Fuzzy Sphere Research Articles

Thermodynamics on fuzzy spacetime

We investigate the thermodynamics of non-relativistic and relativistic ideal gases on the spacetime with noncommutative fuzzy geometry. We first find that the heat capacities of the non-relativistic ideal boson and fermion on the fuzzy two-sphere have different values, contrast to that on the commutative geometry. We calculate the ``statistical interparticle potential'' therein and interprete this property as a result that the non-commutativity of the fuzzy sphere has an inclination to enhance the statistical ``attraction (repulsion) interparticle potential'' between boson (fermion). We also see that at high temperature the heat capacity approaches to zero. We next evaluate the heat capacities of the non-relativistic ideal boson and fermion on the product of the 1+D (with D=2,3) Minkowski spacetime by a fuzzy two-sphere and see that the fermion capacity could be a decreasing function of temperature in high-temperature limit, contrast to that always being an increasing function on the commutative geometry. Also, the boson and fermion heat capacities both approach to that on the 1+D Minkowski spacetime in high-temperature limit. We discuss these results and mention that the properties may be traced to the mechanism of ``thermal reduction of the fuzzy space''. We also investigate the same problems in the relativistic system with free Klein-Gordon field and Dirac field and find the similar properties.

OSp(4|2) superconformal mechanics

A new superconformal mechanics with OSp(4|2) symmetry is obtained by gauging the U(1) isometry of a superfield model. It is the one-particle case of the new = 4 super Calogero model recently proposed in arXiv:0812.4276 [hep-th]. Classical and quantum generators of the osp(4|2) superalgebra are constructed on physical states. As opposed to other realizations of = 4 superconformal algebras, all supertranslation generators are linear in the odd variables, similarly to the = 2 case. The bosonic sector of the component action is standard one-particle (dilatonic) conformal mechanics accompanied by an SU(2)/U(1) Wess-Zumino term, which gives rise to a fuzzy sphere upon quantization. The strength of the conformal potential is quantized.

SIMULATION OF A SCALAR FIELD ON A FUZZY SPHERE

The ϕ4 real scalar field theory on a fuzzy sphere is studied numerically. We refine the phase diagram for this model where three distinct phases are known to exist: a uniformly ordered phase, a disordered phase, and a nonuniformly ordered phase where the spatial SO(3) symmetry of the round sphere is spontaneously broken and which has no classical equivalent. The three coexistence lines between these phases, which meet at a triple point, are carefully located, with particular attention paid to the one between the two ordered phases and the triple point itself. In the neighborhood of the triple point all phase boundaries are well approximated by straight lines which, surprisingly, have the same scaling. We argue that unless an extra term is added to enhance the effect of the kinetic term the infinite matrix limit of this model will not correspond to a real scalar field on the commutative sphere or plane.

BPS vortices in nonrelativistic M2-brane Chern-Simons-matter theory

We study BPS vortices in the mass-deformed nonrelativistic N=6 U(N){sub k}xU(N){sub -k} Chern-Simons-matter theory. We focus on the massive deformation that preserves the maximal N=6 supersymmetry and consider a nonrelativistic limit that carries 14 supercharges. In this nonrelativistic field theory we find Jackiw-Pi type exact vortex solutions combined with S{sup 3} fuzzy sphere geometry. We analyze their properties and show that they preserve one dynamical, one conformal, and five kinematical supersymmetries among the full super Schroedinger symmetry.

Spectral action on a fuzzy sphere

In this paper, the spectral action for the free Dirac operator on a fuzzy 2-sphere is considered, based on the known results. We show that from the spectral point of view, the behavior of the fuzzy sphere can be divided into three regimes: the classical regime, the quasi-classical regime and the fuzzy regime. We also consider its application to quantum cosmology.

Equivariant reduction of Yang–Mills theory over the fuzzy sphere and the emergent vortices

We consider a U ( 2 ) Yang–Mills theory on M × S F 2 where M is a Riemannian manifold and S F 2 is the fuzzy sphere. Using essentially the representation theory of SU ( 2 ) we determine the most general SU ( 2 ) -equivariant gauge field on M × S F 2 . This allows us to reduce the Yang–Mills theory on M × S F 2 down to an Abelian Higgs-type model over M . Depending on the enforcement (or non-enforcement) of a “constraint” term, the latter may (or may not) lead to the standard critically-coupled Abelian Higgs model in the commutative limit, S F 2 → S 2 . For M = R 2 , we find that the Abelian Higgs-type model admits vortex solutions corresponding to instantons in the original Yang–Mills theory. Vortices are in general no longer BPS, but may attract or repel according to the values of parameters.

The fuzzyS2structure of M2-M5 systems in ABJM membrane theories

We analyse the fluctuations of the ground-state/funnel solutions proposed to describe M2-M5 systems in the level-k mass-deformed/pure Chern-Simons-matter ABJM theory of multiple membranes. We show that in the large N limit the fluctuations approach the space of functions on the 2-sphere rather than the naively expected 3-sphere. This is a novel realisation of the fuzzy 2-sphere in the context of Matrix Theories, which uses bifundamental instead of adjoint scalars. Starting from the multiple M2-brane action, a U(1) Yang-Mills theory on 2,1 × S2 is recovered at large N, which is consistent with a single D4-brane interpretation in Type IIA string theory. This is as expected at large k, where the semiclassical analysis is valid. Several aspects of the fluctuation analysis, the ground-state/funnel solutions and the mass-deformed/pure ABJM equations can be understood in terms of a discrete noncommutative realisation of the Hopf fibration. We discuss the implications for the possibility of finding an M2-brane worldvolume derivation of the classical S3 geometry of the M2-M5 system. Using a rewriting of the equations of the SO(4)-covariant fuzzy 3-sphere construction, we also directly compare this fuzzy 3-sphere against the ABJM ground-state/funnel solutions and show them to be different.

Fuzzy spaces topology change as a possible solution to the black hole information loss paradox

The black hole information loss paradox is one of the most intricate problems in modern theoretical physics. A proposal to solve this is one related with topology change. However it has found some obstacles related to unitarity and cluster decomposition (locality). In this Letter we argue that modelling the black hole's event horizon as a noncommutative manifold – the fuzzy sphere – we can solve the problems with topology change, getting a possible solution to the black hole information loss paradox.

Finite-matrix formulation of gauge theories on a non-commutative torus with twisted boundary conditions

We present a novel finite-matrix formulation of gauge theories on a non-commutative torus. Unlike the previous formulation based on a map from a square matrix to a field on a discretized torus with periodic boundary conditions, our formulation is based on the algebraic characterization of the configuration space. This enables us to describe the twisted boundary conditions in terms of finite matrices and hence to realize the Morita equivalence at a fully regularized level. Matter fields in the fundamental representation turn out to be represented by rectangular matrices for twisted boundary conditions analogously to the matrix spherical harmonics on the fuzzy sphere with the monopole background. The corresponding Ginsparg-Wilson Dirac operator defines an index, which can be used to classify gauge field configurations into topological sectors. We also perform Monte Carlo calculations for the index as a consistency check. Our formulation is expected to be useful for applications of non-commutative geometry to various problems related to topological aspects of field theories and string theories.

Dimensional reduction, monopoles and dynamical symmetry breaking

We consider SU(2)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M x CP(1), with emphasis on the effects of non-trivial magnetic flux on CP(1). The reduction of Yang-Mills fields gives a chain of coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking, as a consequence of the monopole fields. The reduction of SU(2)-symmetric fermions gives massless Dirac fermions on M transforming under the low-energy gauge group with Yukawa couplings, again as a result of the internal U(1) fluxes. The tower of massive fermionic Kaluza-Klein states also has Yukawa interactions and admits a natural SU(2)-equivariant truncation by replacing CP(1) with a fuzzy sphere. In this approach it is possible to obtain exactly massless chiral fermions in the effective field theory with Yukawa interactions, without any further requirements. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples.

Smearing effect in plane-wave matrix model

Motivated by the usual D2-D0 system, we consider a configuration composed of flat membrane and fuzzy sphere membrane in plane-wave matrix model, and investigate the interaction between them. The configuration is shown to lead to a non-trivial interaction potential, which indicates that the fuzzy sphere membrane really behaves like a graviton, giant graviton. Interestingly, the interaction is of r−3 type rather than r−5 type. We interpret it as the interaction incorporating the smearing effect due to the fact that the considered supersymmetric flat membrane should span and spin in four dimensional subspace of plane-wave geometry.

From a giant M2 to a giant D2

We study the M2-brane giant graviton of M-theory from the point of view of type IIA theory. We reduce the Abelian dynamics of this giant graviton to that of a spherical D2-brane with worldvolume magnetic flux in type IIA theory. Alternatively, we describe it as a system of D0-branes expanding into a fuzzy sphere in a background with magnetic 3-form flux. The two descriptions agree in the limit of large number of D0-branes.

Multi-matrix models and emergent geometry

Encouraged by the AdS/CFT correspondence, we study emergent local geometry in large N multi-matrix models from the perspective of a strong coupling expansion. By considering various solvable interacting models we show how the emergence or non-emergence of local geometry at strong coupling is captured by observables that effectively measure the mass of off-diagonal excitations about a semiclassical eigenvalue background. We find emergent geometry at strong coupling in models where a mass term regulates an infrared divergence. We also show that our notion of emergent geometry can be usefully applied to fuzzy spheres. Although most of our results are analytic, we have found numerical input valuable in guiding and checking our results.

Time-dependent and non-BPS solutions in 𝒩 = 6 superconformal Chern-Simons theory

We study a class of classical solutions of three-dimensional N=6 superconformal Chern-Simons theory coupled with U(N) \times U(N) bi-fundamental matter fields. Especially, time evolutions of fuzzy spheres are discussed for both massless and massive cases. For the massive case, there are a variety of solutions having different behaviors according to the value of the mass. In addition to the time-dependent solutions, we analyze non-BPS static solutions which represent parallel M5/M5 or M5/anti-M5-branes suspended by multiple M2-branes. These solutions are similar to the fundamental strings connecting two parallel (anti) Dp-branes in perturbative string theory. A moving M5-brane and domain wall solutions with constant velocity that are obtained by the Lorentz boost of the known BPS solutions are briefly addressed.

Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere

We consider a gauge-Higgs system on a fuzzy 2-sphere and study the topological structure of gauge configurations, when the U(2) gauge symmetry is spontaneously broken to U(1)xU(1) by the vev of the Higgs field. The topology is classified by the index of the Dirac operator satisfying the Ginsparg-Wilson relation, which turns out to be a noncommutative analog of the topological charge introduced by 't Hooft. It can be rewritten as a form whose commutative limit becomes the winding number of the Higgs field. We also study conditions which assure the validity of the formulation, and give a generalization of the admissibility condition. Finally we explicitly calculate the topological charge of a one-parameter family of configurations.

FINITE TEMPERATURE PHASE TRANSITION OF A SCALAR FIELD ON A FUZZY SPHERE

We study finite temperature phase transition of neutral scalar field on a fuzzy sphere using Monte Carlo simulations. We work with the zero mode in the temporal directions, while the effects of the higher modes are taken care by the temperature dependence of r. In the numerical calculations we use "pseudo-heatbath" method which reduces the auto-correlation considerably. Our results agree with the conventional calculations. We report some new results which show the presence of meta-stable states, first order symmetry breaking transition and existence of multiple triple points in the phase diagram..

Matrix equations and trilinear commutation relations

In this paper we discuss an algebraic approach to treating static equations of matrix models. In this approach, the equations of motion are considered as a consequence of parafermi-like trilinear commutation relations. In this context we consider several solutions, including the construction of noncommutative spheres. The equivalence of fuzzy spheres and parafermions is underlined.

Probing the fuzzy sphere regularisation in simulations of the 3d λϕ4model

We regularise the 3d λ4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent role. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m2 and λ. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c = 1-model in string theory. This observation enables a conjecture about the thermodynamic limit.

The instability of intersecting fuzzy spheres

We discuss the classical and quantum stability of general configurations representing many fuzzy spheres in dimensionally reduced Yang-Mills-Chern-Simons models with and without supersymmetry. By performing one-loop perturbative calculations around such configurations, we find that intersecting fuzzy spheres are classically unstable in the class of models studied in this paper. We also discuss the large-N limit of the one-loop effective action as a function of the distance of fuzzy spheres. This shows, in particular, that concentric fuzzy spheres with different radii, which are identified with the 't Hooft-Polyakov monopoles, are perturbatively stable in the bosonic model and in the D=10 supersymmetric model.

Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions

Finite Unified Theories (FUTs) are N = 1 supersymmetric Grand Unified Theories, which can be made all-loop finite, both in the dimensionless (gauge and Yukawa couplings) and dimensionful (soft supersymmetry breaking terms) sectors. This remarkable property, based on the reduction of couplings at the quantum level, provides a drastic reduc- tion in the number of free parameters, which in turn leads to an accurate prediction of the top quark mass in the dimensionless sector, and predictions for the Higgs boson mass and the supersymmetric spectrum in the dimensionful sector. Here we examine the predictions of two such FUTs. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We reinterpret these gauge theo- ries as four-dimensional theories with Kaluza-Klein modes. We then perform a generalized ` la Forgacs-Manton dimensional reduction. We emphasize some striking features emerging such as (i) the appearance of non-Abelian gauge theories in four dimensions starting from an Abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormaliza- bility of the theory both in higher as well as in four dimensions. Then reversing the above approach we present a renormalizable four dimensional SU(N) gauge theory with a suitable multiplet of scalar fields, which via spontaneous symmetry breaking dynamically develops extra dimensions in the form of a fuzzy sphere S 2 N. We explicitly find the tower of massive Kaluza-Klein modes consistent with an interpretation as gauge theory on M 4 ◊ S 2 , the scalars being interpreted as gauge fields on S 2 . Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n1) ◊ SU(n2) ◊ U(1). Therefore the second picture justifies the first one in a renormalizable framework but in addition has the potential to reveal new aspects of the theory.