Fuzzy Tori Research Articles

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett’s characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action {S(D)= \operatorname{Tr} f(D)} for 2n -dimensional fuzzy geometries. In contrast to the original Chamseddine–Connes spectral action, we take a polynomial f with f(x)\to \infty as \vert x\vert \to\infty in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type {S(D)=N \cdot \operatorname{tr} F+\sum_i \operatorname{tr}A_i \cdot \operatorname{tr} B_i } , being F , \smash{A_i} and B_i noncommutative polynomials in \smash{2^{2n-1}} complex N\times N matrices that parametrize the Dirac operator D . For arbitrary signature—thus for any admissible KO-dimension—formulas for 2 -dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4 -dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F , \smash{A_i} and B_i are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of \smash{A_i} and B_i simultaneously, for fixed i . Collectively, this favors a free probabilistic perspective for the large- N limit we elaborate on.

Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

The covariant Gromov–Hausdorff propinquity

We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting metric is zero between two Lipschitz dynamical systems if and only if there exists an equivariant full quantum isometry between. We also present sufficient conditions for Cauchy sequences to converge for our new metric, thus exhibiting certain complete classes of Lipschitz dynamical systems. We apply our work to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.

Spectral estimators for finite non-commutative geometries

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addresses the question of how to extract information about these geometries from the spectrum of the Dirac operator. Since the Dirac operator is a finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no sense and is replaced by measurements of the spectrum at a finite energy scale. The spectral dimension of the square of the Dirac operator is improved to provide a new spectral measure of the dimension of a space called the spectral variance. Similarly, the volume of a space can be computed from the spectrum once the dimension is known. Two methods of doing this are investigated: the well-known Dixmier trace and a recent improvement due to Abel Stern. Finally, the distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these techniques are tested on the explicit examples of the fuzzy spheres and fuzzy tori, which can be regarded as approximations of the usual Riemannian sphere and flat tori. Then they are applied to characterise some random fuzzy spaces using data generated by a Monte Carlo simulation.

Spinning the fuzzy sphere

We construct various exact analytical solutions of the SO(3) BMN matrix model that correspond to rotating fuzzy spheres and rotating fuzzy tori. These are also solutions of Yang Mills theory compactified on a sphere times time and they are also translationally invariant solutions of the $$ \mathcal{N}={1}^{*} $$ field theory with a non-trivial chargedensity. The solutions we construct have a ℤ N symmetry, where N is the rank of the matrices. After an appropriate ansatz, we reduce the problem to solving a set of polynomial equations in 2N real variables. These equations have a discrete set of solutions for each value of the angular momentum. We study the phase structure of the solutions for various values of N . Also the continuum limit where N → ∞, where the problem reduces to finding periodic solutions of a set of coupled differential equations. We also study the topology change transition from the sphere to the torus.

THE LOWEST MODES AROUND GAUSSIAN SOLUTIONS OF TENSOR MODELS AND THE GENERAL RELATIVITY

In the paper arXiv:0706.1618[hep-th], the number distribution of the low-lying spectra around Gaussian solutions representing various dimensional fuzzy tori of a tensor model was numerically shown to be in accordance with the general relativity on tori. In this paper, I perform more detailed numerical analysis of the properties of the modes for two-dimensional fuzzy tori, and obtain conclusive evidences for the agreement. Under a proposed correspondence between the rank-3 tensor in tensor models and the metric tensor in the general relativity, conclusive agreement is obtained between the profiles of the low-lying modes in a tensor model and the metric modes transverse to the general coordinate transformation. Moreover, the low-lying modes are shown to be well on a massless trajectory with quartic momentum dependence in the tensor model. This is in agreement with that the lowest momentum dependence of metric fluctuations in the general relativity will come from the R2-term, since the R-term is topological in two dimensions. These evidences support the idea that the low-lying low-momentum dynamics around the Gaussian solutions of tensor models is described by the general relativity. I also propose a renormalization procedure for tensor models. A classical application of the procedure makes the patterns of the low-lying spectra drastically clearer, and suggests also the existence of massive trajectories.

Fuzzy torus in matrix model

We have calculated the free energy up to two loop to compare T 2 with T 4 in IIB matrix model. It turns out that T 2 has smaller free energy than T 4 . We have also discussed the generation of the gauge group by considering k-coincident fuzzy tori and found that in this case U ( 1 ) gauge group is favored. This means that if the true vacuum is four-dimensional, it is not a simple fuzzy space considered here.

A fuzzy three sphere and fuzzy tori

A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces.

Deconstructing noncommutativity with a giant fuzzy moose

We argue that the worldvolume theories of D-branes probing orbifolds with discrete torsion develop, in the large quiver limit, new non-commutative directions. This provides an explicit `deconstruction' of a wide class of noncommutative theories. This also provides insight into the physical meaning of discrete torsion and its relation to the T-dual B field. We demonstrate that the strict large quiver limit reproduces the matrix theory construction of higher-dimensional D-branes, and argue that finite `fuzzy moose' theories provide novel regularizations of non-commutative theories and explicit string theory realizations of gauge theories on fuzzy tori. We also comment briefly on the relation to NCOS, (2,0) and little string theories.