Quasi-symmetric invariant properties of Cantor metric spaces

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.Comment: To appear in Annales de l'Institut Fourie

Boundary correlation numbers in one matrix model

We introduce one matrix model coupled to multi-flavor vectors. The two-flavor vector model is demonstrated to reproduce the two-point correlation numbers of boundary primary fields of two dimensional (2, 2p+1) minimal Liouville gravity on disk, generalizing the loop operator (resolvent) description. The model can properly describe non-trivial boundary conditions for the matter Cardy state as well as for the Liouville field. From this we propose that the n-flavor vector model will be suited for producing the boundary correlation numbers with n different boundary conditions on disk.Comment: 16 pages, 3 figures, add elaboration on matter Cardy state and reference

Large N reduction for Chern-Simons theory on S^3

We study a matrix model which is obtained by dimensional reduction of Chern-Simon theory on S^3 to zero dimension. We find that expanded around a particular background consisting of multiple fuzzy spheres, it reproduces the original theory on S^3 in the planar limit. This is viewed as a new type of the large N reduction generalized to curved space.Comment: 4 pages, 2 figures, references added, typos correcte

On comeager sets of metrics whose ranges are disconnected

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space $X$, we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense $G_{\delta}$ subspace in $\mathrm{Met}(X)$. As its application, we show that some sets of universal metrics are meager in spaces of metrics.Comment: 13 pages. This paper is published in Topology and its Applications vol.327 (2023), 10844