Fuzzy Actions and their Continuum Limits

TL;DR

This paper shows how standard scalar and spinor actions on a manifold can be derived from matrix model actions for fuzzy fields in the large matrix limit, linking non-commutative geometry to classical field theories.

Contribution

It demonstrates the continuum limit of fuzzy matrix actions reproduces classical scalar and spinor actions, establishing a connection between non-commutative and classical geometries.

Findings

Standard actions are recovered in the large matrix limit

Proper normalization is crucial for the continuum limit

Fuzzy actions retain topological properties like instantons and anomalies

Abstract

Previously matrix model actions for ``fuzzy'' fields have been proposed using non-commutative geometry. They retained ``topological'' properties extremely well, being capable of describing instantons, $\theta$--states, the chiral anomaly, and even chiral fermions with no ``doubling''. Here, we demonstrate that the standard scalar and spinor actions on a $d$--dimensional manifold are recovered from such actions in the limit of large matrices if their normalizations are correctly scaled as the limit is taken.