Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere

TL;DR

This paper develops a holographic hierarchy connecting matrix models, string worldsheets, and membrane worldvolumes using fuzzy sphere constructions and Ponzano-Regge models, revealing new geometric and algebraic structures in quantum field theories.

Contribution

It introduces a fuzzy sphere approach to express Gaussian matrix model correlators as sums over ribbon graphs, linking matrix models to 3D topological quantum field theories and membranes.

Findings

Correlators expressed via trivalent ribbon graphs and 3j, 6j symbols.

Perturbed Gaussian model as a generating function for Ponzano-Regge partition functions.

Establishment of a holographic hierarchy connecting 0D, 2D, and 3D theories.

Abstract

The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano-Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of a holographic hierarchy linking, via the large N expansion, the zero-dimensional QFT of the Matrix model to 2D strings and 3D membranes.