncommutative spacetime. Especially, we elucidate why the emergent gravity implies that vacuum energy does not gravitate but only fluctuations around the vacuum generate gravity. That is, a flat spacetime emerges from uniform condensation of energy, previously identified with the cosmol…
We study the thermodynamics of the large N noncommutative super Yang-Mills theory in the strong 't Hooft coupling limit in the spirit of AdS/CFT correspondence. It has already been noticed that some thermodynamic quantities of near-extremal D3-branes with NS B fields, which are dual gravity configurations of the noncommutative ${\cal N}$=4 super Yang-Mills theory, are the same as those without B fields. In this paper, (1) we examine the $α'^3 R^4$ corrections to the free energy and find that the part of the tree-level contribution remains unchanged, but the one-loop and the non-perturbative D-…
The contemporary physics has revealed growing evidences that the emergence can be applied to not only biology and condensed matter systems but also gravity and spacetime. We observe that noncommutative spacetime necessarily implies emergent spacetime if spacetime at microscopic scales should be viewed as noncommutative. Since the emergent spacetime is a new fundamental paradigm for quantum gravity, it is necessary to reexamine all the rationales to introduce the multiverse hypothesis from the standpoint of emergent spacetime. We argue that the emergent spacetime certainly opens a new perspecti…
We conjecture an embedding operator which assigns, to any 2n+1 hermitian matrices, a 2n-dimensional hypersurface in flat (2n + 1)-dimensional Euclidean space. This corresponds to precisely defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the emergent hypersurface correspond to zero eigenstates of the embedding operator, which have an interpretation as coherent states underlying the emergent noncommutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed. We apply our conjecture to noncommutative flat and spherical spac…
In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U(1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U(1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U(1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that t…
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 give a brief review of two nonperturbative phenomena typical of noncommutative field theory which are known to lead to the perturbative instability known as the UV-IR mixing. The first phenomena concerns the emergence/evaporation of spacetime geometry in matrix models which describe perturbative noncommutative gauge theory on fuzzy backgrounds. In particular we show that the transition from a geometrical background to a matrix phase makes the description of noncommutative gauge theory in terms of fields via the Weyl map only valid below a critical value g_*. The second phenomena concerns th…
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 map noncommutative (NC) U(1) gauge theory on R^d_C X R^{2n}_{NC} to U(N -> \infty) Yang-Mills theory on R^d_C, where R^d_C is a d-dimensional commutative spacetime while R^{2n}_{NC} is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on R^d_C is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto R^d_C. We show that the gauge-Higgs system (A_μ,Φ^a) in the U(N -> \infty) Yang-Mills theory on R^d_C leads to an emergent geometry in the (d+2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In parti…
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…
Recent progress in the understanding of gravity on noncommutative spaces is discussed. A gravity theory naturally emerges from matrix models of noncommutative gauge theory. The effective metric depends on the dynamical Poisson structure, absorbing the degrees of freedom of the would-be U(1) gauge field. The gravity action is induced upon quantization.
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.
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 demonstrate the existence of entanglement between the spatial degrees of freedom of a system of harmonic oscillators placed in the noncommutative Moyal plane ("exotic oscillators") by computing the entanglement entropy as measured by the von Neumann entropy of the reduced density matrix. It is explicitly verified that the entanglement arises from the noncommutativity, which controls the coupling strength between the spatial modes. This can easily be generalised to the case where the momentum components also satisfy noncommutative relations, so that the entire phase space becomes noncommutat…
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.
Recently Verlinde proposed that gravity can be described as an emergent phenomena arising from changes in the information associated with the positions of material bodies. By using noncommutative geometry as a way to describe the microscopic, microstructure of quantum spacetime, we derive modified Friedmann equation in this setup and study the entropic force modifications to the inflationary dynamics of early universe.
