Fuzzy Scalar Field Theory as a Multitrace Matrix Model

TL;DR

This paper develops an analytical method for scalar field theory on the fuzzy sphere by reducing it to a solvable multitrace matrix model, enabling the study of its phase structure and regularization properties.

Contribution

It introduces a perturbative expansion approach to derive a multitrace matrix model from fuzzy sphere scalar field theory, facilitating analytical solutions.

Findings

Derived a multitrace matrix model from fuzzy sphere scalar field theory.

Solved the model using saddle-point approximation in the large N limit.

Applied the method to a fuzzy geometry regularization of scalar field theory on the plane.

Abstract

We develop an analytical approach to scalar field theory on the fuzzy sphere based on considering a perturbative expansion of the kinetic term. This expansion allows us to integrate out the angular degrees of freedom in the hermitian matrices encoding the scalar field. The remaining model depends only on the eigenvalues of the matrices and corresponds to a multitrace hermitian matrix model. Such a model can be solved by standard techniques as e.g. the saddle-point approximation. We evaluate the perturbative expansion up to second order and present the one-cut solution of the saddle-point approximation in the large N limit. We apply our approach to a model which has been proposed as an appropriate regularization of scalar field theory on the plane within the framework of fuzzy geometry.