We analyze the expectation value of observables in a scalar theory on the fuzzy two sphere, represented as a generalized hermitian matrix model. We calculate explicitly the form of the expectation values in the large-N limit and demonstrate that, for any single kind of field (matrix), the distribution of its eigenvalues is still a Wigner semicircle but with a renormalized radius. For observables involving more than one type of matrix we obtain a new distribution corresponding to correlated Wigner semicircles.arXiv:1109.3349v3 [hep-th]

We analyze two types of hermitian matrix models with asymmetric solutions. One type breaks the symmetry explicitly with an asymmetric quartic potential. We give the phase diagram of this model with two different phase transitions between the one cut and two cut solutions. The second type, describing real scalar field theory on fuzzy spaces, breaks the symmetry spontaneously with multitrace terms. We present two methods to study this model, one direct and one using a connection with the first type of models. We analyze the model for the fuzzy sphere and obtain a phase diagram with the location of the triple point in a good agreement with the most recent numerical simulations.

We review the interplay of fuzzy field theories and matrix models, with an emphasis on the phase structure of fuzzy scalar field theories. We give a self-contained introduction to these topics and give the details concerning the saddle point approach for the usual single trace and multitrace matrix models. We then review the attempts to explain the phase structure of the fuzzy field theory using a corresponding random matrix ensemble, showing the strength and weaknesses of this approach. We conclude with a list of challenges one needs to overcome and the most interesting open problems one can try to solve.Multitrace matrix models, Noncommutative geometry, Fuzzy field theory, Phase diagram of fuzzy field theory Phase structure of fuzzy field theories and matrix models multitrace models is known, so we have hope to analyze the fuzzy field theory via this matrix model.But to be able to do that, we have a long way to cover. Non-commutative spaces in physicsThe idea of the non-commutative geometry arises in the correspondence of commutative C * algebras and differentiable manifolds. Every manifold comes with a naturally defined algebra of functions and every C * algebra is an algebra of functions on some manifold. One then defines non-commutative spaces as spaces that correspond to a non-commutative algebra [6,7]. One introduces a spectral triple of a C * algebra A, a Hilbert space H on which this algebra can be realized as bounded operators and a special Dirac operator D, which will characterize the geometry. Using these three ingredients, it is possible to define differential calculus on a manifold. More specifically, in a finite dimensional case one arrives at a notion of a fuzzy space, which will be central for this presentation. Here, the Hilbert space H is finite dimensional and algebra A can be realized as an algebra of N × N matrices. This can be thought of as a generalization of a well-known notion from quantum mechanics. With large number of quanta the quantum theory is well approximated by the classical theory, with the functions on the classical phase space representing the linear Hermitian operators. The fuzzy space defined by (A, H, ∆ N ), with ∆ N the matrix Laplacian, are finite state-approximation to the classical phase space manifold. The finite dimensionality of H amounts for compactness of the classical version of the manifold.The original motivation for considering non-commutative manifolds in physics dates back to the early days of quantum field theories. It was suggested by Heisenberg and later formalized by Snyder [8] that the divergences which plague the qft's can be regularized by the space-time noncommutativity. Opposing to other methods, non-commutative space-time keeps its Lorentz invariance. However, since renormalization proved to be effective in providing accurate numerical results, this idea was abandoned.More recently, it has been shown that putting the quantum theory and gravity together introduces some kind of nontrivial short distance structure to the theory [9]. Noncommutative spa...

We study the phase structure of the scalar field theory on fuzzy CP n in the large N limit. Considering the theory as a hermitian matrix model we compute the perturbative expansion of the kinetic term effective action under the assumption of distributions being close to the semicircle. We show that this model admits also a uniform order phase, corresponding to the asymmetric one-cut distribution, and we find the phase boundary. We compute a non-perturbative approximation to the effective action which enables us to identify the disorder and the non-uniform order phases and the phase transition between them. We locate the triple point of the theory and find an agreement with previous numerical studies for the case of the fuzzy sphere.

We solve a multitrace matrix model approximating the real quartic scalar field theory on the fuzzy sphere and obtain its phase diagram. We generalize this method to models with modified kinetic terms and demonstrate its use by investigating models related to the removal of the UV/IR mixing. We show that for the fuzzy sphere a modification of the kinetic part of the action by higher derivative term can change the phase diagram of the theory such that the triple point moves further from the origin.