arXiv subjects

Badis Ydri

Publications and source records attributed to Badis Ydri.

At least 55 records · Page 3Linked to original sources

arXiv2012

The fate of the Wilson-Fisher fixed point in non-commutative ϕ^4

In this article we study non-commutative vector sigma model with the most general ϕ^4 interaction on Moyal-Weyl spaces. We compute the 2- and 4-point functions to all orders in the large N limit and then apply the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. The non-commutative Wilson-Fisher fixed point interpolates between the commutative Wilson-Fisher fixed point of the Ising universality class which is found to lie at zero value of the critical coupling constant a_* of the zero dimensional reduction of the theory, and a novel strongly interacting fixed point which lies at infinite value of a_* corresponding to maximal non-commutativity beyond which the two-sheeted structure of a_* as a function of the dilation parameter disappears.

hep-th↗

arXiv2012

Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models

We present a study of D=4 supersymmetric Yang-Mills matrix models with SO(3) mass terms based on the cohomological approach and the Monte Carlo method. In the bosonic models we show the existence of an exotic first/second order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first order due to the discontinuity in the action whereas it looks second order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a non-polynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this non-polynomial law. We also discuss the D=3 models and by using cohomological deformation, localization techniques and the saddle-point method we give a derivation of the D=3 eigenvalues distribution starting from a particular D=4 model.

hep-th↗

arXiv2012

Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II

We present a study of D=4 supersymmetric Yang-Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first/second order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first order due to the discontinuity in the action whereas it looks second order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a non-polynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this non-polynomial law.

hep-th↗

arXiv2010

New Phenomena in NC Field Theory and Emergent Spacetime Geometry

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 the appearance of a nonuniform ordered phase in noncommutative scalar ϕ^4 field theory and the spontaneous symmetry breaking of translational/rotational invariance which happens even in two dimensions. We argue that this phenomena also originates in the underlying matrix degrees of freedom of the noncommutative field theory. Furthermore it is conjectured that in addition to the usual WF fixed point at $θ=0$ there must exist a novel fixed point at θ=\infty corresponding to the quartic hermitian matrix model.

hep-th↗

arXiv2009

Matrix Models, Gauge Theory and Emergent Geometry

We present, theoretical predictions and Monte Carlo simulations, for a simple three matrix model that exhibits an exotic phase transition. The nature of the transition is very different if approached from the high or low temperature side. The high temperature phase is described by three self interacting random matrices with no background spacetime geometry. As the system cools there is a phase transition in which a classical two-sphere condenses to form the background geometry. The transition has an entropy jump or latent heat, yet the specific heat diverges as the transition is approached from low temperatures. We find no divergence or evidence of critical fluctuations when the transition is approached from the high temperature phase. At sufficiently low temperatures the system is described by small fluctuations, on a background classical two-sphere, of a U(1) gauge field coupled to a massive scalar field. The critical temperature is pushed upwards as the scalar field mass is increased. Once the geometrical phase is well established the specific heat takes the value 1 with the gauge and scalar fields each contributing 1/2.

hep-th↗

arXiv2008

Geometry in transition: A model of emergent geometry

We study a three matrix model with global SO(3) symmetry containing at most quartic powers of the matrices. We find an exotic line of discontinuous transitions with a jump in the entropy, characteristic of a 1st order transition, yet with divergent critical fluctuations and a divergent specific heat with critical exponent $α=1/2$. The low temperature phase is a geometrical one with gauge fields fluctuating on a round sphere. As the temperature increased the sphere evaporates in a transition to a pure matrix phase with no background geometrical structure. Both the geometry and gauge fields are determined dynamically. It is not difficult to invent higher dimensional models with essentially similar phenomenology. The model presents an appealing picture of a geometrical phase emerging as the system cools and suggests a scenario for the emergence of geometry in the early universe.

hep-th↗

arXiv2007

A Proposal for a Non-Perturbative Regularization of {\cal N}=2 SUSY 4D Gauge Theory

In this letter we show that supersymmetry like geometry can be approximated using finite dimensional matrix models and fuzzy manifolds. In particular we propose a non-perturbative regularization of {\cal N}=2 supersymmetric U(n) gauge action in 4D. In some planar large N limits we recover exact SUSY together with the smooth geometry of R^4_{theta}.

hep-th↗

arXiv2007

Notes on noncommutative supersymmetric gauge theory on the fuzzy supersphere

In these notes we review Klimcik's construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this article a novel fuzzy supersymmetric scalar action on the fuzzy supersphere.

hep-th↗

arXiv2007

Topology Change From Quantum Instability of Gauge Theory on Fuzzy CP^2

Many gauge theory models on fuzzy complex projective spaces will contain a strong instability in the quantum field theory leading to topology change. This can be thought of as due to the interaction between spacetime via its noncommutativity and the fields (matrices) and it is related to the perturbative UV-IR mixing. We work out in detail the example of fuzzy CP^2 and discuss at the level of the phase diagram the quantum transitions between the 3 spaces (spacetimes) CP^2, S^2 and the 0-dimensional space consisting of a single point {0}.

hep-th↗

arXiv2007

Quantum Equivalence of NC and YM Gauge Theories in 2 D and Matrix Theory

We construct noncommutative U(1) gauge theory on the fuzzy sphere S^2_N as a unitary 2N x 2N matrix model. In the quantum theory the model is equivalent to a nonabelian U(N) Yang-Mills theory on a 2 dimensional lattice with 2 plaquettes. This equivalence holds in the " fuzzy sphere" phase where we observe a 3rd order phase transition between weak-coupling and strong-coupling phases of the gauge theory. In the ``matrix'' phase we have a U(N) gauge theory on a single point.

hep-th↗

arXiv2006

Towards Noncommutative Fuzzy QED

We study in one-loop perturbation theory noncommutative fuzzy quenched QED_4. We write down the effective action on fuzzy S**2 x S**2 and show the existence of a gauge-invariant UV-IR mixing in the model in the large N planar limit. We also give a derivation of the beta function and comment on the limit of large mass of the normal scalar fields. We also discuss topology change in this 4 fuzzy dimensions arising from the interaction of fields (matrices) with spacetime through its noncommutativity.

hep-th↗

arXiv2006

The One-Plaquette Model Limit of NC Gauge Theory in 2D

It is found that noncommutative U(1) gauge field on the fuzzy sphere S^2_N is equivalent in the quantum theory to a commutative 2-dimensional U(N) gauge field on a lattice with two plaquettes in the axial gauge A_1=0. This quantum equivalence holds in the fuzzy sphere-weak coupling phase in the limit of infinite mass of the scalar normal component of the gauge field. The doubling of plaquettes is a natural consequence of the model and it is reminiscent of the usual doubling of points in Connes standard model. In the continuum large N limit the plaquette variable W approaches the identity 1_{2N} and as a consequence the model reduces to a simple matrix model which can be easily solved. We compute the one-plaquette critical point and show that it agrees with the observed value \barα_*=3.35. We compute the quantum effective potential and the specific heat for U(1) gauge field on the fuzzy sphere S^2_{N} in the 1/N expansion using this one-plaquette model. In particular the specific heat per one degree of freedom was found to be equal to 1 in the fuzzy sphere-weak coupling phase of the gauge field which agrees with the observed value 1 seen in Monte Carlo simulation. This value of 1 comes precisely because we have 2 plaquettes approximating the NC U(1) gauge field on the fuzzy sphere.

hep-th↗

arXiv2006

Entanglement Entropy on Fuzzy Spaces

We study the entanglement entropy of a scalar filed in 2+1 spacetime where space is modeled by a fuzzy sphere and a fuzzy disc. In both models we evaluate numerically the resulting entropies and find that they are proportional to the number of boundary degrees of freedom. In the Moyal plan limit of the fuzzy disc the entanglement entropy per unit area (length) diverges if the ignored region is of infinite size. The divergence is (interpreted) of IR-UV mixing origin. In general we expect the entanglement entropy per unit area to be finite on a non-commutative space if the ignored region is of finite size.

gr-qc↗

arXiv2006

Monte Carlo Simulation of a NC Gauge Theory on The Fuzzy Sphere

We find using Monte Carlo simulation the phase structure of noncommutative U(1) gauge theory in two dimensions with the fuzzy sphere S^2_N as a non-perturbative regulator. There are three phases of the model. i) A matrix phase where the theory is essentially SU(N) Yang-Mills reduced to zero dimension . ii) A weak coupling fuzzy sphere phase with constant specific heat and iii) A strong coupling fuzzy sphere phase with non-constant specific heat. The order parameter distinguishing the matrix phase from the sphere phase is the radius of the fuzzy sphere. The three phases meet at a triple point. We also give the theoretical one-loop and 1/N expansion predictions for the transition lines which are in good agreement with the numerical data. A Monte Carlo measurement of the triple point is also given.

hep-lat↗

arXiv2005

Quantum effective potential for U(1) fields on S^2_L X S^2_L

We compute the one-loop effective potential for noncommutative U(1) gauge fields on S^2_L X S^2_L. We show the existence of a novel phase transition in the model from the 4-dimensional space S^2_L X S^2_L to a matrix phase where the spheres collapse under the effect of quantum fluctuations. It is also shown that the transition to the matrix phase occurs at infinite value of the gauge coupling constant when the mass of the two normal components of the gauge field on S^2_L X S^2_L is sent to infinity.

hep-th↗

arXiv2004

A Gauge-Invariant UV-IR Mixing and The Corresponding Phase Transition For U(1) Fields on the Fuzzy Sphere

From a string theory point of view the most natural gauge action on the fuzzy sphere {\bf S}^2_L is the Alekseev-Recknagel-Schomerus action which is a particular combination of the Yang-Mills action and the Chern-Simons term . Since the differential calculus on the fuzzy sphere is 3-dimensional the field content of this model consists naturally of a 2-dimensional gauge field together with a scalar fluctuation normal to the sphere . For U(1) gauge theory we compute the quadratic effective action and shows explicitly that the tadpole diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR mixing in the continuum limit L{\longrightarrow}{\infty} where L is the matrix size of the fuzzy sphere. In other words the quantum U(1) effective action does not vanish in the commutative limit and a noncommutative anomaly survives . We compute the scalar effective potential and prove the gauge-fixing-independence of the limiting model L={\infty} and then show explicitly that the one-loop result predicts a first order phase transition which was observed recently in simulation . The one-loop result for the U(1) theory is exact in this limit . It is also argued that if we add a large mass term for the scalar mode the UV-IR mixing will be completely removed from the gauge sector . It is found in this case to be confined to the scalar sector only. This is in accordance with the large L analysis of the model . Finally we show that the phase transition becomes harder to reach starting from small couplings when we increase M .

hep-th↗

arXiv2004

Exact Solution of Noncommutative U(1) Gauge Theory in 4-Dimensions

Noncommutative U(1) gauge theory on the Moyal-Weyl space ${\bf R}^2{\times}{\bf R}^2_θ$ is regularized by approximating the noncommutative spatial slice ${\bf R}^2_θ$ by a fuzzy sphere of matrix size $L$ and radius $R$ . Classically we observe that the field theory on the fuzzy space ${\bf R}^2{\times}{\bf S}^2_L$ reduces to the field theory on the Moyal-Weyl plane ${\bf R}^2{\times}{\bf R}^2_θ$ in the flattening continuum planar limits $R,L{\longrightarrow}{\infty}$ where $R^2/L^{2q}{\simeq}θ^2/4^q$ and $q>{3/2}$ . The effective noncommutativity parameter is found to be given by $θ_{eff}^2{\sim}2θ^2(\frac{L}{2})^{2q-1}$ and thus it corresponds to a strongly noncommuting space. In the quantum theory it turns out that this prescription is also equivalent to a dimensional reduction of the model where the noncommutative U(1) gauge theory in 4 dimensions is shown to be equivalent in the large $L$ limit to an ordinary $O(M)$ non-linear sigma model in 2 dimensions where $M{\sim}3L^2$ . The Moyal-Weyl model defined this way is also seen to be an ordinary renormalizable theory which can be solved exactly using the method of steepest descents . More precisely we find for a fixed renormalization scale $μ$ and a fixed renormalized coupling constant $g_r^2$ an $O(M)-$symmetric mass, for the different components of the sigma field, which is non-zero for all values of $g_r^2$ and hence the $O(M)$ symmetry is never broken in this solution . We obtain also an exact representation of the beta function of the theory which agrees with the known one-loop perturbative result .

hep-th↗

arXiv2004

Noncommutative U(1) Gauge Theory As a Non-Linear Sigma Model

Noncommutative U(1) gauge theory in 4-dimensions is shown to be equivalent in some scaling limit to an ordinary non-linear sigma model in 2-dimensions . The model in this regime is solvable and the corresponding exact beta function is found. We also show that classical U(n) gauge theory on {R}^{d-2}{\times}{R}^2_θ can be approximated by a sequence of ordinary (d-2)-dimensional Georgi-Glashow models with gauge groups U(n(L+1)) where L+1 is the matrix size of the regularized noncommutative plane {R}^2_θ.

hep-th↗