We formulate theory of interacting scalar field on the fuzzy sphere as a random matrix model. We then analyze the expectation values of observables of the theory in the large N limit and we demonstrate that the eigenvalue distribution of the matrix M remains the polynomially deformed Wigner semicircle. We also compute distributions involving the matrix Laplacian of M and we show that the correlation between the eigenvalues of these two is different from the free field case.
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.
We study the phase diagram of the scalar field theory on the fuzzy sphere described as a particular multitrace matrix model. We consider perturbative and nonperturbative terms in the kinetic term effective action and describe consequences for the asymmetric regime, the free energy and the location of the triple point within the approximation.
Fuzzy spaces like the fuzzy sphere or the fuzzy torus have received remarkable attention, since they appeared as objects in string theory. Although there are many higher dimensional examples, the most known and most studied fuzzy spaces are realized as matrix algebras defined by three Hermitian matrices, which may be seen as fuzzy membranes or fuzzy surfaces. We give a mapping between directed graphs and matrix algebras defined by three Hermitian matrices and show that the matrix algebras of known two-dimensional fuzzy spaces are associated with unbranched graphs. By including branchings into …
We review the description of scalar field theories on fuzzy spaces by Hermitian random matrix models. After reminding the reader of the relevant aspects of the random matrix theory and construction of the fuzzy spaces, we summarize the most important results for the scalar fields on such spaces. We then introduce the multi-trace matrix models relevant for the analytical description of scalar field theories on fuzzy spaces and show to what extent they do, and to what extent they do not, describe the know phase structure of $ϕ^4$ theory on the fuzzy sphere.
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 …
We briefly review the connection between the fuzzy field theories and matrix models and describe the main features of the models that appear. We summarize the different approaches to their analysis, some of the recent results and the challenges to be addressed in the future.
We review analytical approaches to scalar field theory on fuzzy spaces. We briefly outline the matrix description of these theories and describe various approximations to the relevant matrix model. We discuss the challenge of obtaining a consistent approximation that includes the higher moments of the theory.
We analyze fuzzy configurations of D0-branes in BFSS matrix model as microstates of black hole. Fuzzy configurations of D0-branes consist of localized fuzzy objects in 3 spatial directions which are smeared into 6 internal directions. Since the solutions are time dependent, these are non-BPS configurations and have internal energy compared with static case. Especially we examine the smeared fuzzy sphere in the BFSS matrix model, which will correspond to the microstate of the charged black hole in 4 dimensions, and compare the effective potential in that background with the result obtained by t…
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.
We revisit an orbifold matrix model obtained as a restriction of the type IIB matrix model on a Z_3-invariant sector. An investigation of its moduli space of vacua is performed and issues related to chiral gauge theory and gravity are discussed. Modifications of the orbifolded model triggered by Chern-Simons or mass deformations are also analyzed. Certain vacua of the modified models exhibit higher-dimensional behaviour with internal geometries related to fuzzy spheres.
We present a manifestly Spin(5) invariant construction of squashed fuzzy $CP^3$ as a fuzzy $S^2$ bundle over fuzzy $S^4$. We develop the necessary projectors and exhibit the squashing in terms of the radii of the $S^2$ and $S^4$. Our analysis allows us give both scalar and spinor fuzzy action functionals whose low lying modes are truncated versions of those of a commutative $S^4$.
We consider the information metric and Berry connection in the context of noncommutative matrix geometry. We propose that these objects give a new method of characterizing the fuzzy geometry of matrices. We first give formal definitions of these geometric objects and then explicitly calculate them for the well-known matrix configurations of fuzzy $S^2$ and fuzzy $S^4$. We find that the information metrics are given by the usual round metrics for both examples, while the Berry connections coincide with the configurations of the Wu-Yang monopole and the Yang monopole for fuzzy $S^2$ and fuzzy $S…
Quantization of spacetime by means of finite dimensional matrices is the basic idea of fuzzy spaces. There remains an issue of quantizing time, however, the idea is simple and it provides an interesting interplay of various ideas in mathematics and physics. Shedding some light on such an interplay is the main theme of this dissertation. The dissertation roughly separates into two parts. In the first part, we consider a mathematical aspect of fuzzy spaces, namely, their construction. We begin with a review of the construction of fuzzy complex projective spaces CP^k (k=1,2,...) in relation to ge…
We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter $τ$. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator …
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 …
We investigate the phase structure of a special class of multi-trace hermitian matrix models, which are candidates for the description of scalar field theory on fuzzy spaces. We include up to the fourth moment of the eigenvalue distribution into the multi-trace part of the probability distribution, which stems from the kinetic term of the field theory action. We show that by considering different multi-trace behavior in the large moment and in the small moment regimes of the model, it is possible to obtain a matrix model, which describes the numerically observed phase structure of fuzzy field …
Quantum field theory defined on a noncommutative space is a useful toy model of quantum gravity and is known to have several intriguing properties, such as nonlocality and UV/IR mixing. They suggest novel types of correlation among the degrees of freedom of different energy scales. In this paper, we investigate such correlations by the use of entanglement entropy in the momentum space. We explicitly evaluate the entanglement entropy of scalar field theory on a fuzzy sphere and find that it exhibits different behaviors from that on the usual continuous sphere. We argue that these differences wo…
Stability of Fuzzy S^2 x S^2 x S^2 in IIB Type Matrix Models
本文研究了三种不同的IIB型矩阵模型中模糊S^2 x S^2 x S^2背景在两圈水平上关于每个S^2自旋变化的稳定性。我们发现S^2 x S^2 x S^2背景是亚稳态的，并且在具有Myers项的模型中，与剩余的S^2 x S^2相比，有效作用量有利于单个大的S^2。另一方面，我们发现在IIB矩阵模型本身中，与剩余的S^2相比，较大的S^2 x S^2是有利的。进一步研究了IIB矩阵模型中模糊S^2 × S^2背景相对于各S^2尺度因子的稳定性。在这种情况下，我们找到了不稳定的方向，这些方向降低了远离最对称的模糊S^2 x S^2背景的有效作用。
We consider the matrix regularization of scalar fields on a Riemann surface with a general gauge-field background. We propose a construction of the fuzzy version of the Laplacian.
We investigate mass deformation of twisted superalgebra of U(N) super Yang-Mills (SYM) theories in several models and in several dimensions, motivated by the method formulated in [1]. We show that there are several ways to perform the deformation, if a model possesses four scalar supercharges except for two dimensional B-model. We also evaluate classical vacuum solutions of the potentials of scalar fields for each model. We then find that it is always possible to find fuzzy sphere solution in the theories.
Diffeomorphisms can be seen as automorphisms of the algebra of functions. In the matrix regularization, functions on a smooth compact manifold are mapped to finite size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through the matrix regularization. For the case of the fuzzy $S^2$, we construct the matrix regularization in terms of the Berezin-Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $S^2$. We also propose thre…
