论文标题
在由随机非交通性几何I：功能重归其化基团作为自由代数中的功能重归其化基团的动机上的多层次模型
On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
论文作者
论文摘要
随机非交通性几何形状可以看作是欧几里得路径综合方法，用于量化由非交通性几何形状（NCG）中光谱作用定义的理论。为了研究任意维度随机NCG中的相变，我们研究了多层次模型的非扰动功能重新归一化组，其作用由遗式和反赫米特式矩阵中的非共同多项式组成。这种结构是由狄拉克操作员在Barrett的光谱三重配方中的光谱作用决定的。当前的数学严格处理提出了“无坐标”语言，在其他地方也可能有用，这更是如此，因为我们的方法适用于一般多膜模型。该工具包是自由代数上的非交通算体，它允许用Voiculescu的环状梯度和Rota-Sagan-Stein-Stein-Stein非交易式衍生剂来描述重新归一化组流量的发生（这里引入了此处引入的非交易性laplacian）。我们探讨了功能重新归一化组方程的代数结构，并且，作为这种形式主义的应用，我们发现了$β$函数，在两个不同的签名中确定了$ n $限制中的固定点，并获得$ 2 $二维几何的关键指数。
Random noncommutative geometry can be seen as a Euclidean path-integral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the non-perturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and anti-Hermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barrett's spectral triple formulation of fuzzy spaces.The present mathematically rigorous treatment puts forward "coordinate-free" language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow -- a noncommutative Laplacian introduced here -- in terms of Voiculescu's cyclic gradient and Rota-Sagan-Stein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group Equation and, as an application of this formalism, we find the $β$-functions, identify the fixed points in the large-$N$ limit and obtain the critical exponents of $2$-dimensional geometries in two different signatures.
