A Perturbative Approach to Fuzzifying Field Theories

TL;DR

This paper introduces a perturbative method to compute noncommutative corrections to the metric tensor in scalar field theories on manifolds, demonstrating its application to fuzzy spheres with different star products.

Contribution

It presents a novel perturbative approach for calculating noncommutative metric corrections, applicable to scalar fields on curved manifolds with various star products.

Findings

Derived lowest order fuzzy corrections for scalar fields on a sphere

Compared star product independent and dependent corrections

Applied method to stereographically projected sphere geometry

Abstract

We propose a procedure for computing noncommutative corrections to the metric tensor, and apply it to scalar field theory written on coordinate patches of smooth manifolds. The procedure involves finding maps to the noncommutative plane where differentiation and integration are easily defined, and introducing a star product. There are star product independent, as well as dependent, corrections. Applying the procedure for two different star products, we find the lowest order fuzzy corrections to scalar field theory on a sphere which is sterographically projected to the plane.