Fuzzy Sphere Research Articles

Noncommutative spherically symmetric spacetimes at semiclassical order

Working within the recent formalism of Poisson–Riemannian geometry, we completely solve the case of generic spherically symmetric metric and spherically symmetric Poisson-bracket to find a unique answer for the quantum differential calculus, quantum metric and quantum Levi-Civita connection at semiclassical order . Here λ is the deformation parameter, plausibly the Planck scale. We find that are all forced to be central, i.e. undeformed at order λ, while for each value of r, t we are forced to have a fuzzy sphere of radius r with a unique differential calculus which is necessarily nonassociative at order . We give the spherically symmetric quantisation of the FLRW cosmology in detail and also recover a previous analysis for the Schwarzschild black hole, now showing that the quantum Ricci tensor for the latter vanishes at order λ. The quantum Laplace–Beltrami operator for spherically symmetric models turns out to be undeformed at order λ while more generally in Poisson–Riemannian geometry we show that it deforms to□f+λ2ωαβ(Ricγα−Sγ;α)(∇^βdf)γ+O(λ2)in terms of the classical Levi-Civita connection , the contorsion tensor S, the Poisson-bivector ω and the Ricci curvature of the Poisson-connection that controls the quantum differential structure. The Majid–Ruegg spacetime with its standard calculus and unique quantum metric provides an example with nontrivial correction to the Laplacian at order λ.

Correlation functions and renormalization in a scalar field theory on the fuzzy sphere

We study renormalization in a scalar field theory on the fuzzy sphere. The theory is realized by a matrix model, where the matrix size plays the role of an ultraviolet cutoff. We define correlation functions by using the Berezin symbol identified with a field and calculate them nonperturbatively by Monte Carlo simulation. We find that the 2-point and 4-point functions are made independent of the matrix size by tuning a parameter and performing a wave function renormalization. The results strongly suggest that the theory is nonperturbatively renormalizable in the ordinary sense.

A generalized volume law for entanglement entropy on the fuzzy sphere

We investigate entanglement entropy in a scalar field theory on the fuzzy sphere. The theory is realized by a matrix model. In our previous study, we confirmed that entanglement entropy in the free case is proportional to the square of the boundary area of a focused region. Here we argue that this behavior of entanglement entropy can be understood by the fact that the theory is regularized by matrices, and further examine the dependence of entanglement entropy on the matrix size. In the interacting case, by performing Monte Carlo simulations, we observe a transition from a generalized volume law, which is obtained by integrating the square of area law, to the square of area law.

The trinification model SU(3)3 from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)3 model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)3 spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)3.

Emergent fuzzy geometry and fuzzy physics in four dimensions

A detailed Monte Carlo calculation of the phase diagram of bosonic mass-deformed IKKT Yang–Mills matrix models in three and six dimensions with quartic mass deformations is given. Background emergent fuzzy geometries in two and four dimensions are observed with a fluctuation given by a noncommutative U(1) gauge theory very weakly coupled to normal scalar fields. The geometry, which is determined dynamically, is given by the fuzzy spheres SN2 and SN2×SN2 respectively. The three and six matrix models are effectively in the same universality class. For example, in two dimensions the geometry is completely stable, whereas in four dimensions the geometry is stable only in the limit M⟶∞, where M is the mass of the normal fluctuations. The behaviors of the eigenvalue distribution in the two theories are also different. We also sketch how we can obtain a stable fuzzy four-sphere SN2×SN2 in the large N limit for all values of M as well as models of topology change in which the transition between spheres of different dimensions is observed. The stable fuzzy spheres in two and four dimensions act precisely as regulators which is the original goal of fuzzy geometry and fuzzy physics. Fuzzy physics and fuzzy field theory on these spaces are briefly discussed.

Non-topological Vortex Configurations in the ABJM Model

In this paper we study the existence of vortex-type solutions for a system of self-dual equations deduced from the mass-deformed Aharony--Bergman--Jafferis--Maldacena (ABJM) model. The governing equations, derived by Mohammed, Murugan, and Nastse under suitable ansatz involving fuzzy sphere matrices, have the new feature that they can support only non-topological vortex solutions. After transforming the self-dual equations into a nonlinear elliptic $2\times 2$ system we prove first an existence result by means of a perturbation argument based on a new and appropriate scaling for the solutions. Subsequently, we prove a more complete existence result by using a dynamical analysis together with a blow-up argument. In this way we establish that, any positive energy level is attained by a 1-parameter family of vortex solutions which also correspond to (constraint) energy minimizers. In other words, we register the exceptional fact in a BPS-setting that, neither a "quantization" effect nor an energy gap is induced upon the system by the rigid "critical" coupling of the self-dual regime.

On tuning microgel character and softness of cross-linked polystyrene particles.

Polystyrene (PS) microgel colloids have often been used successfully to model hard sphere behaviour even though the term "gel" invokes inevitably the notion of a more or less soft, deformable object. Here we systematically study the effect of reducing the cross-link density from 1 : 10 (1 cross-link per 10 monomers) to 1 : 100 on particle interactions and "softness". We report on the synthesis and purification of 1 : 10, 1 : 25, 1 : 50, 1 : 75 and 1 : 100 cross-linked PS particles and their characterization in terms of single particle properties, as well as the behaviour of concentrated dispersions. We are able to tune particle softness in the range between soft PNiPAM-microgels and hard PMMA particles while still allowing the mapping of the microgels onto hard sphere behavior with respect to phase diagram and static structure factors. This is mainly due to a rather homogeneous radial distribution of cross-links in contrast to PNiPAM microgels where the cross-link density decreases radially. We find that up to a cross-link density of 1 : 50 particle form factors are perfectly described by a homogeneous sphere model whereas 1 : 75 and 1 : 100 cross-linked spheres are slightly better described as fuzzy spheres. However the fuzziness is rather small compared to typical PNiPAM microgels so that a hard sphere mapping still holds even for these low cross-link densities. Finally, by varying the reaction conditions - changing from batch to semibatch emulsion polymerization and varying the feed rate or by adjusting the monomer to initiator ratio we can tune the fuzziness or significantly alter the inner structure to a more open, star-like architecture.

PH Responsiveness of hydrogels formed by telechelic polyampholytes

We investigate the influence of pH on the rheological and structural properties of hydrogels formed by hydrophobic association of the sticky ends of the triblock terpolymer poly(methyl methacrylate)-b-poly(2-(diethylamino)ethyl methacrylate-co-methacrylic acid)-b-poly(methyl methacrylate) (PMMA-b-P(DEA-co-MAA)-b-PMMA). The middle block is a weak polyampholyte having a pH dependent charge density and sign, which enables tuning of the rheological and structural properties by pH variation. Small-angle neutron scattering (SANS) studies of solutions in D2O at 0.05 wt% and pH 3.0 reveal clusters of interconnected spherical micelles having PMMA cores, stabilized by repulsive ionic interactions in the middle polyampholyte block. With increasing pH, the degree of ionization of the DEA units decreases, whereas the one of the MAA units increases, resulting in a complete loss of the correlation between these micelles. At a concentration of 3 wt% at low pH values, the system forms a gel with charged fuzzy spheres from PMMA interacting via a screened Coulomb potential. With increasing pH, the gel disintegrates due to the decrease in the effective charge on the micelles. At both concentrations, the hydrophobic aggregation of micelles is observed near the isoelectric point. At pH 3.0-7.4, the autocorrelation functions measured by rotational dynamic light scattering at 3 wt% exhibit a decay steeper than single exponential, which confirms that the gels are frozen, presumably due to the glassy PMMA cores and hydrophobic interpolyelectrolyte complexes. At pH 11, the diffusion of single micelles is observed in addition to the frozen dynamics.

Poisson–Riemannian geometry

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter λ , using a functorial approach. This leads us to field equations of ‘Poisson–Riemannian geometry’ between the classical metric, the Poisson bracket and a certain Poisson-compatible connection needed as initial data for the quantisation of the differential structure. We use such data to define a functor Q to O ( λ 2 ) from the monoidal category of all classical vector bundles equipped with connections to the monoidal category of bimodules equipped with bimodule connections over the quantised algebra. This is used to ‘semiquantise’ the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection in the sense of constructing a noncommutative geometry to O ( λ 2 ) . We solve our field equations for the Schwarzschild black-hole metric under the assumption of spherical symmetry and classical dimension, finding a unique solution and the necessity of nonassociativity at order λ 2 , which is similar to previous results for quantum groups. The paper also includes a nonassociative hyperboloid, nonassociative fuzzy sphere and our previously algebraic bicrossproduct model.

Emergent gravity on covariant quantum spaces in the IKKT model

We study perturbations of 4-dimensional fuzzy spheres as backgrounds in the IKKT or IIB matrix model. Gauge fields and metric fluctuations are identified among the excitation modes with lowest spin, supplemented by a tower of higher-spin fields. They arise from an internal structure which can be viewed as a twisted bundle over $S^4$, leading to a covariant noncommutative geometry. The linearized 4-dimensional Einstein equations are obtained from the classical matrix model action under certain conditions, modified by an IR cutoff. Some one-loop contributions to the effective action are computed using the formalism of string states.

U(3)gauge theory on fuzzy extra dimensions

In this article, we explore the low energy structure of a $U(3)$ gauge theory over spaces with fuzzy sphere(s) as extra dimensions. In particular, we determine the equivariant parametrization of the gauge fields, which transform either invariantly or as vectors under the combined action of $SU(2)$ rotations of the fuzzy spheres and those $U(3)$ gauge transformations generated by $SU(2) \subset U(3)$ carrying the spin $1$ irreducible representation of $SU(2)$. The cases of a single fuzzy sphere $S_F^2$ and a particular direct sum of concentric fuzzy spheres, $S_F^{2 \, Int}$, covering the monopole bundle sectors with windings $\pm 1$ are treated in full and the low energy degrees of freedom for the gauge fields are obtained. Employing the parametrizations of the fields in the former case, we determine a low energy action by tracing over the fuzzy sphere and show that the emerging model is abelian Higgs type with $U(1) \times U(1)$ gauge symmetry and possess vortex solutions on ${\mathbb R}^2$, which we discuss in some detail. Generalization of our formulation to the equivariant parametrization of gauge fields in $U(n)$ theories is also briefly addressed.

Relativistic Landau models and generation of fuzzy spheres

Noncommutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac–Landau operator eigenstates is obtained based on algebraic methods. We establish [Formula: see text] “gauge” transformation between the relativistic Landau model and the Pauli–Schrödinger nonrelativistic quantum mechanics. After the [Formula: see text] transformation, the Dirac operator and the angular momentum operators are found to satisfy the [Formula: see text] algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac–Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres.

Equivariant fields in anSU(N)gauge theory with new spontaneously generated fuzzy extra dimensions

We find new spontaneously generated fuzzy extra dimensions emerging from a certain deformation of $N=4$ supersymmetric Yang-Mills (SYM) theory with cubic soft supersymmetry breaking and mass deformation terms. First, we determine a particular four dimensional fuzzy vacuum that may be expressed in terms of a direct sum of product of two fuzzy spheres, and denote it in short as $S_F^{2\, Int}\times S_F^{2\, Int}$. The direct sum structure of the vacuum is revealed by a suitable splitting of the scalar fields in the model in a manner that generalizes our approach in \cite{Seckinson}. Fluctuations around this vacuum have the structure of gauge fields over $S_F^{2\, Int}\times S_F^{2\, Int}$, and this enables us to conjecture the spontaneous broken model as an effective $U(n)$ $(n < {\cal N})$ gauge theory on the product manifold $M^4 \times S_F^{2\, Int} \times S_F^{2\, Int}$. We support this interpretation by examining the $U(4)$ theory and determining all of the $SU(2)\times SU(2)$ equivariant fields in the model, characterizing its low energy degrees of freedom. Monopole sectors with winding numbers $(\pm 1,0),\,(0,\pm1),\,(\pm1,\pm 1)$ are accessed from $S_F^{2\, Int}\times S_F^{2\, Int}$ after suitable projections and subsequently equivariant fields in these sectors are obtained. We indicate how Abelian Higgs type models with vortex solutions emerge after dimensionally reducing over the fuzzy monopole sectors as well. A family of fuzzy vacua is determined by giving a systematic treatment for the splitting of the scalar fields and it is made manifest that suitable projections of these vacuum solutions yield all higher winding number fuzzy monopole sectors. We observe that the vacuum configuration $S_F^{2\, Int}\times S_F^{2\, Int}$ identifies with the bosonic part of the product of two fuzzy superspheres with $OSP(2,2)\times OSP(2,2)$ supersymmetry and elaborate on this feature.

Exact partition functions for gauge theories on [formula omitted

The noncommutative space Rλ3, a deformation of R3, supports a 3-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced partition functions, each one corresponding to the reduced gauge theory on one of the fuzzy spheres entering the decomposition of Rλ3. For a particular sub-family of gauge theories, each reduced partition function is exactly expressible as a ratio of determinants. A relation with integrable 2-D Toda lattice hierarchy is indicated.

A one-loop test for construction of 4DN= 4 SYM from 2D SYM via fuzzy-sphere geometry

As a perturbative check of the construction of four-dimensional (4D) ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) from mass deformed ${\cal N}=(8,8)$ SYM on the two-dimensional (2D) lattice, the one-loop effective action for scalar kinetic terms is computed in ${\cal N}=4$ $U(k)$ SYM on ${\mathbb R}^2 \times$ (fuzzy $S^2$), which is obtained by expanding 2D ${\cal N}=(8,8)$ $U(N)$ SYM with mass deformation around its fuzzy sphere classical solution. The radius of the fuzzy sphere is proportional to the inverse of the mass. We consider two successive limits: (1) decompactify the fuzzy sphere to a noncommutative (Moyal) plane and (2) turn off the noncommutativity of the Moyal plane. It is straightforward at the classical level to obtain the ordinary ${\cal N}=4$ SYM on ${\mathbb R}^4$ in the limits, while it is nontrivial at the quantum level. The one-loop effective action for $SU(k)$ sector of the gauge group $U(k)$ coincides with that of the ordinary 4D ${\cal N}=4$ SYM in the above limits. Although "noncommutative anomaly" appears in the overall $U(1)$ sector of the $U(k)$ gauge group, this can be expected to be a gauge artifact not affecting gauge invariant observables.

Phase diagrams of the multitrace quartic matrix models of noncommutativeΦ4theory

We report a direct and robust calculation, free from ergodic problems, of the non-uniform-to-uniform (stripe) transition line of noncommutative $\Phi^4_2$ by means of an exact Metropolis algorithm applied to the first non-trivial multitrace correction of this theory on the fuzzy sphere. In fact, we reconstruct the entire phase diagram including the Ising, matrix and stripe boundaries together with the triple point. We also report that the measured critical exponents of the Ising transition line agrees with the Onsager values in two dimensions. The triple point is identified as a termination point of the one-cut-to-two-cut transition line and is located at $(\tilde{b},\tilde{c})=(-1.55,0.4)$ which compares favorably with previous Monte Carlo estimate.

Multitrace approach to noncommutativeΦ24theory

In this article we provide a multitrace analysis of the theory of noncommutative $\Phi^4$ in two dimensions on the fuzzy sphere ${\bf S}^2_{N,\Omega}$, and on the Moyal-Weyl plane ${\bf R}^{2}_{\theta, \Omega}$, with a non-zero harmonic oscillator term added. The doubletrace matrix model symmetric under $M\longrightarrow -M$ is solved in closed form. An analytical prediction for the disordered-to-non-uniform-ordered phase transition and an estimation of the triple point, from the termination point of the critical boundary, are derived and compared with previous Monte Carlo measurement.

20pAK-12 ゆらいだ非可換球面上のケーラー構造について

20pAK-12 ゆらいだ非可換球面上のケーラー構造について

Entanglement entropy in scalar field theory on the fuzzy sphere

We study entanglement entropy on the fuzzy sphere. We calculate it in a scalar field theory on the fuzzy sphere, which is given by a matrix model. We use a method that is based on the replica method and applicable to interacting fields as well as free fields. For free fields, we obtain the results consistent with the previous study, which serves as a test of the validity of the method. For interacting fields, we perform Monte Carlo simulations at strong coupling and see a novel behavior of entanglement entropy.

Structural characterization of aqueous solution poly(oligo(ethylene oxide) monomethyl methacrylate)-grafted silica nanoparticles.

The structure of aqueous dispersions of poly(oligo(ethylene oxide) monomethyl methacrylate)-grafted silica nanoparticles was characterized using contrast variation small-angle neutron scattering studies. Modeling the low hybrid concentration dispersion scattering data using a fuzzy sphere and a polydisperse core-shell model, demonstrated that the polymer chains are highly swollen in the dispersions as compared to the dimensions of the free polymer chains in dilute solution. At higher hybrid concentrations, the dispersions were well described using a Percus-Yevick approximation to describe the structure factor. These structural characterization tools are excellent starting points for effective molecular level descriptors of dewetting and macroscopic phase transitions for polymer tethered hybrid nanoparticle systems.